Properties

Label 91.8.a.d.1.8
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $10$
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] = [91,8,Mod(1,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, 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("91.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.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: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-6.95530\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.95530 q^{2} -32.1557 q^{3} -79.6238 q^{4} -381.581 q^{5} -223.652 q^{6} -343.000 q^{7} -1444.09 q^{8} -1153.01 q^{9} +O(q^{10})\) \(q+6.95530 q^{2} -32.1557 q^{3} -79.6238 q^{4} -381.581 q^{5} -223.652 q^{6} -343.000 q^{7} -1444.09 q^{8} -1153.01 q^{9} -2654.01 q^{10} +1248.93 q^{11} +2560.36 q^{12} -2197.00 q^{13} -2385.67 q^{14} +12270.0 q^{15} +147.808 q^{16} -12454.1 q^{17} -8019.55 q^{18} +7192.86 q^{19} +30382.9 q^{20} +11029.4 q^{21} +8686.66 q^{22} +14380.5 q^{23} +46435.5 q^{24} +67478.9 q^{25} -15280.8 q^{26} +107400. q^{27} +27311.0 q^{28} -13777.5 q^{29} +85341.4 q^{30} -87132.7 q^{31} +185871. q^{32} -40160.1 q^{33} -86622.3 q^{34} +130882. q^{35} +91807.4 q^{36} -107500. q^{37} +50028.4 q^{38} +70646.0 q^{39} +551035. q^{40} -451487. q^{41} +76712.7 q^{42} +145439. q^{43} -99444.4 q^{44} +439968. q^{45} +100021. q^{46} +220745. q^{47} -4752.86 q^{48} +117649. q^{49} +469336. q^{50} +400471. q^{51} +174934. q^{52} -516722. q^{53} +747001. q^{54} -476567. q^{55} +495321. q^{56} -231291. q^{57} -95826.6 q^{58} +384664. q^{59} -976983. q^{60} -1.96831e6 q^{61} -606034. q^{62} +395484. q^{63} +1.27387e6 q^{64} +838333. q^{65} -279325. q^{66} -852766. q^{67} +991647. q^{68} -462415. q^{69} +910325. q^{70} +5.65372e6 q^{71} +1.66505e6 q^{72} -1.67822e6 q^{73} -747697. q^{74} -2.16983e6 q^{75} -572723. q^{76} -428382. q^{77} +491364. q^{78} +3.43630e6 q^{79} -56400.7 q^{80} -931888. q^{81} -3.14023e6 q^{82} -1.39897e6 q^{83} -878202. q^{84} +4.75226e6 q^{85} +1.01157e6 q^{86} +443024. q^{87} -1.80356e6 q^{88} +1.78133e6 q^{89} +3.06011e6 q^{90} +753571. q^{91} -1.14503e6 q^{92} +2.80181e6 q^{93} +1.53535e6 q^{94} -2.74466e6 q^{95} -5.97680e6 q^{96} -6.05505e6 q^{97} +818284. q^{98} -1.44003e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9} + 2548 q^{10} + 451 q^{11} - 16241 q^{12} - 21970 q^{13} + 1029 q^{14} + 27184 q^{15} + 11897 q^{16} - 8654 q^{17} + 159348 q^{18} + 10130 q^{19} - 82012 q^{20} + 34643 q^{21} - 57863 q^{22} - 52155 q^{23} - 49227 q^{24} + 47190 q^{25} + 6591 q^{26} - 155171 q^{27} - 123823 q^{28} + 