Properties

Label 387.8.a.b.1.8
Level $387$
Weight $8$
Character 387.1
Self dual yes
Analytic conductor $120.893$
Analytic rank $0$
Dimension $11$
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] = [387,8,Mod(1,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.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: \(120.893004862\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} + \cdots + 238240894976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(11.4671\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+13.4671 q^{2} +53.3630 q^{4} -131.138 q^{5} -712.280 q^{7} -1005.14 q^{8} +O(q^{10})\) \(q+13.4671 q^{2} +53.3630 q^{4} -131.138 q^{5} -712.280 q^{7} -1005.14 q^{8} -1766.05 q^{10} -6103.13 q^{11} -7927.36 q^{13} -9592.35 q^{14} -20366.9 q^{16} +16334.5 q^{17} +39583.9 q^{19} -6997.93 q^{20} -82191.5 q^{22} -29296.2 q^{23} -60927.8 q^{25} -106759. q^{26} -38009.4 q^{28} -71148.2 q^{29} +107693. q^{31} -145624. q^{32} +219979. q^{34} +93407.1 q^{35} +386329. q^{37} +533080. q^{38} +131813. q^{40} +768609. q^{41} +79507.0 q^{43} -325681. q^{44} -394535. q^{46} -929733. q^{47} -316201. q^{49} -820521. q^{50} -423028. q^{52} +1.50722e6 q^{53} +800354. q^{55} +715944. q^{56} -958161. q^{58} +1.77396e6 q^{59} -1.22781e6 q^{61} +1.45031e6 q^{62} +645821. q^{64} +1.03958e6 q^{65} -831926. q^{67} +871658. q^{68} +1.25792e6 q^{70} +21047.1 q^{71} -631213. q^{73} +5.20274e6 q^{74} +2.11231e6 q^{76} +4.34713e6 q^{77} -3.76509e6 q^{79} +2.67087e6 q^{80} +1.03509e7 q^{82} -1.71861e6 q^{83} -2.14208e6 q^{85} +1.07073e6 q^{86} +6.13453e6 q^{88} +6.08728e6 q^{89} +5.64650e6 q^{91} -1.56333e6 q^{92} -1.25208e7 q^{94} -5.19096e6 q^{95} -7.22067e6 q^{97} -4.25831e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.4671 1.19034 0.595168 0.803602i \(-0.297086\pi\)
0.595168 + 0.803602i \(0.297086\pi\)
\(3\) 0 0
\(4\) 53.3630 0.416898
\(5\) −131.138 −0.469174 −0.234587 0.972095i \(-0.575374\pi\)
−0.234587 + 0.972095i \(0.575374\pi\)
\(6\) 0 0
\(7\) −712.280 −0.784887 −0.392444 0.919776i \(-0.628370\pi\)
−0.392444 + 0.919776i \(0.628370\pi\)
\(8\) −1005.14 −0.694086
\(9\) 0 0
\(10\) −1766.05 −0.558475
\(11\) −6103.13 −1.38254 −0.691271 0.722596i \(-0.742949\pi\)
−0.691271 + 0.722596i \(0.742949\pi\)
\(12\) 0 0
\(13\) −7927.36 −1.00075 −0.500377 0.865808i \(-0.666805\pi\)
−0.500377 + 0.865808i \(0.666805\pi\)
\(14\) −9592.35 −0.934279
\(15\) 0 0
\(16\) −20366.9 −1.24309
\(17\) 16334.5 0.806371 0.403185 0.915118i \(-0.367903\pi\)
0.403185 + 0.915118i \(0.367903\pi\)
\(18\) 0 0
\(19\) 39583.9 1.32398 0.661989 0.749514i \(-0.269713\pi\)
0.661989 + 0.749514i \(0.269713\pi\)
\(20\) −6997.93 −0.195598
\(21\) 0 0
\(22\) −82191.5 −1.64569
\(23\) −29296.2 −0.502069 −0.251035 0.967978i \(-0.580771\pi\)
−0.251035 + 0.967978i \(0.580771\pi\)
\(24\) 0 0
\(25\) −60927.8 −0.779875
\(26\) −106759. −1.19123
\(27\) 0 0
\(28\) −38009.4 −0.327218
\(29\) −71148.2 −0.541715 −0.270858 0.962619i \(-0.587307\pi\)
−0.270858 + 0.962619i \(0.587307\pi\)
\(30\) 0 0
\(31\) 107693. 0.649264 0.324632 0.945840i \(-0.394760\pi\)
0.324632 + 0.945840i \(0.394760\pi\)
\(32\) −145624. −0.785613
\(33\) 0 0
\(34\) 219979. 0.959852
\(35\) 93407.1 0.368249
\(36\) 0 0
\(37\) 386329. 1.25387 0.626933 0.779073i \(-0.284310\pi\)
0.626933 + 0.779073i \(0.284310\pi\)
\(38\) 533080. 1.57598
\(39\) 0 0
\(40\) 131813. 0.325648
\(41\) 768609. 1.74165 0.870827 0.491590i \(-0.163584\pi\)
0.870827 + 0.491590i \(0.163584\pi\)
\(42\) 0 0
\(43\) 79507.0 0.152499
\(44\) −325681. −0.576380
\(45\) 0 0
\(46\) −394535. −0.597631
\(47\) −929733. −1.30622 −0.653109 0.757264i \(-0.726536\pi\)
−0.653109 + 0.757264i \(0.726536\pi\)
\(48\) 0 0
\(49\) −316201. −0.383952
\(50\) −820521. −0.928313
\(51\) 0 0
\(52\) −423028. −0.417213
\(53\) 1.50722e6 1.39063 0.695314 0.718707i \(-0.255266\pi\)
0.695314 + 0.718707i \(0.255266\pi\)
\(54\) 0 0
\(55\) 800354. 0.648653
\(56\) 715944. 0.544780
\(57\) 0 0
\(58\) −958161. −0.644823
\(59\) 1.77396e6 1.12450 0.562252 0.826966i \(-0.309935\pi\)
0.562252 + 0.826966i \(0.309935\pi\)
\(60\) 0 0
\(61\) −1.22781e6 −0.692591 −0.346295 0.938125i \(-0.612561\pi\)
−0.346295 + 0.938125i \(0.612561\pi\)
\(62\) 1.45031e6 0.772842
\(63\) 0 0
\(64\) 645821. 0.307952
\(65\) 1.03958e6 0.469528
\(66\) 0 0
\(67\) −831926. −0.337927 −0.168963 0.985622i \(-0.554042\pi\)
−0.168963 + 0.985622i \(0.554042\pi\)
\(68\) 871658. 0.336175
\(69\) 0 0
\(70\) 1.25792e6 0.438340
\(71\) 21047.1 0.00697891 0.00348945 0.999994i \(-0.498889\pi\)
0.00348945 + 0.999994i \(0.498889\pi\)
\(72\) 0 0
\(73\) −631213. −0.189909 −0.0949545 0.995482i \(-0.530271\pi\)
