Properties

Label 69.8.a.d.1.1
Level $69$
Weight $8$
Character 69.1
Self dual yes
Analytic conductor $21.555$
Analytic rank $0$
Dimension $8$
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] = [69,8,Mod(1,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, 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("69.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.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: \(21.5545667584\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.5036\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.5036 q^{2} +27.0000 q^{3} +144.368 q^{4} -425.514 q^{5} -445.597 q^{6} -1501.62 q^{7} -270.137 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-16.5036 q^{2} +27.0000 q^{3} +144.368 q^{4} -425.514 q^{5} -445.597 q^{6} -1501.62 q^{7} -270.137 q^{8} +729.000 q^{9} +7022.50 q^{10} -566.851 q^{11} +3897.95 q^{12} -10492.6 q^{13} +24782.1 q^{14} -11488.9 q^{15} -14020.9 q^{16} +8189.40 q^{17} -12031.1 q^{18} -48781.8 q^{19} -61430.7 q^{20} -40543.8 q^{21} +9355.07 q^{22} +12167.0 q^{23} -7293.70 q^{24} +102937. q^{25} +173166. q^{26} +19683.0 q^{27} -216787. q^{28} -78559.1 q^{29} +189608. q^{30} +182771. q^{31} +265973. q^{32} -15305.0 q^{33} -135154. q^{34} +638961. q^{35} +105245. q^{36} -397244. q^{37} +805075. q^{38} -283301. q^{39} +114947. q^{40} +567191. q^{41} +669118. q^{42} +751273. q^{43} -81835.3 q^{44} -310199. q^{45} -200799. q^{46} -748990. q^{47} -378565. q^{48} +1.43133e6 q^{49} -1.69883e6 q^{50} +221114. q^{51} -1.51480e6 q^{52} +377925. q^{53} -324840. q^{54} +241203. q^{55} +405644. q^{56} -1.31711e6 q^{57} +1.29651e6 q^{58} +2.10007e6 q^{59} -1.65863e6 q^{60} +792539. q^{61} -3.01638e6 q^{62} -1.09468e6 q^{63} -2.59483e6 q^{64} +4.46476e6 q^{65} +252587. q^{66} -241805. q^{67} +1.18229e6 q^{68} +328509. q^{69} -1.05451e7 q^{70} -4.72270e6 q^{71} -196930. q^{72} -2.33919e6 q^{73} +6.55596e6 q^{74} +2.77930e6 q^{75} -7.04255e6 q^{76} +851196. q^{77} +4.67549e6 q^{78} +794806. q^{79} +5.96609e6 q^{80} +531441. q^{81} -9.36068e6 q^{82} +5.22492e6 q^{83} -5.85324e6 q^{84} -3.48470e6 q^{85} -1.23987e7 q^{86} -2.12110e6 q^{87} +153127. q^{88} -5.72503e6 q^{89} +5.11940e6 q^{90} +1.57560e7 q^{91} +1.75653e6 q^{92} +4.93483e6 q^{93} +1.23610e7 q^{94} +2.07573e7 q^{95} +7.18127e6 q^{96} -7.01309e6 q^{97} -2.36220e7 q^{98} -413234. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.5036 −1.45872 −0.729362 0.684128i \(-0.760183\pi\)
−0.729362 + 0.684128i \(0.760183\pi\)
\(3\) 27.0000 0.577350
\(4\) 144.368 1.12788
\(5\) −425.514 −1.52236 −0.761182 0.648538i \(-0.775381\pi\)
−0.761182 + 0.648538i \(0.775381\pi\)
\(6\) −445.597 −0.842195
\(7\) −1501.62 −1.65469 −0.827347 0.561692i \(-0.810151\pi\)
−0.827347 + 0.561692i \(0.810151\pi\)
\(8\) −270.137 −0.186539
\(9\) 729.000 0.333333
\(10\) 7022.50 2.22071
\(11\) −566.851 −0.128409 −0.0642043 0.997937i \(-0.520451\pi\)
−0.0642043 + 0.997937i \(0.520451\pi\)
\(12\) 3897.95 0.651181
\(13\) −10492.6 −1.32459 −0.662297 0.749241i \(-0.730418\pi\)
−0.662297 + 0.749241i \(0.730418\pi\)
\(14\) 24782.1 2.41374
\(15\) −11488.9 −0.878937
\(16\) −14020.9 −0.855769
\(17\) 8189.40 0.404279 0.202139 0.979357i \(-0.435211\pi\)
0.202139 + 0.979357i \(0.435211\pi\)
\(18\) −12031.1 −0.486242
\(19\) −48781.8 −1.63162 −0.815812 0.578317i \(-0.803710\pi\)
−0.815812 + 0.578317i \(0.803710\pi\)
\(20\) −61430.7 −1.71704
\(21\) −40543.8 −0.955338
\(22\) 9355.07 0.187313
\(23\) 12167.0 0.208514
\(24\) −7293.70 −0.107698
\(25\) 102937. 1.31759
\(26\) 173166. 1.93222
\(27\) 19683.0 0.192450
\(28\) −216787. −1.86629
\(29\) −78559.1 −0.598141 −0.299071 0.954231i \(-0.596677\pi\)
−0.299071 + 0.954231i \(0.596677\pi\)
\(30\) 189608. 1.28213
\(31\) 182771. 1.10190 0.550950 0.834538i \(-0.314265\pi\)
0.550950 + 0.834538i \(0.314265\pi\)
\(32\) 265973. 1.43487
\(33\) −15305.0 −0.0741368
\(34\) −135154. −0.589731
\(35\) 638961. 2.51905
\(36\) 105245. 0.375959
\(37\) −397244. −1.28929 −0.644646 0.764481i \(-0.722995\pi\)
−0.644646 + 0.764481i \(0.722995\pi\)
\(38\) 805075. 2.38009
\(39\) −283301. −0.764755
\(40\) 114947. 0.283980
\(41\) 567191. 1.28524 0.642622 0.766183i \(-0.277846\pi\)
0.642622 + 0.766183i \(0.277846\pi\)
\(42\) 669118. 1.39357
\(43\) 751273. 1.44098 0.720491 0.693464i \(-0.243917\pi\)
0.720491 + 0.693464i \(0.243917\pi\)
\(44\) −81835.3 −0.144829
\(45\) −310199. −0.507455
\(46\) −200799. −0.304165
\(47\) −748990. −1.05229 −0.526143 0.850396i \(-0.676362\pi\)
−0.526143 + 0.850396i \(0.676362\pi\)
\(48\) −378565. −0.494079
\(49\) 1.43133e6 1.73801
\(50\) −1.69883e6 −1.92200
\(51\) 221114. 0.233410
\(52\) −1.51480e6 −1.49398
\(53\) 377925. 0.348691 0.174345 0.984685i \(-0.444219\pi\)
0.174345 + 0.984685i \(0.444219\pi\)
\(54\) −324840. −0.280732
\(55\) 241203. 0.195485
\(56\) 405644. 0.308665
\(57\) −1.31711e6 −0.942019
\(58\) 1.29651e6 0.872523
\(59\) 2.10007e6 1.33122 0.665612 0.746298i \(-0.268171\pi\)
0.665612 + 0.746298i \(0.268171\pi\)
\(60\) −1.65863e6 −0.991334
\(61\) 792539. 0.447060 0.223530 0.974697i \(-0.428242\pi\)
0.223530 + 0.974697i \(0.428242\pi\)
\(62\) −3.01638e6 −1.60737
\(63\) −1.09468e6 −0.551564
\(64\) −2.59483e6 −1.23731
\(65\) 4.46476e6 2.01651
\(66\) 252587. 0.108145
\(67\) −241805. −0.0982209 −0.0491105 0.998793i \(-0.515639\pi\)
−0.0491105 + 0.998793i \(0.515639\pi\)
\(68\) 1.18229e6 0.455977