520154 q^{29} + 1070236 q^{30} + 692605 q^{31} + 149835 q^{32} + 436053 q^{33} + 1059060 q^{34} - 77518 q^{35} + 2843742 q^{36} - 20511 q^{37} + 1905286 q^{38} + 221897 q^{39} + 636320 q^{40} + 355049 q^{41} - 379015 q^{42} + 1256772 q^{43} - 687913 q^{44} + 1259926 q^{45} + 4043075 q^{46} + 1260721 q^{47} + 1128551 q^{48} + 1176490 q^{49} + 609035 q^{50} + 1411976 q^{51} - 793117 q^{52} + 928854 q^{53} + 6642607 q^{54} + 3423196 q^{55} - 99813 q^{56} + 3014966 q^{57} + 1612588 q^{58} + 3144446 q^{59} + 7738848 q^{60} + 6322923 q^{61} + 6545331 q^{62} - 4200721 q^{63} - 6629943 q^{64} - 496522 q^{65} - 14343317 q^{66} + 3944507 q^{67} - 1787356 q^{68} - 148281 q^{69} - 873964 q^{70} + 6032248 q^{71} + 9760866 q^{72} + 1248533 q^{73} - 8263279 q^{74} + 1573413 q^{75} + 1788254 q^{76} - 154693 q^{77} - 2427685 q^{78} - 14947605 q^{79} - 9147616 q^{80} + 25716334 q^{81} - 6987095 q^{82} - 14177784 q^{83} + 5570663 q^{84} - 11788444 q^{85} + 8748840 q^{86} - 29484448 q^{87} - 15390723 q^{88} + 6734836 q^{89} + 5994972 q^{90} + 7535710 q^{91} - 24493215 q^{92} + 17307847 q^{93} - 22760149 q^{94} - 9329708 q^{95} - 36488483 q^{96} - 12365397 q^{97} - 352947 q^{98} - 43198042 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.95530 0.614767 0.307384 0.951586i \(-0.400547\pi\)
0.307384 + 0.951586i \(0.400547\pi\)
\(3\) −32.1557 −0.687595 −0.343798 0.939044i \(-0.611713\pi\)
−0.343798 + 0.939044i \(0.611713\pi\)
\(4\) −79.6238 −0.622061
\(5\) −381.581 −1.36518 −0.682592 0.730799i \(-0.739148\pi\)
−0.682592 + 0.730799i \(0.739148\pi\)
\(6\) −223.652 −0.422711
\(7\) −343.000 −0.377964
\(8\) −1444.09 −0.997190
\(9\) −1153.01 −0.527212
\(10\) −2654.01 −0.839271
\(11\) 1248.93 0.282919 0.141460 0.989944i \(-0.454820\pi\)
0.141460 + 0.989944i \(0.454820\pi\)
\(12\) 2560.36 0.427727
\(13\) −2197.00 −0.277350
\(14\) −2385.67 −0.232360
\(15\) 12270.0 0.938695
\(16\) 147.808 0.00902148
\(17\) −12454.1 −0.614812 −0.307406 0.951578i \(-0.599461\pi\)
−0.307406 + 0.951578i \(0.599461\pi\)
\(18\) −8019.55 −0.324113
\(19\) 7192.86 0.240582 0.120291 0.992739i \(-0.461617\pi\)
0.120291 + 0.992739i \(0.461617\pi\)
\(20\) 30382.9 0.849229
\(21\) 11029.4 0.259887
\(22\) 8686.66 0.173930
\(23\) 14380.5 0.246449 0.123225 0.992379i \(-0.460676\pi\)
0.123225 + 0.992379i \(0.460676\pi\)
\(24\) 46435.5 0.685663
\(25\) 67478.9 0.863730
\(26\) −15280.8 −0.170506
\(27\) 107400. 1.05010
\(28\) 27311.0 0.235117
\(29\) −13777.5 −0.104900 −0.0524502 0.998624i \(-0.516703\pi\)
−0.0524502 + 0.998624i \(0.516703\pi\)
\(30\) 85341.4 0.577079