−0.0949545 + 0.995482i \(0.530271\pi\)
\(74\) 5.20274e6 1.49252
\(75\) 0 0
\(76\) 2.11231e6 0.551964
\(77\) 4.34713e6 1.08514
\(78\) 0 0
\(79\) −3.76509e6 −0.859173 −0.429587 0.903026i \(-0.641341\pi\)
−0.429587 + 0.903026i \(0.641341\pi\)
\(80\) 2.67087e6 0.583228
\(81\) 0 0
\(82\) 1.03509e7 2.07315
\(83\) −1.71861e6 −0.329916 −0.164958 0.986301i \(-0.552749\pi\)
−0.164958 + 0.986301i \(0.552749\pi\)
\(84\) 0 0
\(85\) −2.14208e6 −0.378329
\(86\) 1.07073e6 0.181524
\(87\) 0 0
\(88\) 6.13453e6 0.959604
\(89\) 6.08728e6 0.915288 0.457644 0.889135i \(-0.348693\pi\)
0.457644 + 0.889135i \(0.348693\pi\)
\(90\) 0 0
\(91\) 5.64650e6 0.785479
\(92\) −1.56333e6 −0.209312
\(93\) 0 0
\(94\) −1.25208e7 −1.55484
\(95\) −5.19096e6 −0.621176
\(96\) 0 0
\(97\) −7.22067e6 −0.803298 −0.401649 0.915794i \(-0.631563\pi\)
−0.401649 + 0.915794i \(0.631563\pi\)
\(98\) −4.25831e6 −0.457032
\(99\) 0 0
\(100\) −3.25129e6 −0.325129
\(101\) −9.01277e6 −0.870429 −0.435214 0.900327i \(-0.643327\pi\)
−0.435214 + 0.900327i \(0.643327\pi\)
\(102\) 0 0
\(103\) −3.97490e6 −0.358423 −0.179211 0.983811i \(-0.557355\pi\)
−0.179211 + 0.983811i \(0.557355\pi\)
\(104\) 7.96815e6 0.694609
\(105\) 0 0
\(106\) 2.02979e7 1.65531
\(107\) 2.21874e7 1.75091 0.875454 0.483301i \(-0.160562\pi\)
0.875454 + 0.483301i \(0.160562\pi\)
\(108\) 0 0
\(109\) 1.24187e7 0.918506 0.459253 0.888305i \(-0.348117\pi\)
0.459253 + 0.888305i \(0.348117\pi\)
\(110\) 1.07785e7 0.772115
\(111\) 0 0
\(112\) 1.45069e7 0.975689
\(113\) −2.56219e7 −1.67046 −0.835231 0.549899i \(-0.814666\pi\)
−0.835231 + 0.549899i \(0.814666\pi\)
\(114\) 0 0
\(115\) 3.84185e6 0.235558
\(116\) −3.79668e6 −0.225840
\(117\) 0 0
\(118\) 2.38901e7 1.33854
\(119\) −1.16347e7 −0.632910
\(120\) 0 0
\(121\) 1.77610e7 0.911422
\(122\) −1.65351e7 −0.824416
\(123\) 0 0
\(124\) 5.74682e6 0.270677
\(125\) 1.82351e7 0.835072
\(126\) 0 0
\(127\) 3.48811e7 1.51104 0.755522 0.655124i \(-0.227383\pi\)
0.755522 + 0.655124i \(0.227383\pi\)
\(128\) 2.73372e7 1.15218
\(129\) 0 0
\(130\) 1.40001e7 0.558896
\(131\) −7.90037e6 −0.307042 −0.153521 0.988145i \(-0.549061\pi\)
−0.153521 + 0.988145i \(0.549061\pi\)
\(132\) 0 0
\(133\) −2.81948e7 −1.03917
\(134\) −1.12036e7 −0.402246
\(135\) 0 0
\(136\) −1.64185e7 −0.559691
\(137\) 2.61766e7 0.869744 0.434872 0.900492i \(-0.356794\pi\)
0.434872 + 0.900492i \(0.356794\pi\)
\(138\) 0 0
\(139\) −3.82597e7 −1.20834 −0.604171 0.796855i \(-0.706496\pi\)
−0.604171 + 0.796855i \(0.706496\pi\)
\(140\) 4.98448e6 0.153522
\(141\) 0 0
\(142\) 283443. 0.00830724
\(143\) 4.83817e7 1.38358
\(144\) 0 0
\(145\) 9.33025e6 0.254159
\(146\) −8.50061e6 −0.226055
\(147\) 0 0
\(148\) 2.06157e7 0.522735
\(149\) 7.50017e6 0.185746 0.0928729 0.995678i \(-0.470395\pi\)
0.0928729 + 0.995678i \(0.470395\pi\)
\(150\) 0 0
\(151\) 7.15245e7 1.69058 0.845290 0.534308i \(-0.179428\pi\)
0.845290 + 0.534308i \(0.179428\pi\)
\(152\) −3.97875e7 −0.918955
\(153\) 0 0
\(154\) 5.85433e7 1.29168
\(155\) −1.41227e7 −0.304618
\(156\) 0 0
\(157\) −1.03218e7 −0.212865 −0.106433 0.994320i \(-0.533943\pi\)
−0.106433 + 0.994320i \(0.533943\pi\)
\(158\) −5.07049e7 −1.02270
\(159\) 0 0
\(160\) 1.90969e7 0.368589
\(161\) 2.08671e7 0.394068
\(162\) 0 0
\(163\) −9.73117e7 −1.75998 −0.879992 0.474989i \(-0.842452\pi\)
−0.879992 + 0.474989i \(0.842452\pi\)
\(164\) 4.10153e7 0.726093
\(165\) 0 0
\(166\) −2.31447e7 −0.392711
\(167\) 1.15556e8 1.91992 0.959961 0.280133i \(-0.0903787\pi\)
0.959961 + 0.280133i \(0.0903787\pi\)
\(168\) 0 0
\(169\) 94596.7 0.00150755
\(170\) −2.88476e7 −0.450338
\(171\) 0 0
\(172\) 4.24273e6 0.0635764
\(173\) −4.06579e6 −0.0597012 −0.0298506 0.999554i \(-0.509503\pi\)
−0.0298506 + 0.999554i \(0.509503\pi\)
\(174\) 0 0
\(175\) 4.33976e7 0.612114
\(176\) 1.24302e8 1.71863
\(177\) 0 0
\(178\) 8.19780e7 1.08950
\(179\) 8.18724e7 1.06697 0.533485 0.845810i \(-0.320882\pi\)
0.533485 + 0.845810i \(0.320882\pi\)
\(180\) 0 0
\(181\) −1.09002e8 −1.36634 −0.683168 0.730261i \(-0.739398\pi\)
−0.683168 + 0.730261i \(0.739398\pi\)
\(182\) 7.60420e7 0.934983
\(183\) 0 0
\(184\) 2.94469e7 0.348479
\(185\) −5.06625e7 −0.588282
\(186\) 0 0
\(187\) −9.96917e7 −1.11484
\(188\) −4.96133e7 −0.544560
\(189\) 0 0
\(190\) −6.99072e7 −0.739408
\(191\) 2.66371e7 0.276612 0.138306 0.990390i \(-0.455834\pi\)
0.138306 + 0.990390i \(0.455834\pi\)
\(192\) 0 0
\(193\) 1.74825e8 1.75047 0.875233 0.483701i \(-0.160708\pi\)
0.875233 + 0.483701i \(0.160708\pi\)
\(194\) −9.72416e7 −0.956194
\(195\) 0 0
\(196\) −1.68734e7 −0.160069
\(197\) 5.70061e7 0.531238 0.265619 0.964078i \(-0.414424\pi\)
0.265619 + 0.964078i \(0.414424\pi\)
\(198\) 0 0