\(69\) 328509. 0.120386
\(70\) −1.05451e7 −3.67459
\(71\) −4.72270e6 −1.56598 −0.782990 0.622034i \(-0.786307\pi\)
−0.782990 + 0.622034i \(0.786307\pi\)
\(72\) −196930. −0.0621796
\(73\) −2.33919e6 −0.703779 −0.351889 0.936042i \(-0.614461\pi\)
−0.351889 + 0.936042i \(0.614461\pi\)
\(74\) 6.55596e6 1.88072
\(75\) 2.77930e6 0.760712
\(76\) −7.04255e6 −1.84027
\(77\) 851196. 0.212477
\(78\) 4.67549e6 1.11557
\(79\) 794806. 0.181370 0.0906851 0.995880i \(-0.471094\pi\)
0.0906851 + 0.995880i \(0.471094\pi\)
\(80\) 5.96609e6 1.30279
\(81\) 531441. 0.111111
\(82\) −9.36068e6 −1.87482
\(83\) 5.22492e6 1.00301 0.501506 0.865154i \(-0.332780\pi\)
0.501506 + 0.865154i \(0.332780\pi\)
\(84\) −5.85324e6 −1.07750
\(85\) −3.48470e6 −0.615459
\(86\) −1.23987e7 −2.10200
\(87\) −2.12110e6 −0.345337
\(88\) 153127. 0.0239532
\(89\) −5.72503e6 −0.860821 −0.430411 0.902633i \(-0.641631\pi\)
−0.430411 + 0.902633i \(0.641631\pi\)
\(90\) 5.11940e6 0.740237
\(91\) 1.57560e7 2.19180
\(92\) 1.75653e6 0.235179
\(93\) 4.93483e6 0.636182
\(94\) 1.23610e7 1.53499
\(95\) 2.07573e7 2.48393
\(96\) 7.18127e6 0.828423
\(97\) −7.01309e6 −0.780204 −0.390102 0.920772i \(-0.627560\pi\)
−0.390102 + 0.920772i \(0.627560\pi\)
\(98\) −2.36220e7 −2.53528
\(99\) −413234. −0.0428029
\(100\) 1.48608e7 1.48608
\(101\) 1.91067e7 1.84527 0.922637 0.385670i \(-0.126030\pi\)
0.922637 + 0.385670i \(0.126030\pi\)
\(102\) −3.64917e6 −0.340482
\(103\) −1.99719e7 −1.80090 −0.900448 0.434964i \(-0.856761\pi\)
−0.900448 + 0.434964i \(0.856761\pi\)
\(104\) 2.83445e6 0.247088
\(105\) 1.72519e7 1.45437
\(106\) −6.23712e6 −0.508644
\(107\) −1.88125e7 −1.48458 −0.742288 0.670081i \(-0.766259\pi\)
−0.742288 + 0.670081i \(0.766259\pi\)
\(108\) 2.84160e6 0.217060
\(109\) −1.74721e6 −0.129227 −0.0646133 0.997910i \(-0.520581\pi\)
−0.0646133 + 0.997910i \(0.520581\pi\)
\(110\) −3.98071e6 −0.285158
\(111\) −1.07256e7 −0.744374
\(112\) 2.10541e7 1.41604
\(113\) −3.56073e6 −0.232148 −0.116074 0.993241i \(-0.537031\pi\)
−0.116074 + 0.993241i \(0.537031\pi\)
\(114\) 2.17370e7 1.37415
\(115\) −5.17722e6 −0.317435
\(116\) −1.13415e7 −0.674630
\(117\) −7.64913e6 −0.441531
\(118\) −3.46586e7 −1.94189
\(119\) −1.22974e7 −0.668957
\(120\) 3.10357e6 0.163956
\(121\) −1.91659e7 −0.983511
\(122\) −1.30797e7 −0.652138
\(123\) 1.53142e7 0.742036
\(124\) 2.63864e7 1.24281
\(125\) −1.05578e7 −0.483490
\(126\) 1.80662e7 0.804581
\(127\) −4.10534e6 −0.177843 −0.0889213 0.996039i \(-0.528342\pi\)
−0.0889213 + 0.996039i \(0.528342\pi\)
\(128\) 8.77948e6 0.370027
\(129\) 2.02844e7 0.831951
\(130\) −7.36846e7 −2.94154
\(131\) −2.59973e6 −0.101037 −0.0505183 0.998723i \(-0.516087\pi\)
−0.0505183 + 0.998723i \(0.516087\pi\)
\(132\) −2.20955e6 −0.0836173
\(133\) 7.32518e7 2.69984
\(134\) 3.99065e6 0.143277
\(135\) −8.37539e6 −0.292979
\(136\) −2.21226e6 −0.0754137
\(137\) −9.66370e6 −0.321086 −0.160543 0.987029i \(-0.551325\pi\)
−0.160543 + 0.987029i \(0.551325\pi\)
\(138\) −5.42158e6 −0.175610
\(139\) 2.95184e7 0.932268 0.466134 0.884714i \(-0.345646\pi\)
0.466134 + 0.884714i \(0.345646\pi\)
\(140\) 9.22457e7 2.84118
\(141\) −2.02227e7 −0.607537
\(142\) 7.79415e7 2.28434
\(143\) 5.94776e6 0.170089
\(144\) −1.02213e7 −0.285256
\(145\) 3.34280e7 0.910588
\(146\) 3.86051e7 1.02662
\(147\) 3.86458e7 1.00344
\(148\) −5.73495e7 −1.45417
\(149\) 3.91998e7 0.970805 0.485403 0.874291i \(-0.338673\pi\)
0.485403 + 0.874291i \(0.338673\pi\)
\(150\) −4.58683e7 −1.10967
\(151\) −6.10718e7 −1.44351 −0.721757 0.692146i \(-0.756665\pi\)
−0.721757 + 0.692146i \(0.756665\pi\)
\(152\) 1.31778e7 0.304361
\(153\) 5.97007e6 0.134760
\(154\) −1.40478e7 −0.309945
\(155\) −7.77717e7 −1.67749
\(156\) −4.08997e7 −0.862550
\(157\) 5.82798e7 1.20190 0.600951 0.799286i \(-0.294789\pi\)
0.600951 + 0.799286i \(0.294789\pi\)
\(158\) −1.31171e7 −0.264569
\(159\) 1.02040e7 0.201317
\(160\) −1.13175e8 −2.18440
\(161\) −1.82702e7 −0.345027
\(162\) −8.77068e6 −0.162081
\(163\) 1.17687e7 0.212850 0.106425 0.994321i \(-0.466060\pi\)
0.106425 + 0.994321i \(0.466060\pi\)
\(164\) 8.18844e7 1.44960
\(165\) 6.51247e6 0.112863
\(166\) −8.62298e7 −1.46312
\(167\) 2.36080e7 0.392239 0.196120 0.980580i \(-0.437166\pi\)
0.196120 + 0.980580i \(0.437166\pi\)
\(168\) 1.09524e7 0.178208
\(169\) 4.73469e7 0.754550
\(170\) 5.75101e7 0.897786
\(171\) −3.55619e7 −0.543875
\(172\) 1.08460e8 1.62525
\(173\) −2.90114e7 −0.425997 −0.212999 0.977052i \(-0.568323\pi\)
−0.212999 + 0.977052i \(0.568323\pi\)
\(174\) 3.50057e7 0.503752
\(175\) −1.54572e8 −2.18021
\(176\) 7.94777e6 0.109888
\(177\) 5.67018e7 0.768582
\(178\) 9.44836e7 1.25570
\(179\) −6.83488e7 −0.890728 −0.445364 0.895350i \(-0.646926\pi\)
−0.445364 + 0.895350i \(0.646926\pi\)
\(180\) −4.47830e7 −0.572347
\(181\) −5.06098e7 −0.634395 −0.317197 0.948360i \(-0.602742\pi\)
−0.317197 + 0.948360i \(0.602742\pi\)
\(182\) −2.60030e8 −3.19723
\(183\) 2.13986e7 0.258110
\(184\) −3.28676e6 −0.0388960
\(185\) 1.69033e8 1.96277
\(186\) −8.14423e7 −0.928015
\(187\) −4.64217e6 −0.0519129
\(188\) −1.08130e8 −1.18685
\(189\) −2.95564e7 −0.318446
\(190\) −3.42570e8 −3.62336
\(191\) −1.11632e8 −1.15923 −0.579616 0.814890i \(-0.696797\pi\)
−0.579616 + 0.814890i \(0.696797\pi\)
\(192\) −7.00604e7 −0.714362
\(193\) 1.30760e8 1.30926 0.654630 0.755950i \(-0.272825\pi\)
0.654630 + 0.755950i \(0.272825\pi\)
\(194\) 1.15741e8 1.13810