\(31\) −87132.7 −0.525309 −0.262655 0.964890i \(-0.584598\pi\)
−0.262655 + 0.964890i \(0.584598\pi\)
\(32\) 185871. 1.00274
\(33\) −40160.1 −0.194534
\(34\) −86622.3 −0.377966
\(35\) 130882. 0.515991
\(36\) 91807.4 0.327958
\(37\) −107500. −0.348902 −0.174451 0.984666i \(-0.555815\pi\)
−0.174451 + 0.984666i \(0.555815\pi\)
\(38\) 50028.4 0.147902
\(39\) 70646.0 0.190705
\(40\) 551035. 1.36135
\(41\) −451487. −1.02306 −0.511531 0.859265i \(-0.670921\pi\)
−0.511531 + 0.859265i \(0.670921\pi\)
\(42\) 76712.7 0.159770
\(43\) 145439. 0.278960 0.139480 0.990225i \(-0.455457\pi\)
0.139480 + 0.990225i \(0.455457\pi\)
\(44\) −99444.4 −0.175993
\(45\) 439968. 0.719742
\(46\) 100021. 0.151509
\(47\) 220745. 0.310133 0.155067 0.987904i \(-0.450441\pi\)
0.155067 + 0.987904i \(0.450441\pi\)
\(48\) −4752.86 −0.00620313
\(49\) 117649. 0.142857
\(50\) 469336. 0.530993
\(51\) 400471. 0.422742
\(52\) 174934. 0.172529
\(53\) −516722. −0.476751 −0.238375 0.971173i \(-0.576615\pi\)
−0.238375 + 0.971173i \(0.576615\pi\)
\(54\) 747001. 0.645570
\(55\) −476567. −0.386237
\(56\) 495321. 0.376902
\(57\) −231291. −0.165423
\(58\) −95826.6 −0.0644893
\(59\) 384664. 0.243837 0.121919 0.992540i \(-0.461095\pi\)
0.121919 + 0.992540i \(0.461095\pi\)
\(60\) −976983. −0.583926
\(61\) −1.96831e6 −1.11030 −0.555148 0.831752i \(-0.687338\pi\)
−0.555148 + 0.831752i \(0.687338\pi\)
\(62\) −606034. −0.322943
\(63\) 395484. 0.199268
\(64\) 1.27387e6 0.607428
\(65\) 838333. 0.378634
\(66\) −279325. −0.119593
\(67\) −852766. −0.346392 −0.173196 0.984887i \(-0.555409\pi\)
−0.173196 + 0.984887i \(0.555409\pi\)
\(68\) 991647. 0.382451
\(69\) −462415. −0.169457
\(70\) 910325. 0.317215
\(71\) 5.65372e6 1.87469 0.937346 0.348401i \(-0.113275\pi\)
0.937346 + 0.348401i \(0.113275\pi\)
\(72\) 1.66505e6 0.525731
\(73\) −1.67822e6 −0.504917 −0.252458 0.967608i \(-0.581239\pi\)
−0.252458 + 0.967608i \(0.581239\pi\)
\(74\) −747697. −0.214494
\(75\) −2.16983e6 −0.593897
\(76\) −572723. −0.149657
\(77\) −428382. −0.106933
\(78\) 491364. 0.117239
\(79\) 3.43630e6 0.784145 0.392073 0.919934i \(-0.371758\pi\)
0.392073 + 0.919934i \(0.371758\pi\)
\(80\) −56400.7 −0.0123160
\(81\) −931888. −0.194835
\(82\) −3.14023e6 −0.628945
\(83\) −1.39897e6 −0.268555 −0.134278 0.990944i \(-0.542871\pi\)
−0.134278 + 0.990944i \(0.542871\pi\)
\(84\) −878202. −0.161665
\(85\) 4.75226e6 0.839333
\(86\) 1.01157e6 0.171496
\(87\) 443024. 0.0721291
\(88\) −1.80356e6 −0.282124
\(89\) 1.78133e6 0.267843 0.133921 0.990992i \(-0.457243\pi\)
0.133921 + 0.990992i \(0.457243\pi\)
\(90\) 3.06011e6 0.442474