\(199\) −3.66798e7 −0.329945 −0.164972 0.986298i \(-0.552753\pi\)
−0.164972 + 0.986298i \(0.552753\pi\)
\(200\) 6.12412e7 0.541301
\(201\) 0 0
\(202\) −1.21376e8 −1.03610
\(203\) 5.06774e7 0.425185
\(204\) 0 0
\(205\) −1.00794e8 −0.817139
\(206\) −5.35304e7 −0.426643
\(207\) 0 0
\(208\) 1.61455e8 1.24403
\(209\) −2.41585e8 −1.83045
\(210\) 0 0
\(211\) −1.29934e8 −0.952213 −0.476106 0.879388i \(-0.657952\pi\)
−0.476106 + 0.879388i \(0.657952\pi\)
\(212\) 8.04297e7 0.579750
\(213\) 0 0
\(214\) 2.98800e8 2.08417
\(215\) −1.04264e7 −0.0715484
\(216\) 0 0
\(217\) −7.67075e7 −0.509599
\(218\) 1.67243e8 1.09333
\(219\) 0 0
\(220\) 4.27093e7 0.270423
\(221\) −1.29490e8 −0.806979
\(222\) 0 0
\(223\) −2.02158e8 −1.22074 −0.610370 0.792116i \(-0.708979\pi\)
−0.610370 + 0.792116i \(0.708979\pi\)
\(224\) 1.03725e8 0.616617
\(225\) 0 0
\(226\) −3.45053e8 −1.98841
\(227\) 3.00569e6 0.0170551 0.00852753 0.999964i \(-0.497286\pi\)
0.00852753 + 0.999964i \(0.497286\pi\)
\(228\) 0 0
\(229\) 2.04655e8 1.12616 0.563078 0.826404i \(-0.309617\pi\)
0.563078 + 0.826404i \(0.309617\pi\)
\(230\) 5.17386e7 0.280393
\(231\) 0 0
\(232\) 7.15142e7 0.375997
\(233\) −5.47940e7 −0.283784 −0.141892 0.989882i \(-0.545318\pi\)
−0.141892 + 0.989882i \(0.545318\pi\)
\(234\) 0 0
\(235\) 1.21924e8 0.612844
\(236\) 9.46637e7 0.468804
\(237\) 0 0
\(238\) −1.56686e8 −0.753376
\(239\) 2.12739e8 1.00799 0.503994 0.863707i \(-0.331863\pi\)
0.503994 + 0.863707i \(0.331863\pi\)
\(240\) 0 0
\(241\) 1.29896e8 0.597774 0.298887 0.954289i \(-0.403385\pi\)
0.298887 + 0.954289i \(0.403385\pi\)
\(242\) 2.39190e8 1.08490
\(243\) 0 0
\(244\) −6.55196e7 −0.288740
\(245\) 4.14660e7 0.180140
\(246\) 0 0
\(247\) −3.13796e8 −1.32497
\(248\) −1.08247e8 −0.450645
\(249\) 0 0
\(250\) 2.45575e8 0.994016
\(251\) −4.05186e8 −1.61732 −0.808661 0.588274i \(-0.799807\pi\)
−0.808661 + 0.588274i \(0.799807\pi\)
\(252\) 0 0
\(253\) 1.78798e8 0.694132
\(254\) 4.69747e8 1.79865
\(255\) 0 0
\(256\) 2.85488e8 1.06353
\(257\) 4.32001e8 1.58752 0.793758 0.608233i \(-0.208121\pi\)
0.793758 + 0.608233i \(0.208121\pi\)
\(258\) 0 0
\(259\) −2.75174e8 −0.984144
\(260\) 5.54751e7 0.195745
\(261\) 0 0
\(262\) −1.06395e8 −0.365483
\(263\) −9.96029e7 −0.337619 −0.168809 0.985649i \(-0.553992\pi\)
−0.168809 + 0.985649i \(0.553992\pi\)
\(264\) 0 0
\(265\) −1.97654e8 −0.652447
\(266\) −3.79702e8 −1.23696
\(267\) 0 0
\(268\) −4.43941e7 −0.140881
\(269\) 4.54907e8 1.42492 0.712459 0.701713i \(-0.247581\pi\)
0.712459 + 0.701713i \(0.247581\pi\)
\(270\) 0 0
\(271\) −2.69553e8 −0.822718 −0.411359 0.911473i \(-0.634946\pi\)
−0.411359 + 0.911473i \(0.634946\pi\)
\(272\) −3.32683e8 −1.00240
\(273\) 0 0
\(274\) 3.52523e8 1.03529
\(275\) 3.71850e8 1.07821
\(276\) 0 0
\(277\) −9.07532e7 −0.256556 −0.128278 0.991738i \(-0.540945\pi\)
−0.128278 + 0.991738i \(0.540945\pi\)
\(278\) −5.15248e8 −1.43833
\(279\) 0 0
\(280\) −9.38876e7 −0.255597
\(281\) 2.39385e7 0.0643614 0.0321807 0.999482i \(-0.489755\pi\)
0.0321807 + 0.999482i \(0.489755\pi\)
\(282\) 0 0
\(283\) −3.89906e8 −1.02261 −0.511303 0.859401i \(-0.670837\pi\)
−0.511303 + 0.859401i \(0.670837\pi\)
\(284\) 1.12313e6 0.00290950
\(285\) 0 0
\(286\) 6.51562e8 1.64693
\(287\) −5.47464e8 −1.36700
\(288\) 0 0
\(289\) −1.43522e8 −0.349766
\(290\) 1.25651e8 0.302534
\(291\) 0 0
\(292\) −3.36834e7 −0.0791728
\(293\) 3.58463e8 0.832544 0.416272 0.909240i \(-0.363336\pi\)
0.416272 + 0.909240i \(0.363336\pi\)
\(294\) 0 0
\(295\) −2.32634e8 −0.527589
\(296\) −3.88317e8 −0.870292
\(297\) 0 0
\(298\) 1.01006e8 0.221100
\(299\) 2.32242e8 0.502448
\(300\) 0 0
\(301\) −5.66312e7 −0.119694
\(302\) 9.63228e8 2.01236
\(303\) 0 0
\(304\) −8.06199e8 −1.64583
\(305\) 1.61013e8 0.324946
\(306\) 0 0
\(307\) 3.60914e8 0.711901 0.355951 0.934505i \(-0.384157\pi\)
0.355951 + 0.934505i \(0.384157\pi\)
\(308\) 2.31976e8 0.452393
\(309\) 0 0
\(310\) −1.90191e8 −0.362598
\(311\) −5.61989e8 −1.05942 −0.529708 0.848180i \(-0.677698\pi\)
−0.529708 + 0.848180i \(0.677698\pi\)
\(312\) 0 0
\(313\) 5.21602e8 0.961466 0.480733 0.876867i \(-0.340371\pi\)
0.480733 + 0.876867i \(0.340371\pi\)
\(314\) −1.39004e8 −0.253381
\(315\) 0 0
\(316\) −2.00917e8 −0.358188
\(317\) 1.50341e8 0.265076 0.132538 0.991178i \(-0.457687\pi\)
0.132538 + 0.991178i \(0.457687\pi\)
\(318\) 0 0
\(319\) 4.34227e8 0.748944
\(320\) −8.46919e7 −0.144483
\(321\) 0 0
\(322\) 2.81019e8 0.469073
\(323\) 6.46583e8 1.06762
\(324\) 0 0
\(325\) 4.82997e8 0.780463
\(326\) −1.31051e9 −2.09497
\(327\) 0 0
\(328\) −7.72563e8 −1.20886
\(329\) 6.62230e8 1.02523
\(330\) 0 0
\(331\) 7.31126e8 1.10814 0.554070 0.832470i \(-0.313074\pi\)