\(195\) 1.20549e8 1.16424
\(196\) 2.06638e8 1.96026
\(197\) 1.10071e7 0.102575 0.0512877 0.998684i \(-0.483667\pi\)
0.0512877 + 0.998684i \(0.483667\pi\)
\(198\) 6.81985e6 0.0624376
\(199\) 1.33730e8 1.20294 0.601469 0.798896i \(-0.294582\pi\)
0.601469 + 0.798896i \(0.294582\pi\)
\(200\) −2.78071e7 −0.245782
\(201\) −6.52874e6 −0.0567079
\(202\) −3.15329e8 −2.69175
\(203\) 1.17966e8 0.989740
\(204\) 3.19218e7 0.263259
\(205\) −2.41347e8 −1.95661
\(206\) 3.29608e8 2.62701
\(207\) 8.86974e6 0.0695048
\(208\) 1.47116e8 1.13355
\(209\) 2.76520e7 0.209515
\(210\) −2.84719e8 −2.12153
\(211\) 2.22573e8 1.63111 0.815555 0.578679i \(-0.196432\pi\)
0.815555 + 0.578679i \(0.196432\pi\)
\(212\) 5.45605e7 0.393281
\(213\) −1.27513e8 −0.904119
\(214\) 3.10473e8 2.16559
\(215\) −3.19677e8 −2.19370
\(216\) −5.31711e6 −0.0358994
\(217\) −2.74454e8 −1.82331
\(218\) 2.88352e7 0.188506
\(219\) −6.31582e7 −0.406327
\(220\) 3.48221e7 0.220483
\(221\) −8.59284e7 −0.535505
\(222\) 1.77011e8 1.08584
\(223\) −2.43448e7 −0.147007 −0.0735036 0.997295i \(-0.523418\pi\)
−0.0735036 + 0.997295i \(0.523418\pi\)
\(224\) −3.99391e8 −2.37427
\(225\) 7.50410e7 0.439197
\(226\) 5.87648e7 0.338640
\(227\) −2.94104e8 −1.66882 −0.834411 0.551143i \(-0.814192\pi\)
−0.834411 + 0.551143i \(0.814192\pi\)
\(228\) −1.90149e8 −1.06248
\(229\) 5.37734e7 0.295899 0.147949 0.988995i \(-0.452733\pi\)
0.147949 + 0.988995i \(0.452733\pi\)
\(230\) 8.54428e7 0.463050
\(231\) 2.29823e7 0.122674
\(232\) 2.12217e7 0.111577
\(233\) 2.87694e8 1.49000 0.744999 0.667065i \(-0.232450\pi\)
0.744999 + 0.667065i \(0.232450\pi\)
\(234\) 1.26238e8 0.644073
\(235\) 3.18705e8 1.60196
\(236\) 3.03183e8 1.50146
\(237\) 2.14598e7 0.104714
\(238\) 2.02951e8 0.975825
\(239\) 2.96078e8 1.40286 0.701428 0.712740i \(-0.252546\pi\)
0.701428 + 0.712740i \(0.252546\pi\)
\(240\) 1.61085e8 0.752167
\(241\) −1.39493e8 −0.641937 −0.320968 0.947090i \(-0.604008\pi\)
−0.320968 + 0.947090i \(0.604008\pi\)
\(242\) 3.16305e8 1.43467
\(243\) 1.43489e7 0.0641500
\(244\) 1.14418e8 0.504230
\(245\) −6.09049e8 −2.64588
\(246\) −2.52738e8 −1.08243
\(247\) 5.11850e8 2.16124
\(248\) −4.93733e7 −0.205547
\(249\) 1.41073e8 0.579089
\(250\) 1.74241e8 0.705280
\(251\) 2.67558e8 1.06797 0.533986 0.845493i \(-0.320694\pi\)
0.533986 + 0.845493i \(0.320694\pi\)
\(252\) −1.58038e8 −0.622097
\(253\) −6.89687e6 −0.0267751
\(254\) 6.77528e7 0.259423
\(255\) −9.40870e7 −0.355336
\(256\) 1.87246e8 0.697544
\(257\) −2.10540e8 −0.773691 −0.386846 0.922144i \(-0.626435\pi\)
−0.386846 + 0.922144i \(0.626435\pi\)
\(258\) −3.34765e8 −1.21359
\(259\) 5.96511e8 2.13338
\(260\) 6.44570e8 2.27438
\(261\) −5.72696e7 −0.199380
\(262\) 4.29049e7 0.147385
\(263\) 2.37688e8 0.805680 0.402840 0.915271i \(-0.368023\pi\)
0.402840 + 0.915271i \(0.368023\pi\)
\(264\) 4.13444e6 0.0138294
\(265\) −1.60812e8 −0.530834
\(266\) −1.20892e9 −3.93832
\(267\) −1.54576e8 −0.496995
\(268\) −3.49090e7 −0.110781
\(269\) 1.97716e8 0.619310 0.309655 0.950849i \(-0.399786\pi\)
0.309655 + 0.950849i \(0.399786\pi\)
\(270\) 1.38224e8 0.427376
\(271\) 4.72382e8 1.44179 0.720893 0.693046i \(-0.243732\pi\)
0.720893 + 0.693046i \(0.243732\pi\)
\(272\) −1.14823e8 −0.345969
\(273\) 4.25411e8 1.26543
\(274\) 1.59486e8 0.468376
\(275\) −5.83499e7 −0.169190
\(276\) 4.74263e7 0.135781
\(277\) 1.22279e8 0.345680 0.172840 0.984950i \(-0.444706\pi\)
0.172840 + 0.984950i \(0.444706\pi\)
\(278\) −4.87159e8 −1.35992
\(279\) 1.33240e8 0.367300
\(280\) −1.72607e8 −0.469900
\(281\) 3.64898e8 0.981069 0.490534 0.871422i \(-0.336802\pi\)
0.490534 + 0.871422i \(0.336802\pi\)
\(282\) 3.33748e8 0.886230
\(283\) −7.69088e7 −0.201708 −0.100854 0.994901i \(-0.532158\pi\)
−0.100854 + 0.994901i \(0.532158\pi\)
\(284\) −6.81809e8 −1.76624
\(285\) 5.60448e8 1.43410
\(286\) −9.81594e7 −0.248114
\(287\) −8.51706e8 −2.12669
\(288\) 1.93894e8 0.478290
\(289\) −3.43272e8 −0.836559
\(290\) −5.51682e8 −1.32830
\(291\) −1.89353e8 −0.450451
\(292\) −3.37706e8 −0.793777
\(293\) −2.84862e8 −0.661603 −0.330802 0.943700i \(-0.607319\pi\)
−0.330802 + 0.943700i \(0.607319\pi\)
\(294\) −6.37794e8 −1.46374
\(295\) −8.93607e8 −2.02661
\(296\) 1.07310e8 0.240503
\(297\) −1.11573e7 −0.0247123
\(298\) −6.46938e8 −1.41614
\(299\) −1.27664e8 −0.276197
\(300\) 4.01242e8 0.857990
\(301\) −1.12813e9 −2.38438
\(302\) 1.00790e9 2.10569
\(303\) 5.15881e8 1.06537
\(304\) 6.83966e8 1.39629
\(305\) −3.37236e8 −0.680589
\(306\) −9.85276e7 −0.196577
\(307\) −6.24159e8 −1.23115 −0.615575 0.788078i \(-0.711076\pi\)
−0.615575 + 0.788078i \(0.711076\pi\)
\(308\) 1.22886e8 0.239648
\(309\) −5.39241e8 −1.03975
\(310\) 1.28351e9 2.44700
\(311\) −4.84237e8 −0.912845 −0.456422 0.889763i \(-0.650869\pi\)
−0.456422 + 0.889763i \(0.650869\pi\)
\(312\) 7.65301e7 0.142656
\(313\) −4.94156e8 −0.910874 −0.455437 0.890268i \(-0.650517\pi\)
−0.455437 + 0.890268i \(0.650517\pi\)
\(314\) −9.61826e8 −1.75325
\(315\) 4.65802e8 0.839682
\(316\) 1.14745e8 0.204564
\(317\) 8.96816e8 1.58123 0.790616 0.612312i \(-0.209760\pi\)
0.790616 + 0.612312i \(0.209760\pi\)
\(318\) −1.68402e8 −0.293666
\(319\) 4.45313e7 0.0768065
\(320\) 1.10414e9 1.88364
\(321\) −5.07937e8 −0.857121
\(322\) 3.01524e8 0.503300
\(323\) −3.99494e8 −0.659631
\(324\) 7.67233e7 0.125320
\(325\) −1.08008e9 −1.74527
\(326\) −1.94226e8 −0.310489
\(327\) −4.71746e7 −0.0746090