\(91\) 753571. 0.104828
\(92\) −1.14503e6 −0.153306
\(93\) 2.80181e6 0.361200
\(94\) 1.53535e6 0.190660
\(95\) −2.74466e6 −0.328439
\(96\) −5.97680e6 −0.689477
\(97\) −6.05505e6 −0.673623 −0.336811 0.941572i \(-0.609348\pi\)
−0.336811 + 0.941572i \(0.609348\pi\)
\(98\) 818284. 0.0878239
\(99\) −1.44003e6 −0.149159
\(100\) −5.37293e6 −0.537293
\(101\) −1.29334e7 −1.24907 −0.624534 0.780997i \(-0.714711\pi\)
−0.624534 + 0.780997i \(0.714711\pi\)
\(102\) 2.78540e6 0.259888
\(103\) −1.19105e7 −1.07399 −0.536995 0.843586i \(-0.680440\pi\)
−0.536995 + 0.843586i \(0.680440\pi\)
\(104\) 3.17266e6 0.276571
\(105\) −4.20860e6 −0.354793
\(106\) −3.59396e6 −0.293091
\(107\) −1.83303e7 −1.44652 −0.723262 0.690573i \(-0.757358\pi\)
−0.723262 + 0.690573i \(0.757358\pi\)
\(108\) −8.55163e6 −0.653229
\(109\) −9.64748e6 −0.713545 −0.356772 0.934191i \(-0.616123\pi\)
−0.356772 + 0.934191i \(0.616123\pi\)
\(110\) −3.31466e6 −0.237446
\(111\) 3.45674e6 0.239904
\(112\) −50698.1 −0.00340980
\(113\) 2.31706e7 1.51065 0.755323 0.655352i \(-0.227480\pi\)
0.755323 + 0.655352i \(0.227480\pi\)
\(114\) −1.60870e6 −0.101697
\(115\) −5.48733e6 −0.336449
\(116\) 1.09702e6 0.0652545
\(117\) 2.53317e6 0.146222
\(118\) 2.67545e6 0.149903
\(119\) 4.27177e6 0.232377
\(120\) −1.77189e7 −0.936057
\(121\) −1.79274e7 −0.919957
\(122\) −1.36902e7 −0.682573
\(123\) 1.45179e7 0.703453
\(124\) 6.93784e6 0.326775
\(125\) 4.06236e6 0.186034
\(126\) 2.75071e6 0.122503
\(127\) 3.14779e7 1.36362 0.681808 0.731531i \(-0.261194\pi\)
0.681808 + 0.731531i \(0.261194\pi\)
\(128\) −1.49314e7 −0.629309
\(129\) −4.67670e6 −0.191812
\(130\) 5.83085e6 0.232772
\(131\) −1.65637e7 −0.643736 −0.321868 0.946785i \(-0.604311\pi\)
−0.321868 + 0.946785i \(0.604311\pi\)
\(132\) 3.19770e6 0.121012
\(133\) −2.46715e6 −0.0909316
\(134\) −5.93124e6 −0.212951
\(135\) −4.09819e7 −1.43359
\(136\) 1.79848e7 0.613085
\(137\) 9.39642e6 0.312205 0.156103 0.987741i \(-0.450107\pi\)
0.156103 + 0.987741i \(0.450107\pi\)
\(138\) −3.21624e6 −0.104177
\(139\) 6.18591e7 1.95367 0.976837 0.213986i \(-0.0686448\pi\)
0.976837 + 0.213986i \(0.0686448\pi\)
\(140\) −1.04213e7 −0.320978
\(141\) −7.09820e6 −0.213246
\(142\) 3.93233e7 1.15250
\(143\) −2.74389e6 −0.0784677
\(144\) −170425. −0.00475624
\(145\) 5.25723e6 0.143208
\(146\) −1.16725e7 −0.310406
\(147\) −3.78308e6 −0.0982279
\(148\) 8.55959e6 0.217039
\(149\) 2.15148e7 0.532827 0.266413 0.963859i \(-0.414161\pi\)
0.266413 + 0.963859i \(0.414161\pi\)
\(150\) −1.50918e7 −0.365108
\(151\) 3.93596e7 0.930318 0.465159 0.885227i \(-0.345997\pi\)