0.554070 + 0.832470i \(0.313074\pi\)
\(332\) −9.17100e7 −0.137541
\(333\) 0 0
\(334\) 1.55620e9 2.28535
\(335\) 1.09097e8 0.158547
\(336\) 0 0
\(337\) −3.27389e8 −0.465972 −0.232986 0.972480i \(-0.574850\pi\)
−0.232986 + 0.972480i \(0.574850\pi\)
\(338\) 1.27394e6 0.00179449
\(339\) 0 0
\(340\) −1.14308e8 −0.157725
\(341\) −6.57264e8 −0.897635
\(342\) 0 0
\(343\) 8.11816e8 1.08625
\(344\) −7.99160e7 −0.105847
\(345\) 0 0
\(346\) −5.47544e7 −0.0710645
\(347\) −7.03971e8 −0.904485 −0.452243 0.891895i \(-0.649376\pi\)
−0.452243 + 0.891895i \(0.649376\pi\)
\(348\) 0 0
\(349\) −3.71304e8 −0.467563 −0.233782 0.972289i \(-0.575110\pi\)
−0.233782 + 0.972289i \(0.575110\pi\)
\(350\) 5.84440e8 0.728621
\(351\) 0 0
\(352\) 8.88763e8 1.08614
\(353\) −1.27959e9 −1.54832 −0.774160 0.632990i \(-0.781828\pi\)
−0.774160 + 0.632990i \(0.781828\pi\)
\(354\) 0 0
\(355\) −2.76008e6 −0.00327432
\(356\) 3.24835e8 0.381582
\(357\) 0 0
\(358\) 1.10258e9 1.27005
\(359\) 3.54999e8 0.404946 0.202473 0.979288i \(-0.435102\pi\)
0.202473 + 0.979288i \(0.435102\pi\)
\(360\) 0 0
\(361\) 6.73010e8 0.752915
\(362\) −1.46794e9 −1.62640
\(363\) 0 0
\(364\) 3.01314e8 0.327465
\(365\) 8.27761e7 0.0891005
\(366\) 0 0
\(367\) 1.73415e9 1.83128 0.915639 0.402001i \(-0.131685\pi\)
0.915639 + 0.402001i \(0.131685\pi\)
\(368\) 5.96671e8 0.624119
\(369\) 0 0
\(370\) −6.82278e8 −0.700253
\(371\) −1.07356e9 −1.09149
\(372\) 0 0
\(373\) −3.01486e8 −0.300805 −0.150403 0.988625i \(-0.548057\pi\)
−0.150403 + 0.988625i \(0.548057\pi\)
\(374\) −1.34256e9 −1.32704
\(375\) 0 0
\(376\) 9.34516e8 0.906628
\(377\) 5.64018e8 0.542123
\(378\) 0 0
\(379\) −9.79815e8 −0.924499 −0.462250 0.886750i \(-0.652958\pi\)
−0.462250 + 0.886750i \(0.652958\pi\)
\(380\) −2.77005e8 −0.258967
\(381\) 0 0
\(382\) 3.58725e8 0.329261
\(383\) 4.89981e8 0.445639 0.222820 0.974860i \(-0.428474\pi\)
0.222820 + 0.974860i \(0.428474\pi\)
\(384\) 0 0
\(385\) −5.70076e8 −0.509120
\(386\) 2.35439e9 2.08364
\(387\) 0 0
\(388\) −3.85317e8 −0.334894
\(389\) 1.81229e9 1.56101 0.780504 0.625151i \(-0.214963\pi\)
0.780504 + 0.625151i \(0.214963\pi\)
\(390\) 0 0
\(391\) −4.78539e8 −0.404854
\(392\) 3.17828e8 0.266496
\(393\) 0 0
\(394\) 7.67707e8 0.632352
\(395\) 4.93748e8 0.403102
\(396\) 0 0
\(397\) −1.72818e9 −1.38619 −0.693096 0.720845i \(-0.743754\pi\)
−0.693096 + 0.720845i \(0.743754\pi\)
\(398\) −4.93971e8 −0.392745
\(399\) 0 0
\(400\) 1.24091e9 0.969458
\(401\) 8.93294e8 0.691814 0.345907 0.938269i \(-0.387571\pi\)
0.345907 + 0.938269i \(0.387571\pi\)
\(402\) 0 0
\(403\) −8.53721e8 −0.649753
\(404\) −4.80948e8 −0.362880
\(405\) 0 0
\(406\) 6.82478e8 0.506113
\(407\) −2.35782e9 −1.73352
\(408\) 0 0
\(409\) −7.17563e8 −0.518595 −0.259298 0.965797i \(-0.583491\pi\)
−0.259298 + 0.965797i \(0.583491\pi\)
\(410\) −1.35740e9 −0.972670
\(411\) 0 0
\(412\) −2.12113e8 −0.149426
\(413\) −1.26355e9 −0.882610
\(414\) 0 0
\(415\) 2.25375e8 0.154788
\(416\) 1.15442e9 0.786205
\(417\) 0 0
\(418\) −3.25346e9 −2.17885
\(419\) 8.34959e8 0.554519 0.277259 0.960795i \(-0.410574\pi\)
0.277259 + 0.960795i \(0.410574\pi\)
\(420\) 0 0
\(421\) 2.75014e6 0.00179626 0.000898128 1.00000i \(-0.499714\pi\)
0.000898128 1.00000i \(0.499714\pi\)
\(422\) −1.74983e9 −1.13345
\(423\) 0 0
\(424\) −1.51497e9 −0.965215
\(425\) −9.95225e8 −0.628869
\(426\) 0 0
\(427\) 8.74544e8 0.543606
\(428\) 1.18399e9 0.729951
\(429\) 0 0
\(430\) −1.40414e8 −0.0851666
\(431\) 1.21897e9 0.733370 0.366685 0.930345i \(-0.380493\pi\)
0.366685 + 0.930345i \(0.380493\pi\)
\(432\) 0 0
\(433\) −1.12096e9 −0.663561 −0.331781 0.943357i \(-0.607649\pi\)
−0.331781 + 0.943357i \(0.607649\pi\)
\(434\) −1.03303e9 −0.606594
\(435\) 0 0
\(436\) 6.62697e8 0.382924
\(437\) −1.15966e9 −0.664728
\(438\) 0 0
\(439\) −5.95770e8 −0.336088 −0.168044 0.985779i \(-0.553745\pi\)
−0.168044 + 0.985779i \(0.553745\pi\)
\(440\) −8.04471e8 −0.450221
\(441\) 0 0
\(442\) −1.74385e9 −0.960575
\(443\) −2.21773e9 −1.21198 −0.605991 0.795472i \(-0.707223\pi\)
−0.605991 + 0.795472i \(0.707223\pi\)
\(444\) 0 0
\(445\) −7.98275e8 −0.429430
\(446\) −2.72248e9 −1.45309
\(447\) 0 0
\(448\) −4.60005e8 −0.241707
\(449\) 7.44092e8 0.387940 0.193970 0.981007i \(-0.437864\pi\)
0.193970 + 0.981007i \(0.437864\pi\)
\(450\) 0 0
\(451\) −4.69092e9 −2.40791
\(452\) −1.36726e9 −0.696413
\(453\) 0 0
\(454\) 4.04779e7 0.0203012
\(455\) −7.40472e8 −0.368527
\(456\) 0 0
\(457\) 7.56617e8 0.370825 0.185413 0.982661i \(-0.440638\pi\)
0.185413 + 0.982661i \(0.440638\pi\)
\(458\) 2.75612e9 1.34050
\(459\) 0 0
\(460\) 2.05013e8 0.0982038
\(461\) 8.02106e7 0.0381310 0.0190655 0.999818i \(-0.493931\pi\)