\(328\) −1.53219e8 −0.239748
\(329\) 1.12470e9 1.74121
\(330\) −1.07479e8 −0.164636
\(331\) 8.34881e8 1.26540 0.632699 0.774398i \(-0.281947\pi\)
0.632699 + 0.774398i \(0.281947\pi\)
\(332\) 7.54313e8 1.13127
\(333\) −2.89591e8 −0.429764
\(334\) −3.89616e8 −0.572169
\(335\) 1.02891e8 0.149528
\(336\) 5.68461e8 0.817548
\(337\) −1.13042e9 −1.60892 −0.804460 0.594007i \(-0.797545\pi\)
−0.804460 + 0.594007i \(0.797545\pi\)
\(338\) −7.81394e8 −1.10068
\(339\) −9.61397e7 −0.134031
\(340\) −5.03081e8 −0.694163
\(341\) −1.03604e8 −0.141494
\(342\) 5.86899e8 0.793364
\(343\) −9.12660e8 −1.22118
\(344\) −2.02947e8 −0.268799
\(345\) −1.39785e8 −0.183271
\(346\) 4.78792e8 0.621413
\(347\) 5.09953e8 0.655204 0.327602 0.944816i \(-0.393759\pi\)
0.327602 + 0.944816i \(0.393759\pi\)
\(348\) −3.06219e8 −0.389498
\(349\) −6.69343e8 −0.842868 −0.421434 0.906859i \(-0.638473\pi\)
−0.421434 + 0.906859i \(0.638473\pi\)
\(350\) 2.55100e9 3.18033
\(351\) −2.06527e8 −0.254918
\(352\) −1.50767e8 −0.184250
\(353\) −5.36062e8 −0.648639 −0.324320 0.945948i \(-0.605135\pi\)
−0.324320 + 0.945948i \(0.605135\pi\)
\(354\) −9.35783e8 −1.12115
\(355\) 2.00957e9 2.38399
\(356\) −8.26514e8 −0.970901
\(357\) −3.32029e8 −0.386223
\(358\) 1.12800e9 1.29933
\(359\) 6.72522e8 0.767142 0.383571 0.923511i \(-0.374694\pi\)
0.383571 + 0.923511i \(0.374694\pi\)
\(360\) 8.37964e7 0.0946600
\(361\) 1.48579e9 1.66220
\(362\) 8.35243e8 0.925408
\(363\) −5.17478e8 −0.567830
\(364\) 2.27466e9 2.47208
\(365\) 9.95359e8 1.07141
\(366\) −3.53153e8 −0.376512
\(367\) −1.82244e9 −1.92452 −0.962261 0.272128i \(-0.912272\pi\)
−0.962261 + 0.272128i \(0.912272\pi\)
\(368\) −1.70593e8 −0.178440
\(369\) 4.13482e8 0.428415
\(370\) −2.78965e9 −2.86315
\(371\) −5.67501e8 −0.576976
\(372\) 7.12433e8 0.717536
\(373\) −8.88447e8 −0.886442 −0.443221 0.896412i \(-0.646164\pi\)
−0.443221 + 0.896412i \(0.646164\pi\)
\(374\) 7.66124e7 0.0757266
\(375\) −2.85060e8 −0.279143
\(376\) 2.02330e8 0.196292
\(377\) 8.24292e8 0.792294
\(378\) 4.87787e8 0.464525
\(379\) 1.36718e9 1.29000 0.645000 0.764183i \(-0.276858\pi\)
0.645000 + 0.764183i \(0.276858\pi\)
\(380\) 2.99670e9 2.80157
\(381\) −1.10844e8 −0.102678
\(382\) 1.84232e9 1.69100
\(383\) −5.13649e7 −0.0467165 −0.0233583 0.999727i \(-0.507436\pi\)
−0.0233583 + 0.999727i \(0.507436\pi\)
\(384\) 2.37046e8 0.213635
\(385\) −3.62195e8 −0.323467
\(386\) −2.15802e9 −1.90985
\(387\) 5.47678e8 0.480327
\(388\) −1.01247e9 −0.879975
\(389\) 1.19577e9 1.02997 0.514985 0.857199i \(-0.327798\pi\)
0.514985 + 0.857199i \(0.327798\pi\)
\(390\) −1.98948e9 −1.69830
\(391\) 9.96404e7 0.0842979
\(392\) −3.86654e8 −0.324206
\(393\) −7.01927e7 −0.0583335
\(394\) −1.81657e8 −0.149629
\(395\) −3.38201e8 −0.276112
\(396\) −5.96580e7 −0.0482764
\(397\) −3.11811e8 −0.250106 −0.125053 0.992150i \(-0.539910\pi\)
−0.125053 + 0.992150i \(0.539910\pi\)
\(398\) −2.20703e9 −1.75476
\(399\) 1.97780e9 1.55875
\(400\) −1.44327e9 −1.12755
\(401\) 2.16496e8 0.167665 0.0838327 0.996480i \(-0.473284\pi\)
0.0838327 + 0.996480i \(0.473284\pi\)
\(402\) 1.07748e8 0.0827212
\(403\) −1.91775e9 −1.45957
\(404\) 2.75840e9 2.08124
\(405\) −2.26135e8 −0.169152
\(406\) −1.94686e9 −1.44376
\(407\) 2.25178e8 0.165556
\(408\) −5.97310e7 −0.0435401
\(409\) −1.11135e9 −0.803195 −0.401598 0.915816i \(-0.631545\pi\)
−0.401598 + 0.915816i \(0.631545\pi\)
\(410\) 3.98310e9 2.85416
\(411\) −2.60920e8 −0.185379
\(412\) −2.88331e9 −2.03119
\(413\) −3.15351e9 −2.20277
\(414\) −1.46383e8 −0.101388
\(415\) −2.22327e9 −1.52695
\(416\) −2.79076e9 −1.90062
\(417\) 7.96996e8 0.538245
\(418\) −4.56357e8 −0.305624
\(419\) 1.46901e9 0.975606 0.487803 0.872954i \(-0.337798\pi\)
0.487803 + 0.872954i \(0.337798\pi\)
\(420\) 2.49063e9 1.64035
\(421\) 3.62126e8 0.236522 0.118261 0.992983i \(-0.462268\pi\)
0.118261 + 0.992983i \(0.462268\pi\)
\(422\) −3.67325e9 −2.37934
\(423\) −5.46014e8 −0.350762
\(424\) −1.02092e8 −0.0650444
\(425\) 8.42991e8 0.532674
\(426\) 2.10442e9 1.31886
\(427\) −1.19009e9 −0.739748
\(428\) −2.71593e9 −1.67442
\(429\) 1.60590e8 0.0982012
\(430\) 5.27582e9 3.20000
\(431\) 1.70696e9 1.02696 0.513480 0.858102i \(-0.328356\pi\)
0.513480 + 0.858102i \(0.328356\pi\)
\(432\) −2.75974e8 −0.164693
\(433\) 1.34325e9 0.795151 0.397576 0.917569i \(-0.369852\pi\)
0.397576 + 0.917569i \(0.369852\pi\)
\(434\) 4.52947e9 2.65970
\(435\) 9.02556e8 0.525728
\(436\) −2.52241e8 −0.145752
\(437\) −5.93528e8 −0.340217
\(438\) 1.04234e9 0.592719
\(439\) 1.17304e9 0.661739 0.330870 0.943677i \(-0.392658\pi\)
0.330870 + 0.943677i \(0.392658\pi\)
\(440\) −6.51578e7 −0.0364655
\(441\) 1.04344e9 0.579337
\(442\) 1.41813e9 0.781155
\(443\) 2.77494e9 1.51649 0.758247 0.651967i \(-0.226056\pi\)
0.758247 + 0.651967i \(0.226056\pi\)
\(444\) −1.54844e9 −0.839563
\(445\) 2.43608e9 1.31048
\(446\) 4.01776e8 0.214443
\(447\) 1.05839e9 0.560495
\(448\) 3.89646e9 2.04737
\(449\) −7.73041e8 −0.403033 −0.201516 0.979485i \(-0.564587\pi\)
−0.201516 + 0.979485i \(0.564587\pi\)
\(450\) −1.23845e9 −0.640668
\(451\) −3.21513e8 −0.165037
\(452\) −5.14057e8 −0.261834
\(453\) −1.64894e9 −0.833414
\(454\) 4.85376e9 2.43435
\(455\) −6.70438e9 −3.33671
\(456\) 3.55800e8 0.175723
\(457\) −4.61578e8 −0.226224 −0.113112 0.993582i \(-0.536082\pi\)
−0.113112 + 0.993582i \(0.536082\pi\)
\(458\) −8.87454e8 −0.431635
\(459\) 1.61192e8 0.0778035