0.0190655 + 0.999818i \(0.493931\pi\)
\(462\) 0 0
\(463\) 1.38557e9 0.648778 0.324389 0.945924i \(-0.394841\pi\)
0.324389 + 0.945924i \(0.394841\pi\)
\(464\) 1.44907e9 0.673403
\(465\) 0 0
\(466\) −7.37917e8 −0.337798
\(467\) −4.12145e9 −1.87258 −0.936292 0.351224i \(-0.885766\pi\)
−0.936292 + 0.351224i \(0.885766\pi\)
\(468\) 0 0
\(469\) 5.92564e8 0.265235
\(470\) 1.64196e9 0.729490
\(471\) 0 0
\(472\) −1.78308e9 −0.780504
\(473\) −4.85242e8 −0.210836
\(474\) 0 0
\(475\) −2.41176e9 −1.03254
\(476\) −6.20864e8 −0.263859
\(477\) 0 0
\(478\) 2.86498e9 1.19984
\(479\) 3.39610e8 0.141191 0.0705955 0.997505i \(-0.477510\pi\)
0.0705955 + 0.997505i \(0.477510\pi\)
\(480\) 0 0
\(481\) −3.06257e9 −1.25481
\(482\) 1.74933e9 0.711551
\(483\) 0 0
\(484\) 9.47782e8 0.379970
\(485\) 9.46906e8 0.376887
\(486\) 0 0
\(487\) −2.35025e9 −0.922066 −0.461033 0.887383i \(-0.652521\pi\)
−0.461033 + 0.887383i \(0.652521\pi\)
\(488\) 1.23413e9 0.480718
\(489\) 0 0
\(490\) 5.58428e8 0.214428
\(491\) 5.12182e8 0.195272 0.0976359 0.995222i \(-0.468872\pi\)
0.0976359 + 0.995222i \(0.468872\pi\)
\(492\) 0 0
\(493\) −1.16217e9 −0.436823
\(494\) −4.22592e9 −1.57716
\(495\) 0 0
\(496\) −2.19337e9 −0.807096
\(497\) −1.49914e7 −0.00547766
\(498\) 0 0
\(499\) −2.62730e9 −0.946582 −0.473291 0.880906i \(-0.656934\pi\)
−0.473291 + 0.880906i \(0.656934\pi\)
\(500\) 9.73081e8 0.348140
\(501\) 0 0
\(502\) −5.45669e9 −1.92516
\(503\) 1.27864e9 0.447982 0.223991 0.974591i \(-0.428091\pi\)
0.223991 + 0.974591i \(0.428091\pi\)
\(504\) 0 0
\(505\) 1.18192e9 0.408383
\(506\) 2.40790e9 0.826250
\(507\) 0 0
\(508\) 1.86136e9 0.629952
\(509\) 3.34993e9 1.12596 0.562980 0.826470i \(-0.309655\pi\)
0.562980 + 0.826470i \(0.309655\pi\)
\(510\) 0 0
\(511\) 4.49600e8 0.149057
\(512\) 3.45536e8 0.113776
\(513\) 0 0
\(514\) 5.81780e9 1.88968
\(515\) 5.21261e8 0.168163
\(516\) 0 0
\(517\) 5.67428e9 1.80590
\(518\) −3.70580e9 −1.17146
\(519\) 0 0
\(520\) −1.04493e9 −0.325893
\(521\) −3.15645e9 −0.977837 −0.488918 0.872330i \(-0.662608\pi\)
−0.488918 + 0.872330i \(0.662608\pi\)
\(522\) 0 0
\(523\) 5.04134e8 0.154095 0.0770477 0.997027i \(-0.475451\pi\)
0.0770477 + 0.997027i \(0.475451\pi\)
\(524\) −4.21587e8 −0.128005
\(525\) 0 0
\(526\) −1.34136e9 −0.401880
\(527\) 1.75911e9 0.523548
\(528\) 0 0
\(529\) −2.54656e9 −0.747926
\(530\) −2.66183e9 −0.776630
\(531\) 0 0
\(532\) −1.50456e9 −0.433230
\(533\) −6.09304e9 −1.74297
\(534\) 0 0
\(535\) −2.90962e9 −0.821481
\(536\) 8.36206e8 0.234550
\(537\) 0 0
\(538\) 6.12629e9 1.69613
\(539\) 1.92982e9 0.530830
\(540\) 0 0
\(541\) −5.65936e9 −1.53666 −0.768329 0.640056i \(-0.778911\pi\)
−0.768329 + 0.640056i \(0.778911\pi\)
\(542\) −3.63010e9 −0.979311
\(543\) 0 0
\(544\) −2.37870e9 −0.633495
\(545\) −1.62856e9 −0.430940
\(546\) 0 0
\(547\) 1.34725e9 0.351958 0.175979 0.984394i \(-0.443691\pi\)
0.175979 + 0.984394i \(0.443691\pi\)
\(548\) 1.39686e9 0.362595
\(549\) 0 0
\(550\) 5.00775e9 1.28343
\(551\) −2.81632e9 −0.717218
\(552\) 0 0
\(553\) 2.68180e9 0.674354
\(554\) −1.22218e9 −0.305388
\(555\) 0 0
\(556\) −2.04165e9 −0.503756
\(557\) 6.05230e9 1.48398 0.741989 0.670413i \(-0.233883\pi\)
0.741989 + 0.670413i \(0.233883\pi\)
\(558\) 0 0
\(559\) −6.30281e8 −0.152613
\(560\) −1.90241e9 −0.457768
\(561\) 0 0
\(562\) 3.22383e8 0.0766116
\(563\) 2.52625e9 0.596619 0.298310 0.954469i \(-0.403577\pi\)
0.298310 + 0.954469i \(0.403577\pi\)
\(564\) 0 0
\(565\) 3.36001e9 0.783738
\(566\) −5.25091e9 −1.21724
\(567\) 0 0
\(568\) −2.11553e7 −0.00484396
\(569\) −1.07885e9 −0.245509 −0.122755 0.992437i \(-0.539173\pi\)
−0.122755 + 0.992437i \(0.539173\pi\)
\(570\) 0 0
\(571\) 8.05110e9 1.80979 0.904897 0.425632i \(-0.139948\pi\)
0.904897 + 0.425632i \(0.139948\pi\)
\(572\) 2.58180e9 0.576814
\(573\) 0 0
\(574\) −7.37276e9 −1.62719
\(575\) 1.78495e9 0.391551
\(576\) 0 0
\(577\) 4.42426e9 0.958795 0.479397 0.877598i \(-0.340855\pi\)
0.479397 + 0.877598i \(0.340855\pi\)
\(578\) −1.93283e9 −0.416339
\(579\) 0 0
\(580\) 4.97890e8 0.105958
\(581\) 1.22413e9 0.258947
\(582\) 0 0
\(583\) −9.19875e9 −1.92260
\(584\) 6.34460e8 0.131813
\(585\) 0 0
\(586\) 4.82745e9 0.991006
\(587\) 3.26125e9 0.665503 0.332752 0.943015i \(-0.392023\pi\)
0.332752 + 0.943015i \(0.392023\pi\)
\(588\) 0 0
\(589\) 4.26290e9 0.859611
\(590\) −3.13290e9 −0.628008
\(591\) 0 0
\(592\) −7.86831e9 −1.55867
\(593\) 1.10562e9 0.217727 0.108864 0.994057i \(-0.465279\pi\)
0.108864 + 0.994057i \(0.465279\pi\)
\(594\) 0 0
\(595\) 1.52576e9 0.296945
\(596\) 4.00231e8 0.0774371
\(597\) 0 0
\(598\) 3.12762e9 0.598081
\(599\) −4.87422e9 −0.926640 −0.463320 0.886191i \(-0.653342\pi\)
−0.463320 + 0.886191i \(0.653342\pi\)
\(600\) 0 0
\(601\) 4.41328e9 0.829280 0.414640 0.909986i \(-0.363908\pi\)
0.414640 + 0.909986i \(0.363908\pi\)
\(602\) −7.62659e8 −0.142476
\(603\) 0 0
\(604\) 3.81676e9 0.704800
\(605\) −2.32915e9 −0.427616
\(606\) 0 0
\(607\) 9.64653e9 1.75070 0.875348 0.483494i \(-0.160632\pi\)
0.875348 + 0.483494i \(0.160632\pi\)
\(608\) −5.76436e9 −1.04013
\(609\) 0 0
\(610\) 2.16838e9 0.386795
\(611\) 7.37033e9 1.30720
\(612\) 0 0
\(613\) −1.75264e9 −0.307313 −0.153656 0.988124i \(-0.549105\pi\)
−0.153656 + 0.988124i \(0.549105\pi\)
\(614\) 4.86047e9 0.847401
\(615\) 0 0
\(616\) −4.36950e9 −0.753181
\(617\) −3.17834e9 −0.544757 −0.272378 0.962190i \(-0.587810\pi\)
−0.272378 + 0.962190i \(0.587810\pi\)
\(618\) 0 0
\(619\) 4.18253e9 0.708797 0.354398 0.935095i \(-0.384686\pi\)
0.354398 + 0.935095i \(0.384686\pi\)
\(620\) −7.53628e8 −0.126995
\(621\) 0 0
\(622\) −7.56836e9 −1.26106
\(623\) −4.33584e9 −0.718398
\(624\) 0 0
\(625\) 2.36866e9 0.388081
\(626\) 7.02447e9 1.14447
\(627\) 0 0
\(628\) −5.50800e8 −0.0887433
\(629\) 6.31050e9 1.01108
\(630\) 0 0
\(631\) 3.38318e9 0.536071 0.268035 0.963409i \(-0.413626\pi\)
0.268035 + 0.963409i \(0.413626\pi\)
\(632\) 3.78446e9 0.596341
\(633\) 0 0
\(634\) 2.02466e9 0.315529
\(635\) −4.57424e9 −0.708943
\(636\) 0 0
\(637\) 2.50664e9 0.384241
\(638\) 5.84778e9 0.891494
\(639\) 0 0
\(640\) −3.58496e9 −0.540573
\(641\) −2.99478e9 −0.449120 −0.224560 0.974460i \(-0.572094\pi\)
−0.224560 + 0.974460i \(0.572094\pi\)
\(642\) 0 0
\(643\) 1.17542e9 0.174363 0.0871817 0.996192i \(-0.472214\pi\)
0.0871817 + 0.996192i \(0.472214\pi\)
\(644\) 1.11353e9 0.164286
\(645\) 0 0
\(646\) 8.70760e9 1.27082
\(647\) 5.64077e9 0.818792 0.409396 0.912357i \(-0.365739\pi\)
0.409396 + 0.912357i \(0.365739\pi\)
\(648\) 0 0
\(649\) −1.08267e10 −1.55467
\(650\) 6.50457e9 0.929013
\(651\) 0 0
\(652\) −5.19285e9 −0.733734
\(653\) 7.03306e9 0.988435 0.494217 0.869338i \(-0.335455\pi\)
0.494217 + 0.869338i \(0.335455\pi\)
\(654\) 0 0
\(655\) 1.03604e9 0.144056
\(656\) −1.56541e10 −2.16504
\(657\) 0 0
\(658\) 8.91832e9 1.22037
\(659\) 5.30583e9 0.722194 0.361097 0.932528i \(-0.382402\pi\)
0.361097 + 0.932528i \(0.382402\pi\)
\(660\) 0 0
\(661\) −4.29135e9 −0.577949 −0.288974 0.957337i \(-0.593314\pi\)
−0.288974 + 0.957337i \(0.593314\pi\)
\(662\) 9.84615e9 1.31906
\(663\) 0 0
\(664\) 1.72745e9 0.228990
\(665\) 3.69741e9 0.487553
\(666\) 0 0
\(667\) 2.08437e9 0.271978
\(668\) 6.16640e9 0.800413
\(669\) 0 0
\(670\) 1.46922e9 0.188724
\(671\) 7.49349e9 0.957536
\(672\) 0 0
\(673\) 3.84315e9 0.485998 0.242999 0.970027i \(-0.421869\pi\)
0.242999 + 0.970027i \(0.421869\pi\)
\(674\) −4.40898e9 −0.554662
\(675\) 0 0
\(676\) 5.04796e6 0.000628496 0
\(677\) 1.03024e10 1.27608 0.638041 0.770002i \(-0.279745\pi\)
0.638041 + 0.770002i \(0.279745\pi\)
\(678\) 0 0
\(679\) 5.14314e9 0.630498
\(680\) 2.15310e9 0.262593
\(681\) 0 0
\(682\) −8.85145e9 −1.06849
\(683\) −2.78807e8 −0.0334835 −0.0167417 0.999860i \(-0.505329\pi\)
−0.0167417 + 0.999860i \(0.505329\pi\)
\(684\) 0 0
\(685\) −3.43276e9 −0.408062
\(686\) 1.09328e10 1.29300
\(687\) 0 0
\(688\) −1.61931e9 −0.189570
\(689\) −1.19483e10 −1.39167
\(690\) 0 0
\(691\) 1.57009e9 0.181030 0.0905151 0.995895i \(-0.471149\pi\)
0.0905151 + 0.995895i \(0.471149\pi\)
\(692\) −2.16963e8 −0.0248893
\(693\) 0 0
\(694\) −9.48045e9 −1.07664
\(695\) 5.01731e9 0.566923
\(696\) 0 0
\(697\) 1.25548e10 1.40442
\(698\) −5.00039e9 −0.556557
\(699\) 0 0
\(700\) 2.31583e9 0.255189
\(701\) −1.55305e10 −1.70284 −0.851418 0.524488i \(-0.824257\pi\)
−0.851418 + 0.524488i \(0.824257\pi\)
\(702\) 0 0
\(703\) 1.52924e10 1.66009
\(704\) −3.94153e9 −0.425756
\(705\) 0 0
\(706\) −1.72324e10 −1.84302
\(707\) 6.41961e9 0.683189
\(708\) 0 0
\(709\) 1.58882e10 1.67422 0.837112 0.547032i \(-0.184242\pi\)
0.837112 + 0.547032i \(0.184242\pi\)
\(710\) −3.71702e7 −0.00389754
\(711\) 0 0
\(712\) −6.11859e9 −0.635289
\(713\) −3.15499e9 −0.325975
\(714\) 0 0
\(715\) −6.34470e9 −0.649142
\(716\) 4.36896e9 0.444818
\(717\) 0 0
\(718\) 4.78082e9 0.482021
\(719\) −6.03906e9 −0.605924 −0.302962 0.953003i \(-0.597976\pi\)
−0.302962 + 0.953003i \(0.597976\pi\)
\(720\) 0 0
\(721\) 2.83124e9 0.281322
\(722\) 9.06349e9 0.896222
\(723\) 0 0
\(724\) −5.81665e9 −0.569624
\(725\) 4.33490e9 0.422470
\(726\) 0 0
\(727\) 1.96027e10 1.89211 0.946054 0.324008i \(-0.105030\pi\)
0.946054 + 0.324008i \(0.105030\pi\)
\(728\) −5.67555e9 −0.545190
\(729\) 0 0
\(730\) 1.11475e9 0.106059
\(731\) 1.29871e9 0.122970
\(732\) 0 0
\(733\) −4.74673e9 −0.445174 −0.222587 0.974913i \(-0.571450\pi\)