\(460\) −7.47428e8 −0.358028
\(461\) −1.15391e9 −0.548555 −0.274277 0.961651i \(-0.588439\pi\)
−0.274277 + 0.961651i \(0.588439\pi\)
\(462\) −3.79290e8 −0.178947
\(463\) 3.46111e9 1.62062 0.810312 0.585999i \(-0.199298\pi\)
0.810312 + 0.585999i \(0.199298\pi\)
\(464\) 1.10147e9 0.511871
\(465\) −2.09984e9 −0.968501
\(466\) −4.74799e9 −2.17350
\(467\) 1.74439e9 0.792566 0.396283 0.918129i \(-0.370300\pi\)
0.396283 + 0.918129i \(0.370300\pi\)
\(468\) −1.10429e9 −0.497994
\(469\) 3.63100e8 0.162525
\(470\) −5.25978e9 −2.33682
\(471\) 1.57355e9 0.693919
\(472\) −5.67306e8 −0.248325
\(473\) −4.25860e8 −0.185035
\(474\) −3.54163e8 −0.152749
\(475\) −5.02145e9 −2.14982
\(476\) −1.77535e9 −0.754502
\(477\) 2.75508e8 0.116230
\(478\) −4.88635e9 −2.04638
\(479\) −8.46941e8 −0.352110 −0.176055 0.984380i \(-0.556334\pi\)
−0.176055 + 0.984380i \(0.556334\pi\)
\(480\) −3.05573e9 −1.26116
\(481\) 4.16814e9 1.70779
\(482\) 2.30213e9 0.936409
\(483\) −4.93296e8 −0.199202
\(484\) −2.76694e9 −1.10928
\(485\) 2.98417e9 1.18775
\(486\) −2.36808e8 −0.0935772
\(487\) −9.15119e8 −0.359026 −0.179513 0.983756i \(-0.557452\pi\)
−0.179513 + 0.983756i \(0.557452\pi\)
\(488\) −2.14094e8 −0.0833941
\(489\) 3.17756e8 0.122889
\(490\) 1.00515e10 3.85962
\(491\) 2.09615e9 0.799167 0.399583 0.916697i \(-0.369155\pi\)
0.399583 + 0.916697i \(0.369155\pi\)
\(492\) 2.21088e9 0.836926
\(493\) −6.43352e8 −0.241816
\(494\) −8.44736e9 −3.15266
\(495\) 1.75837e8 0.0651616
\(496\) −2.56262e9 −0.942972
\(497\) 7.09172e9 2.59122
\(498\) −2.32821e9 −0.844731
\(499\) 3.46322e9 1.24775 0.623875 0.781524i \(-0.285557\pi\)
0.623875 + 0.781524i \(0.285557\pi\)
\(500\) −1.52421e9 −0.545318
\(501\) 6.37415e8 0.226459
\(502\) −4.41566e9 −1.55788
\(503\) 3.31882e9 1.16277 0.581387 0.813627i \(-0.302510\pi\)
0.581387 + 0.813627i \(0.302510\pi\)
\(504\) 2.95714e8 0.102888
\(505\) −8.13016e9 −2.80918
\(506\) 1.13823e8 0.0390574
\(507\) 1.27837e9 0.435640
\(508\) −5.92681e8 −0.200585
\(509\) −2.58709e9 −0.869560 −0.434780 0.900537i \(-0.643174\pi\)
−0.434780 + 0.900537i \(0.643174\pi\)
\(510\) 1.55277e9 0.518337
\(511\) 3.51259e9 1.16454
\(512\) −4.21400e9 −1.38755
\(513\) −9.60172e8 −0.314006
\(514\) 3.47466e9 1.12860
\(515\) 8.49831e9 2.74162
\(516\) 2.92842e9 0.938339
\(517\) 4.24566e8 0.135123
\(518\) −9.84457e9 −3.11202
\(519\) −7.83307e8 −0.245950
\(520\) −1.20610e9 −0.376158
\(521\) −3.99114e9 −1.23642 −0.618208 0.786015i \(-0.712141\pi\)
−0.618208 + 0.786015i \(0.712141\pi\)
\(522\) 9.45154e8 0.290841
\(523\) −3.15844e9 −0.965422 −0.482711 0.875780i \(-0.660348\pi\)
−0.482711 + 0.875780i \(0.660348\pi\)
\(524\) −3.75319e8 −0.113957
\(525\) −4.17345e9 −1.25875
\(526\) −3.92271e9 −1.17526
\(527\) 1.49679e9 0.445475
\(528\) 2.14590e8 0.0634440
\(529\) 1.48036e8 0.0434783
\(530\) 2.65398e9 0.774341
\(531\) 1.53095e9 0.443741
\(532\) 1.05752e10 3.04509
\(533\) −5.95133e9 −1.70243
\(534\) 2.55106e9 0.724980
\(535\) 8.00496e9 2.26007
\(536\) 6.53206e7 0.0183220
\(537\) −1.84542e9 −0.514262
\(538\) −3.26302e9 −0.903402
\(539\) −8.11348e8 −0.223176
\(540\) −1.20914e9 −0.330445
\(541\) 1.45168e9 0.394168 0.197084 0.980387i \(-0.436853\pi\)
0.197084 + 0.980387i \(0.436853\pi\)
\(542\) −7.79600e9 −2.10317
\(543\) −1.36646e9 −0.366268
\(544\) 2.17816e9 0.580088
\(545\) 7.43460e8 0.196730
\(546\) −7.02081e9 −1.84592
\(547\) 5.34408e9 1.39610 0.698051 0.716048i \(-0.254051\pi\)
0.698051 + 0.716048i \(0.254051\pi\)
\(548\) −1.39513e9 −0.362146
\(549\) 5.77761e8 0.149020
\(550\) 9.62982e8 0.246802
\(551\) 3.83226e9 0.975942
\(552\) −8.87425e7 −0.0224566
\(553\) −1.19350e9 −0.300112
\(554\) −2.01805e9 −0.504252
\(555\) 4.56389e9 1.13321
\(556\) 4.26152e9 1.05148
\(557\) 5.00428e9 1.22701 0.613506 0.789690i \(-0.289759\pi\)
0.613506 + 0.789690i \(0.289759\pi\)
\(558\) −2.19894e9 −0.535790
\(559\) −7.88284e9 −1.90872
\(560\) −8.95882e9 −2.15572
\(561\) −1.25339e8 −0.0299719
\(562\) −6.02213e9 −1.43111
\(563\) 5.16557e9 1.21994 0.609971 0.792424i \(-0.291181\pi\)
0.609971 + 0.792424i \(0.291181\pi\)
\(564\) −2.91952e9 −0.685228
\(565\) 1.51514e9 0.353413
\(566\) 1.26927e9 0.294237
\(567\) −7.98023e8 −0.183855
\(568\) 1.27578e9 0.292116
\(569\) 2.72876e9 0.620974 0.310487 0.950578i \(-0.399508\pi\)
0.310487 + 0.950578i \(0.399508\pi\)
\(570\) −9.24940e9 −2.09195
\(571\) 8.49748e8 0.191013 0.0955066 0.995429i \(-0.469553\pi\)
0.0955066 + 0.995429i \(0.469553\pi\)
\(572\) 8.58668e8 0.191840
\(573\) −3.01405e9 −0.669283
\(574\) 1.40562e10 3.10225
\(575\) 1.25243e9 0.274737
\(576\) −1.89163e9 −0.412437
\(577\) −6.81404e9 −1.47669 −0.738345 0.674423i \(-0.764392\pi\)
−0.738345 + 0.674423i \(0.764392\pi\)
\(578\) 5.66523e9 1.22031
\(579\) 3.53053e9 0.755901
\(580\) 4.82594e9 1.02703
\(581\) −7.84585e9 −1.65968
\(582\) 3.12501e9 0.657084
\(583\) −2.14227e8 −0.0447749
\(584\) 6.31903e8 0.131282
\(585\) 3.25481e9 0.672172
\(586\) 4.70124e9 0.965097
\(587\) −2.05329e9 −0.419004 −0.209502 0.977808i \(-0.567184\pi\)
−0.209502 + 0.977808i \(0.567184\pi\)
\(588\) 5.57923e9 1.13176
\(589\) −8.91592e9 −1.79789
\(590\) 1.47477e10 2.95626
\(591\) 2.97193e8 0.0592219
\(592\) 5.56973e9 1.10334
\(593\) −3.64815e9 −0.718424 −0.359212 0.933256i \(-0.616954\pi\)
−0.359212 + 0.933256i \(0.616954\pi\)
\(594\) 1.84136e8 0.0360484
\(595\) 5.23270e9 1.01840
\(596\) 5.65921e9 1.09495
\(597\) 3.61071e9 0.694517
\(598\) 2.10691e9 0.402895
\(599\) −2.96538e9 −0.563750 −0.281875 0.959451i \(-0.590956\pi\)
−0.281875 + 0.959451i \(0.590956\pi\)
\(600\) −7.50791e8 −0.141902
\(601\) 6.94942e9 1.30583 0.652917 0.757429i \(-0.273545\pi\)
0.652917 + 0.757429i \(0.273545\pi\)
\(602\) 1.86182e10 3.47816
\(603\) −1.76276e8 −0.0327403
\(604\) −8.81683e9 −1.62811
\(605\) 8.15533e9 1.49726
\(606\) −8.51388e9 −1.55408
\(607\) −5.47955e9 −0.994454 −0.497227 0.867620i \(-0.665648\pi\)
−0.497227 + 0.867620i \(0.665648\pi\)
\(608\) −1.29746e10 −2.34117
\(609\) 3.18509e9 0.571427
\(610\) 5.56561e9 0.992792
\(611\) 7.85888e9 1.39385
\(612\) 8.61890e8 0.151992
\(613\) −5.21036e9 −0.913600 −0.456800 0.889570i \(-0.651004\pi\)
−0.456800 + 0.889570i \(0.651004\pi\)
\(614\) 1.03009e10 1.79591
\(615\) −6.51638e9 −1.12965
\(616\) −2.29940e8 −0.0396352
\(617\) −2.60433e8 −0.0446372 −0.0223186 0.999751i \(-0.507105\pi\)
−0.0223186 + 0.999751i \(0.507105\pi\)
\(618\) 8.89940e9 1.51671
\(619\) 2.85535e9 0.483885 0.241943 0.970291i \(-0.422215\pi\)
0.241943 + 0.970291i \(0.422215\pi\)
\(620\) −1.12278e10 −1.89201
\(621\) 2.39483e8 0.0401286
\(622\) 7.99165e9 1.33159
\(623\) 8.59684e9 1.42440
\(624\) 3.97214e9 0.654454
\(625\) −3.54946e9 −0.581543
\(626\) 8.15534e9 1.32871
\(627\) 7.46604e8 0.120963
\(628\) 8.41376e9 1.35560
\(629\) −3.25319e9 −0.521234
\(630\) −7.68741e9 −1.22486
\(631\) −1.01242e10 −1.60420 −0.802099 0.597191i \(-0.796283\pi\)
−0.802099 + 0.597191i \(0.796283\pi\)
\(632\) −2.14706e8 −0.0338326
\(633\) 6.00946e9 0.941722
\(634\) −1.48007e10 −2.30658
\(635\) 1.74688e9 0.270741
\(636\) 1.47313e9 0.227061
\(637\) −1.50184e10 −2.30216
\(638\) −7.34926e8 −0.112040
\(639\) −3.44285e9 −0.521994
\(640\) −3.73579e9 −0.563316
\(641\) −1.66619e9 −0.249874 −0.124937 0.992165i \(-0.539873\pi\)
−0.124937 + 0.992165i \(0.539873\pi\)
\(642\) 8.38278e9 1.25030
\(643\) −8.04387e9 −1.19324 −0.596619 0.802525i \(-0.703489\pi\)
−0.596619 + 0.802525i \(0.703489\pi\)
\(644\) −2.63764e9 −0.389149
\(645\) −8.63128e9 −1.26653
\(646\) 6.59308e9 0.962220
\(647\) −1.15798e9 −0.168088 −0.0840439 0.996462i \(-0.526784\pi\)
−0.0840439 + 0.996462i \(0.526784\pi\)
\(648\) −1.43562e8 −0.0207265
\(649\) −1.19042e9 −0.170941
\(650\) 1.78252e10 2.54588
\(651\) −7.41024e9 −1.05269
\(652\) 1.69903e9 0.240069
\(653\) 5.67105e9 0.797017 0.398508 0.917165i \(-0.369528\pi\)
0.398508 + 0.917165i \(0.369528\pi\)
\(654\) 7.78550e8 0.108834
\(655\) 1.10622e9 0.153815
\(656\) −7.95254e9 −1.09987
\(657\) −1.70527e9 −0.234593
\(658\) −1.85616e10 −2.53995
\(659\) 2.21730e9 0.301804 0.150902 0.988549i \(-0.451782\pi\)
0.150902 + 0.988549i \(0.451782\pi\)
\(660\) 9.40195e8 0.127296
\(661\) 2.38616e9 0.321362 0.160681 0.987006i \(-0.448631\pi\)
0.160681 + 0.987006i \(0.448631\pi\)
\(662\) −1.37785e10 −1.84587
\(663\) −2.32007e9 −0.309174
\(664\) −1.41144e9 −0.187101
\(665\) −3.11697e10 −4.11014
\(666\) 4.77929e9 0.626908
\(667\) −9.55829e8 −0.124721
\(668\) 3.40824e9 0.442398
\(669\) −6.57309e8 −0.0848747
\(670\) −1.69808e9 −0.218120
\(671\) −4.49251e8 −0.0574064
\(672\) −1.07836e10 −1.37079
\(673\) −7.59718e9 −0.960727 −0.480363 0.877070i \(-0.659495\pi\)
−0.480363 + 0.877070i \(0.659495\pi\)
\(674\) 1.86559e10 2.34697
\(675\) 2.02611e9 0.253571
\(676\) 6.83539e9 0.851040
\(677\) −7.61664e9 −0.943416 −0.471708 0.881755i \(-0.656362\pi\)
−0.471708 + 0.881755i \(0.656362\pi\)
\(678\) 1.58665e9 0.195514
\(679\) 1.05310e10 1.29100
\(680\) 9.41347e8 0.114807
\(681\) −7.94080e9 −0.963494
\(682\) 1.70984e9 0.206400
\(683\) −3.27264e9 −0.393030 −0.196515 0.980501i \(-0.562962\pi\)
−0.196515 + 0.980501i \(0.562962\pi\)
\(684\) −5.13402e9 −0.613424
\(685\) 4.11204e9 0.488810
\(686\) 1.50622e10 1.78136
\(687\) 1.45188e9 0.170837
\(688\) −1.05335e10 −1.23315
\(689\) −3.96543e9 −0.461874
\(690\) 2.30695e9 0.267342
\(691\) 6.63679e9 0.765218 0.382609 0.923910i \(-0.375026\pi\)
0.382609 + 0.923910i \(0.375026\pi\)
\(692\) −4.18832e9 −0.480473
\(693\) 6.20522e8 0.0708257
\(694\) −8.41605e9 −0.955763
\(695\) −1.25605e10 −1.41925
\(696\) 5.72987e8 0.0644187
\(697\) 4.64495e9 0.519597
\(698\) 1.10466e10 1.22951
\(699\) 7.76775e9 0.860251
\(700\) −2.23153e10 −2.45901
\(701\) −1.22424e10 −1.34231 −0.671155 0.741317i \(-0.734201\pi\)
−0.671155 + 0.741317i \(0.734201\pi\)
\(702\) 3.40843e9 0.371856
\(703\) 1.93783e10 2.10364
\(704\) 1.47088e9 0.158882
\(705\) 8.60505e9 0.924893
\(706\) 8.84694e9 0.946186
\(707\) −2.86910e10 −3.05336
\(708\) 8.18595e9 0.866867
\(709\) 1.58035e10 1.66530 0.832648 0.553802i \(-0.186824\pi\)
0.832648 + 0.553802i \(0.186824\pi\)
\(710\) −3.31652e10 −3.47759
\(711\) 5.79413e8 0.0604568
\(712\) 1.54654e9 0.160577
\(713\) 2.22378e9 0.229762
\(714\) 5.47968e9 0.563393
\(715\) −2.53085e9 −0.258938
\(716\) −9.86740e9 −1.00463
\(717\) 7.99410e9 0.809940
\(718\) −1.10990e10 −1.11905
\(719\) 4.98945e9 0.500612 0.250306 0.968167i \(-0.419469\pi\)
0.250306 + 0.968167i \(0.419469\pi\)
\(720\) 4.34928e9 0.434264
\(721\) 2.99902e10 2.97993
\(722\) −2.45209e10 −2.42469
\(723\) −3.76631e9 −0.370622
\(724\) −7.30646e9 −0.715520
\(725\) −8.08663e9 −0.788106
\(726\) 8.54024e9 0.828308
\(727\) −6.01676e9 −0.580754 −0.290377 0.956912i \(-0.593781\pi\)
−0.290377 + 0.956912i \(0.593781\pi\)
\(728\) −4.25627e9 −0.408855
\(729\) 3.87420e8 0.0370370
\(730\) −1.64270e10 −1.56289
\(731\) 6.15248e9 0.582558
\(732\) 3.08927e9 0.291117