−0.222587 + 0.974913i \(0.571450\pi\)
\(734\) 2.33539e10 2.17984
\(735\) 0 0
\(736\) 4.26623e9 0.394432
\(737\) 5.07735e9 0.467198
\(738\) 0 0
\(739\) −9.87972e9 −0.900512 −0.450256 0.892900i \(-0.648667\pi\)
−0.450256 + 0.892900i \(0.648667\pi\)
\(740\) −2.70350e9 −0.245254
\(741\) 0 0
\(742\) −1.44578e10 −1.29923
\(743\) 1.28591e10 1.15014 0.575069 0.818105i \(-0.304975\pi\)
0.575069 + 0.818105i \(0.304975\pi\)
\(744\) 0 0
\(745\) −9.83559e8 −0.0871472
\(746\) −4.06014e9 −0.358059
\(747\) 0 0
\(748\) −5.31985e9 −0.464776
\(749\) −1.58036e10 −1.37427
\(750\) 0 0
\(751\) 7.42726e9 0.639866 0.319933 0.947440i \(-0.396340\pi\)
0.319933 + 0.947440i \(0.396340\pi\)
\(752\) 1.89357e10 1.62375
\(753\) 0 0
\(754\) 7.59569e9 0.645309
\(755\) −9.37960e9 −0.793177
\(756\) 0 0
\(757\) 7.19774e9 0.603060 0.301530 0.953457i \(-0.402503\pi\)
0.301530 + 0.953457i \(0.402503\pi\)
\(758\) −1.31953e10 −1.10046
\(759\) 0 0
\(760\) 5.21766e9 0.431150
\(761\) 1.01426e10 0.834265 0.417132 0.908846i \(-0.363035\pi\)
0.417132 + 0.908846i \(0.363035\pi\)
\(762\) 0 0
\(763\) −8.84556e9 −0.720924
\(764\) 1.42144e9 0.115319
\(765\) 0 0
\(766\) 6.59862e9 0.530460
\(767\) −1.40628e10 −1.12535
\(768\) 0 0
\(769\) −3.34842e9 −0.265521 −0.132760 0.991148i \(-0.542384\pi\)
−0.132760 + 0.991148i \(0.542384\pi\)
\(770\) −7.67727e9 −0.606023
\(771\) 0 0
\(772\) 9.32920e9 0.729767
\(773\) 1.64501e10 1.28097 0.640485 0.767970i \(-0.278733\pi\)
0.640485 + 0.767970i \(0.278733\pi\)
\(774\) 0 0
\(775\) −6.56149e9 −0.506345
\(776\) 7.25782e9 0.557558
\(777\) 0 0
\(778\) 2.44064e10 1.85812
\(779\) 3.04245e10 2.30591
\(780\) 0 0
\(781\) −1.28453e8 −0.00964863
\(782\) −6.44453e9 −0.481912
\(783\) 0 0
\(784\) 6.44002e9 0.477288
\(785\) 1.35358e9 0.0998710
\(786\) 0 0
\(787\) −1.67903e9 −0.122786 −0.0613929 0.998114i \(-0.519554\pi\)
−0.0613929 + 0.998114i \(0.519554\pi\)
\(788\) 3.04202e9 0.221472
\(789\) 0 0
\(790\) 6.64935e9 0.479827
\(791\) 1.82500e10 1.31112
\(792\) 0 0
\(793\) 9.73330e9 0.693113
\(794\) −2.32736e10 −1.65003
\(795\) 0 0
\(796\) −1.95734e9 −0.137553
\(797\) 1.88216e10 1.31690 0.658449 0.752625i \(-0.271213\pi\)
0.658449 + 0.752625i \(0.271213\pi\)
\(798\) 0 0
\(799\) −1.51867e10 −1.05330
\(800\) 8.87255e9 0.612680
\(801\) 0 0
\(802\) 1.20301e10 0.823491
\(803\) 3.85237e9 0.262557
\(804\) 0 0
\(805\) −2.73647e9 −0.184887
\(806\) −1.14972e10 −0.773424
\(807\) 0 0
\(808\) 9.05913e9 0.604153
\(809\) −1.07428e10 −0.713342 −0.356671 0.934230i \(-0.616088\pi\)
−0.356671 + 0.934230i \(0.616088\pi\)
\(810\) 0 0
\(811\) 2.81229e10 1.85135 0.925673 0.378326i \(-0.123500\pi\)
0.925673 + 0.378326i \(0.123500\pi\)
\(812\) 2.70430e9 0.177259
\(813\) 0 0
\(814\) −3.17530e10 −2.06347
\(815\) 1.27613e10 0.825739
\(816\) 0 0
\(817\) 3.14719e9 0.201905
\(818\) −9.66350e9 −0.617302
\(819\) 0 0
\(820\) −5.37867e9 −0.340664
\(821\) −1.62322e10 −1.02371 −0.511853 0.859073i \(-0.671041\pi\)
−0.511853 + 0.859073i \(0.671041\pi\)
\(822\) 0 0
\(823\) −6.36841e9 −0.398228 −0.199114 0.979976i \(-0.563806\pi\)
−0.199114 + 0.979976i \(0.563806\pi\)
\(824\) 3.99535e9 0.248776
\(825\) 0 0
\(826\) −1.70164e10 −1.05060
\(827\) 1.77050e10 1.08850 0.544248 0.838924i \(-0.316815\pi\)
0.544248 + 0.838924i \(0.316815\pi\)
\(828\) 0 0
\(829\) −3.43518e9 −0.209415 −0.104708 0.994503i \(-0.533391\pi\)
−0.104708 + 0.994503i \(0.533391\pi\)
\(830\) 3.03515e9 0.184250
\(831\) 0 0
\(832\) −5.11966e9 −0.308184
\(833\) −5.16499e9 −0.309608
\(834\) 0 0
\(835\) −1.51538e10 −0.900778
\(836\) −1.28917e10 −0.763113
\(837\) 0 0
\(838\) 1.12445e10 0.660063
\(839\) 2.12784e10 1.24386 0.621930 0.783073i \(-0.286349\pi\)
0.621930 + 0.783073i \(0.286349\pi\)
\(840\) 0 0
\(841\) −1.21878e10 −0.706545
\(842\) 3.70365e7 0.00213815
\(843\) 0 0
\(844\) −6.93367e9 −0.396976
\(845\) −1.24052e7 −0.000707305 0
\(846\) 0 0
\(847\) −1.26508e10 −0.715363
\(848\) −3.06973e10 −1.72868
\(849\) 0 0
\(850\) −1.34028e10 −0.748565
\(851\) −1.13180e10 −0.629528
\(852\) 0 0
\(853\) −8.50432e9 −0.469157 −0.234578 0.972097i \(-0.575371\pi\)
−0.234578 + 0.972097i \(0.575371\pi\)
\(854\) 1.17776e10 0.647073
\(855\) 0 0
\(856\) −2.23016e10 −1.21528
\(857\) 8.11675e8 0.0440503 0.0220252 0.999757i \(-0.492989\pi\)
0.0220252 + 0.999757i \(0.492989\pi\)
\(858\) 0 0
\(859\) −1.84624e10 −0.993832 −0.496916 0.867799i \(-0.665534\pi\)
−0.496916 + 0.867799i \(0.665534\pi\)
\(860\) −5.56384e8 −0.0298284
\(861\) 0 0
\(862\) 1.64160e10 0.872957
\(863\) −1.83495e10 −0.971823 −0.485911 0.874008i \(-0.661512\pi\)
−0.485911 + 0.874008i \(0.661512\pi\)
\(864\) 0 0
\(865\) 5.33180e8 0.0280103
\(866\) −1.50960e10 −0.789860
\(867\) 0 0