\(733\) 4.02201e9 0.377207 0.188603 0.982053i \(-0.439604\pi\)
0.188603 + 0.982053i \(0.439604\pi\)
\(734\) 3.00769e10 2.80735
\(735\) −1.64443e10 −1.52760
\(736\) 3.23609e9 0.299191
\(737\) 1.37068e8 0.0126124
\(738\) −6.82394e9 −0.624939
\(739\) −6.62949e9 −0.604261 −0.302130 0.953267i \(-0.597698\pi\)
−0.302130 + 0.953267i \(0.597698\pi\)
\(740\) 2.44030e10 2.21377
\(741\) 1.38199e10 1.24779
\(742\) 9.36580e9 0.841650
\(743\) −1.07995e10 −0.965923 −0.482961 0.875642i \(-0.660439\pi\)
−0.482961 + 0.875642i \(0.660439\pi\)
\(744\) −1.33308e9 −0.118673
\(745\) −1.66801e10 −1.47792
\(746\) 1.46626e10 1.29308
\(747\) 3.80896e9 0.334337
\(748\) −6.70182e8 −0.0585514
\(749\) 2.82492e10 2.45652
\(750\) 4.70452e9 0.407193
\(751\) −4.44278e9 −0.382750 −0.191375 0.981517i \(-0.561295\pi\)
−0.191375 + 0.981517i \(0.561295\pi\)
\(752\) 1.05015e10 0.900513
\(753\) 7.22406e9 0.616594
\(754\) −1.36038e10 −1.15574
\(755\) 2.59869e10 2.19756
\(756\) −4.26701e9 −0.359168
\(757\) 8.62219e9 0.722407 0.361204 0.932487i \(-0.382366\pi\)
0.361204 + 0.932487i \(0.382366\pi\)
\(758\) −2.25634e10 −1.88175
\(759\) −1.86216e8 −0.0154586
\(760\) −5.60732e9 −0.463349
\(761\) 1.61329e10 1.32699 0.663495 0.748181i \(-0.269073\pi\)
0.663495 + 0.748181i \(0.269073\pi\)
\(762\) 1.82933e9 0.149778
\(763\) 2.62364e9 0.213830
\(764\) −1.61161e10 −1.30747
\(765\) −2.54035e9 −0.205153
\(766\) 8.47705e8 0.0681465
\(767\) −2.20352e10 −1.76333
\(768\) 5.05563e9 0.402727
\(769\) 9.08946e9 0.720769 0.360384 0.932804i \(-0.382646\pi\)
0.360384 + 0.932804i \(0.382646\pi\)
\(770\) 5.97752e9 0.471850
\(771\) −5.68457e9 −0.446691
\(772\) 1.88777e10 1.47669
\(773\) −3.76432e9 −0.293129 −0.146564 0.989201i \(-0.546822\pi\)
−0.146564 + 0.989201i \(0.546822\pi\)
\(774\) −9.03866e9 −0.700665
\(775\) 1.88139e10 1.45185
\(776\) 1.89450e9 0.145538
\(777\) 1.61058e10 1.23171
\(778\) −1.97345e10 −1.50244
\(779\) −2.76686e10 −2.09704
\(780\) 1.74034e10 1.31312
\(781\) 2.67707e9 0.201086
\(782\) −1.64442e9 −0.122968
\(783\) −1.54628e9 −0.115112
\(784\) −2.00685e10 −1.48734
\(785\) −2.47989e10 −1.82973
\(786\) 1.15843e9 0.0850926
\(787\) −9.80851e9 −0.717284 −0.358642 0.933475i \(-0.616760\pi\)
−0.358642 + 0.933475i \(0.616760\pi\)
\(788\) 1.58908e9 0.115692
\(789\) 6.41758e9 0.465159
\(790\) 5.58152e9 0.402771
\(791\) 5.34687e9 0.384133
\(792\) 1.11630e8 0.00798440
\(793\) −8.31582e9 −0.592174
\(794\) 5.14599e9 0.364836
\(795\) −4.34194e9 −0.306477
\(796\) 1.93064e10 1.35677
\(797\) 1.37110e10 0.959325 0.479662 0.877453i \(-0.340759\pi\)
0.479662 + 0.877453i \(0.340759\pi\)
\(798\) −3.26408e10 −2.27379
\(799\) −6.13378e9 −0.425417
\(800\) 2.73784e10 1.89057
\(801\) −4.17355e9 −0.286940
\(802\) −3.57295e9 −0.244578
\(803\) 1.32597e9 0.0903713
\(804\) −9.42544e8 −0.0639596
\(805\) 7.77423e9 0.525257
\(806\) 3.16498e10 2.12911
\(807\) 5.33832e9 0.357559
\(808\) −5.16143e9 −0.344215
\(809\) −4.75415e9 −0.315685 −0.157842 0.987464i \(-0.550454\pi\)
−0.157842 + 0.987464i \(0.550454\pi\)
\(810\) 3.73205e9 0.246746
\(811\) 9.06766e9 0.596929 0.298464 0.954421i \(-0.403526\pi\)
0.298464 + 0.954421i \(0.403526\pi\)
\(812\) 1.70306e10 1.11631
\(813\) 1.27543e10 0.832416
\(814\) −3.71625e9 −0.241501
\(815\) −5.00776e9 −0.324035
\(816\) −3.10022e9 −0.199745
\(817\) −3.66485e10 −2.35114
\(818\) 1.83413e10 1.17164
\(819\) 1.14861e10 0.730599
\(820\) −3.48429e10 −2.20682
\(821\) 4.16986e9 0.262979 0.131489 0.991318i \(-0.458024\pi\)
0.131489 + 0.991318i \(0.458024\pi\)
\(822\) 4.30611e9 0.270417
\(823\) 1.16734e10 0.729959 0.364979 0.931016i \(-0.381076\pi\)
0.364979 + 0.931016i \(0.381076\pi\)
\(824\) 5.39514e9 0.335937
\(825\) −1.57545e9 −0.0976820
\(826\) 5.20442e10 3.21323
\(827\) 1.43823e10 0.884220 0.442110 0.896961i \(-0.354230\pi\)
0.442110 + 0.896961i \(0.354230\pi\)
\(828\) 1.28051e9 0.0783929
\(829\) −2.98191e10 −1.81783 −0.908916 0.416980i \(-0.863089\pi\)
−0.908916 + 0.416980i \(0.863089\pi\)
\(830\) 3.66920e10 2.22740
\(831\) 3.30155e9 0.199578
\(832\) 2.72266e10 1.63894
\(833\) 1.17217e10 0.702640
\(834\) −1.31533e10 −0.785152
\(835\) −1.00455e10 −0.597131
\(836\) 3.99208e9 0.236307
\(837\) 3.59749e9 0.212061
\(838\) −2.42439e10 −1.42314
\(839\) −4.47833e9 −0.261788 −0.130894 0.991396i \(-0.541785\pi\)
−0.130894 + 0.991396i \(0.541785\pi\)
\(840\) −4.66039e9 −0.271297
\(841\) −1.10783e10 −0.642227
\(842\) −5.97638e9 −0.345021
\(843\) 9.85225e9 0.566420
\(844\) 3.21325e10 1.83969
\(845\) −2.01468e10 −1.14870
\(846\) 9.01118e9 0.511665
\(847\) 2.87799e10 1.62741
\(848\) −5.29886e9 −0.298399
\(849\) −2.07654e9 −0.116456
\(850\) −1.39124e10 −0.777025
\(851\) −4.83327e9 −0.268836
\(852\) −1.84088e10 −1.01974
\(853\) −9.39831e9 −0.518476 −0.259238 0.965814i \(-0.583471\pi\)
−0.259238 + 0.965814i \(0.583471\pi\)
\(854\) 1.96408e10 1.07909
\(855\) 1.51321e10 0.827975
\(856\) 5.08194e9 0.276931
\(857\) −1.07879e10 −0.585470 −0.292735 0.956194i \(-0.594565\pi\)
−0.292735 + 0.956194i \(0.594565\pi\)
\(858\) −2.65030e9 −0.143248
\(859\) −1.88319e10 −1.01372 −0.506860 0.862028i \(-0.669194\pi\)
−0.506860 + 0.862028i \(0.669194\pi\)
\(860\) −4.61513e10 −2.47422
\(861\) −2.29961e10 −1.22784
\(862\) −2.81710e10 −1.49805
\(863\) 2.24758e10 1.19036 0.595179 0.803593i \(-0.297081\pi\)
0.595179 + 0.803593i \(0.297081\pi\)
\(864\) 5.23515e9 0.276141
\(865\) 1.23447e10 0.648523