\(868\) −4.09334e9 −0.212451
\(869\) 2.29789e10 1.18784
\(870\) 0 0
\(871\) 6.59498e9 0.338182
\(872\) −1.24825e10 −0.637523
\(873\) 0 0
\(874\) −1.56172e10 −0.791250
\(875\) −1.29885e10 −0.655437
\(876\) 0 0
\(877\) 3.41189e10 1.70803 0.854016 0.520247i \(-0.174160\pi\)
0.854016 + 0.520247i \(0.174160\pi\)
\(878\) −8.02330e9 −0.400057
\(879\) 0 0
\(880\) −1.63007e10 −0.806337
\(881\) −1.72277e10 −0.848811 −0.424406 0.905472i \(-0.639517\pi\)
−0.424406 + 0.905472i \(0.639517\pi\)
\(882\) 0 0
\(883\) 2.58479e10 1.26347 0.631733 0.775186i \(-0.282344\pi\)
0.631733 + 0.775186i \(0.282344\pi\)
\(884\) −6.90995e9 −0.336428
\(885\) 0 0
\(886\) −2.98664e10 −1.44266
\(887\) 3.51605e10 1.69170 0.845848 0.533424i \(-0.179095\pi\)
0.845848 + 0.533424i \(0.179095\pi\)
\(888\) 0 0
\(889\) −2.48451e10 −1.18600
\(890\) −1.07505e10 −0.511166
\(891\) 0 0
\(892\) −1.07877e10 −0.508925
\(893\) −3.68024e10 −1.72940
\(894\) 0 0
\(895\) −1.07366e10 −0.500595
\(896\) −1.94718e10 −0.904330
\(897\) 0 0
\(898\) 1.00208e10 0.461778
\(899\) −7.66216e9 −0.351716
\(900\) 0 0
\(901\) 2.46197e10 1.12136
\(902\) −6.31731e10 −2.86622
\(903\) 0 0
\(904\) 2.57537e10 1.15945
\(905\) 1.42943e10 0.641050
\(906\) 0 0
\(907\) 1.84400e8 0.00820607 0.00410303 0.999992i \(-0.498694\pi\)
0.00410303 + 0.999992i \(0.498694\pi\)
\(908\) 1.60392e8 0.00711023
\(909\) 0 0
\(910\) −9.97202e9 −0.438670
\(911\) 1.19294e10 0.522761 0.261380 0.965236i \(-0.415822\pi\)
0.261380 + 0.965236i \(0.415822\pi\)
\(912\) 0 0
\(913\) 1.04889e10 0.456123
\(914\) 1.01894e10 0.441407
\(915\) 0 0
\(916\) 1.09210e10 0.469493
\(917\) 5.62727e9 0.240993
\(918\) 0 0
\(919\) 3.79246e10 1.61182 0.805910 0.592038i \(-0.201677\pi\)
0.805910 + 0.592038i \(0.201677\pi\)
\(920\) −3.86161e9 −0.163498
\(921\) 0 0
\(922\) 1.08020e9 0.0453887
\(923\) −1.66848e8 −0.00698417
\(924\) 0 0
\(925\) −2.35382e10 −0.977860
\(926\) 1.86597e10 0.772263
\(927\) 0 0
\(928\) 1.03609e10 0.425578
\(929\) 3.05711e10 1.25099 0.625497 0.780226i \(-0.284896\pi\)
0.625497 + 0.780226i \(0.284896\pi\)
\(930\) 0 0
\(931\) −1.25165e10 −0.508344
\(932\) −2.92397e9 −0.118309
\(933\) 0 0
\(934\) −5.55040e10 −2.22900
\(935\) 1.30734e10 0.523055
\(936\) 0 0
\(937\) −1.29971e10 −0.516129 −0.258065 0.966128i \(-0.583085\pi\)
−0.258065 + 0.966128i \(0.583085\pi\)
\(938\) 7.98012e9 0.315718
\(939\) 0 0
\(940\) 6.50621e9 0.255494
\(941\) 3.94818e10 1.54466 0.772330 0.635221i \(-0.219091\pi\)
0.772330 + 0.635221i \(0.219091\pi\)
\(942\) 0 0
\(943\) −2.25173e10 −0.874431
\(944\) −3.61299e10 −1.39787
\(945\) 0 0
\(946\) −6.53480e9 −0.250965
\(947\) −1.20683e10 −0.461766 −0.230883 0.972982i \(-0.574161\pi\)
−0.230883 + 0.972982i \(0.574161\pi\)
\(948\) 0 0
\(949\) 5.00385e9 0.190052
\(950\) −3.24794e10 −1.22907
\(951\) 0 0
\(952\) 1.16946e10 0.439294
\(953\) −8.06727e9 −0.301927 −0.150963 0.988539i \(-0.548238\pi\)
−0.150963 + 0.988539i \(0.548238\pi\)
\(954\) 0 0
\(955\) −3.49314e9 −0.129779
\(956\) 1.13524e10 0.420228
\(957\) 0 0
\(958\) 4.57357e9 0.168065
\(959\) −1.86451e10 −0.682651
\(960\) 0 0
\(961\) −1.59148e10 −0.578456
\(962\) −4.12440e10 −1.49365
\(963\) 0 0
\(964\) 6.93165e9 0.249211
\(965\) −2.29263e10 −0.821274
\(966\) 0 0
\(967\) −2.31765e10 −0.824244 −0.412122 0.911129i \(-0.635212\pi\)
−0.412122 + 0.911129i \(0.635212\pi\)
\(968\) −1.78524e10 −0.632606
\(969\) 0 0
\(970\) 1.27521e10 0.448622
\(971\) −6.13552e9 −0.215072 −0.107536 0.994201i \(-0.534296\pi\)
−0.107536 + 0.994201i \(0.534296\pi\)
\(972\) 0 0
\(973\) 2.72516e10 0.948412
\(974\) −3.16510e10 −1.09757
\(975\) 0 0
\(976\) 2.50066e10 0.860956
\(977\) 2.15287e10 0.738563 0.369282 0.929317i \(-0.379604\pi\)
0.369282 + 0.929317i \(0.379604\pi\)
\(978\) 0 0
\(979\) −3.71514e10 −1.26542
\(980\) 2.21275e9 0.0751003
\(981\) 0 0
\(982\) 6.89761e9 0.232439
\(983\) −5.33254e8 −0.0179059 −0.00895296 0.999960i \(-0.502850\pi\)
−0.00895296 + 0.999960i \(0.502850\pi\)
\(984\) 0 0
\(985\) −7.47568e9 −0.249243
\(986\) −1.56511e10 −0.519966
\(987\) 0 0
\(988\) −1.67451e10 −0.552380
\(989\) −2.32925e9 −0.0765648
\(990\) 0 0
\(991\) −2.96780e10 −0.968671 −0.484336 0.874882i \(-0.660939\pi\)
−0.484336 + 0.874882i \(0.660939\pi\)
\(992\) −1.56827e10 −0.510070
\(993\) 0 0
\(994\) −2.01891e8 −0.00652025
\(995\) 4.81012e9 0.154802
\(996\) 0 0
\(997\) 3.33968e10 1.06726 0.533632 0.845717i \(-0.320827\pi\)
0.533632 + 0.845717i \(0.320827\pi\)
\(998\) −3.53822e10 −1.12675
\(999\) 0 0
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 387.8.a.b.1.8 11
3.2 odd 2 43.8.a.a.1.4 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.4 11 3.2 odd 2
387.8.a.b.1.8 11 1.1 even 1 trivial