\(866\) −2.21685e10 −1.15991
\(867\) −9.26835e9 −0.482987
\(868\) −3.96224e10 −2.05647
\(869\) −4.50536e8 −0.0232895
\(870\) −1.48954e10 −0.766893
\(871\) 2.53717e9 0.130103
\(872\) 4.71985e8 0.0241058
\(873\) −5.11254e9 −0.260068
\(874\) 9.79534e9 0.496283
\(875\) 1.58538e10 0.800028
\(876\) −9.11805e9 −0.458287
\(877\) 3.68411e10 1.84431 0.922155 0.386821i \(-0.126427\pi\)
0.922155 + 0.386821i \(0.126427\pi\)
\(878\) −1.93594e10 −0.965295
\(879\) −7.69127e9 −0.381977
\(880\) −3.38189e9 −0.167290
\(881\) −2.17761e10 −1.07291 −0.536457 0.843927i \(-0.680238\pi\)
−0.536457 + 0.843927i \(0.680238\pi\)
\(882\) −1.72204e10 −0.845093
\(883\) 6.13071e9 0.299673 0.149837 0.988711i \(-0.452125\pi\)
0.149837 + 0.988711i \(0.452125\pi\)
\(884\) −1.24053e10 −0.603985
\(885\) −2.41274e10 −1.17006
\(886\) −4.57965e10 −2.21215
\(887\) 8.18008e9 0.393572 0.196786 0.980446i \(-0.436950\pi\)
0.196786 + 0.980446i \(0.436950\pi\)
\(888\) 2.89738e9 0.138855
\(889\) 6.16467e9 0.294275
\(890\) −4.02041e10 −1.91163
\(891\) −3.01248e8 −0.0142676
\(892\) −3.51462e9 −0.165806
\(893\) 3.65371e10 1.71693
\(894\) −1.74673e10 −0.817607
\(895\) 2.90833e10 1.35601
\(896\) −1.31835e10 −0.612282
\(897\) −3.44693e9 −0.159462
\(898\) 1.27580e10 0.587914
\(899\) −1.43584e10 −0.659092
\(900\) 1.08335e10 0.495361
\(901\) 3.09498e9 0.140968
\(902\) 5.30611e9 0.240743
\(903\) −3.04595e10 −1.37662
\(904\) 9.61885e8 0.0433046
\(905\) 2.15352e10 0.965780
\(906\) 2.72134e10 1.21572
\(907\) −4.94515e9 −0.220067 −0.110033 0.993928i \(-0.535096\pi\)
−0.110033 + 0.993928i \(0.535096\pi\)
\(908\) −4.24593e10 −1.88223
\(909\) 1.39288e10 0.615091
\(910\) 1.10646e11 4.86735
\(911\) 2.27032e10 0.994883 0.497442 0.867497i \(-0.334273\pi\)
0.497442 + 0.867497i \(0.334273\pi\)
\(912\) 1.84671e10 0.806151
\(913\) −2.96175e9 −0.128795
\(914\) 7.61770e9 0.329998
\(915\) −9.10538e9 −0.392938
\(916\) 7.76318e9 0.333738
\(917\) 3.90381e9 0.167185
\(918\) −2.66025e9 −0.113494
\(919\) 4.49844e10 1.91187 0.955933 0.293586i \(-0.0948486\pi\)
0.955933 + 0.293586i \(0.0948486\pi\)
\(920\) 1.39856e9 0.0592139
\(921\) −1.68523e10 −0.710805
\(922\) 1.90437e10 0.800190
\(923\) 4.95536e10 2.07429
\(924\) 3.31792e9 0.138361
\(925\) −4.08911e10 −1.69876
\(926\) −5.71208e10 −2.36404
\(927\) −1.45595e10 −0.600299
\(928\) −2.08946e10 −0.858255
\(929\) −3.90782e10 −1.59911 −0.799557 0.600590i \(-0.794932\pi\)
−0.799557 + 0.600590i \(0.794932\pi\)
\(930\) 3.46548e10 1.41278
\(931\) −6.98226e10 −2.83578
\(932\) 4.15340e10 1.68054
\(933\) −1.30744e10 −0.527031
\(934\) −2.87887e10 −1.15614
\(935\) 1.97531e9 0.0790303
\(936\) 2.06631e9 0.0823628
\(937\) 3.64378e10 1.44699 0.723493 0.690332i \(-0.242536\pi\)
0.723493 + 0.690332i \(0.242536\pi\)
\(938\) −5.99245e9 −0.237080
\(939\) −1.33422e10 −0.525894
\(940\) 4.60110e10 1.80682
\(941\) −2.02949e10 −0.794004 −0.397002 0.917818i \(-0.629949\pi\)
−0.397002 + 0.917818i \(0.629949\pi\)
\(942\) −2.59693e10 −1.01224
\(943\) 6.90101e9 0.267992
\(944\) −2.94449e10 −1.13922
\(945\) 1.25767e10 0.484790
\(946\) 7.02822e9 0.269914
\(947\) −3.16533e10 −1.21114 −0.605570 0.795792i \(-0.707055\pi\)
−0.605570 + 0.795792i \(0.707055\pi\)
\(948\) 3.09811e9 0.118105
\(949\) 2.45443e10 0.932222
\(950\) 8.28719e10 3.13599
\(951\) 2.42140e10 0.912925
\(952\) 3.32198e9 0.124787
\(953\) 2.71017e10 1.01431 0.507155 0.861855i \(-0.330697\pi\)
0.507155 + 0.861855i \(0.330697\pi\)
\(954\) −4.54686e9 −0.169548
\(955\) 4.75007e10 1.76477
\(956\) 4.27443e10 1.58225
\(957\) 1.20235e9 0.0443443
\(958\) 1.39776e10 0.513632
\(959\) 1.45112e10 0.531299
\(960\) 2.98117e10 1.08752
\(961\) 5.89276e9 0.214184
\(962\) −6.87893e10 −2.49120
\(963\) −1.37143e10 −0.494859
\(964\) −2.01383e10 −0.724026
\(965\) −5.56403e10 −1.99317
\(966\) 8.14116e9 0.290580
\(967\) 2.22263e10 0.790450 0.395225 0.918584i \(-0.370667\pi\)
0.395225 + 0.918584i \(0.370667\pi\)
\(968\) 5.17741e9 0.183463
\(969\) −1.07863e10 −0.380838
\(970\) −4.92494e10 −1.73261
\(971\) 3.09159e10 1.08371 0.541857 0.840471i \(-0.317721\pi\)
0.541857 + 0.840471i \(0.317721\pi\)
\(972\) 2.07153e9 0.0723534
\(973\) −4.43255e10 −1.54262
\(974\) 1.51028e10 0.523721
\(975\) −2.91621e10 −1.00763
\(976\) −1.11121e10 −0.382581
\(977\) 1.86861e9 0.0641044 0.0320522 0.999486i \(-0.489796\pi\)
0.0320522 + 0.999486i \(0.489796\pi\)
\(978\) −5.24411e9 −0.179261
\(979\) 3.24524e9 0.110537
\(980\) −8.79274e10 −2.98423
\(981\) −1.27371e9 −0.0430755
\(982\) −3.45940e10 −1.16576
\(983\) 6.94471e9 0.233194 0.116597 0.993179i \(-0.462801\pi\)
0.116597 + 0.993179i \(0.462801\pi\)
\(984\) −4.13692e9 −0.138419
\(985\) −4.68369e9 −0.156157
\(986\) 1.06176e10 0.352743
\(987\) 3.03669e10 1.00529
\(988\) 7.38949e10 2.43762
\(989\) 9.14074e9 0.300465
\(990\) −2.90194e9 −0.0950528
\(991\) 1.61609e10 0.527483 0.263742 0.964593i \(-0.415043\pi\)
0.263742 + 0.964593i \(0.415043\pi\)
\(992\) 4.86123e10 1.58108
\(993\) 2.25418e10 0.730577
\(994\) −1.17039e11 −3.77987
\(995\) −5.69040e10 −1.83131
\(996\) 2.03664e10 0.653142
\(997\) 1.20763e10 0.385922 0.192961 0.981206i \(-0.438191\pi\)
0.192961 + 0.981206i \(0.438191\pi\)
\(998\) −5.71555e10 −1.82012
\(999\) −7.81896e9 −0.248125
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 69.8.a.d.1.1 8
3.2 odd 2 207.8.a.e.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.d.1.1 8 1.1 even 1 trivial
207.8.a.e.1.8 8 3.2 odd 2