0.465159 + 0.885227i \(0.345997\pi\)
\(152\) −1.03871e7 −0.239906
\(153\) 1.43598e7 0.324137
\(154\) −2.97952e6 −0.0657392
\(155\) 3.32481e7 0.717144
\(156\) −5.62510e6 −0.118630
\(157\) 6.20466e7 1.27959 0.639793 0.768547i \(-0.279020\pi\)
0.639793 + 0.768547i \(0.279020\pi\)
\(158\) 2.39005e7 0.482067
\(159\) 1.66155e7 0.327812
\(160\) −7.09248e7 −1.36892
\(161\) −4.93252e6 −0.0931490
\(162\) −6.48156e6 −0.119778
\(163\) 5.61916e7 1.01628 0.508141 0.861274i \(-0.330333\pi\)
0.508141 + 0.861274i \(0.330333\pi\)
\(164\) 3.59491e7 0.636407
\(165\) 1.53243e7 0.265575
\(166\) −9.73023e6 −0.165099
\(167\) 2.13130e7 0.354109 0.177054 0.984201i \(-0.443343\pi\)
0.177054 + 0.984201i \(0.443343\pi\)
\(168\) −1.59274e7 −0.259156
\(169\) 4.82681e6 0.0769231
\(170\) 3.30534e7 0.515994
\(171\) −8.29346e6 −0.126838
\(172\) −1.15804e7 −0.173530
\(173\) −4.51220e7 −0.662562 −0.331281 0.943532i \(-0.607481\pi\)
−0.331281 + 0.943532i \(0.607481\pi\)
\(174\) 3.08137e6 0.0443426
\(175\) −2.31453e7 −0.326459
\(176\) 184601. 0.00255235
\(177\) −1.23691e7 −0.167661
\(178\) 1.23897e7 0.164661
\(179\) 8.49518e7 1.10710 0.553550 0.832816i \(-0.313273\pi\)
0.553550 + 0.832816i \(0.313273\pi\)
\(180\) −3.50319e7 −0.447724
\(181\) 6.90967e7 0.866128 0.433064 0.901363i \(-0.357432\pi\)
0.433064 + 0.901363i \(0.357432\pi\)
\(182\) 5.24131e6 0.0644451
\(183\) 6.32922e7 0.763434
\(184\) −2.07667e7 −0.245757
\(185\) 4.10201e7 0.476316
\(186\) 1.94874e7 0.222054
\(187\) −1.55543e7 −0.173942
\(188\) −1.75766e7 −0.192922
\(189\) −3.68383e7 −0.396902
\(190\) −1.90899e7 −0.201914
\(191\) 1.05054e7 0.109092 0.0545462 0.998511i \(-0.482629\pi\)
0.0545462 + 0.998511i \(0.482629\pi\)
\(192\) −4.09621e7 −0.417665
\(193\) −2.74699e7 −0.275047 −0.137523 0.990499i \(-0.543914\pi\)
−0.137523 + 0.990499i \(0.543914\pi\)
\(194\) −4.21147e7 −0.414121
\(195\) −2.69571e7 −0.260347
\(196\) −9.36767e6 −0.0888659
\(197\) 9.00501e7 0.839174 0.419587 0.907715i \(-0.362175\pi\)
0.419587 + 0.907715i \(0.362175\pi\)
\(198\) −1.00158e7 −0.0916978
\(199\) 1.82532e7 0.164193 0.0820964 0.996624i \(-0.473838\pi\)
0.0820964 + 0.996624i \(0.473838\pi\)
\(200\) −9.74453e7 −0.861303
\(201\) 2.74213e7 0.238178
\(202\) −8.99553e7 −0.767887
\(203\) 4.72568e6 0.0396486
\(204\) −3.18871e7 −0.262972
\(205\) 1.72279e8 1.39667
\(206\) −8.28411e7 −0.660253
\(207\) −1.65809e7 −0.129931
\(208\) −324734. −0.00250211
\(209\) 8.98335e6 0.0680654
\(210\) −2.92721e7 −0.218115
\(211\) 1.16380e8 0.852880 0.426440 0.904516i \(-0.359767\pi\)
0.426440 + 0.904516i \(0.359767\pi\)
\(212\) 4.11434e7 0.296568
\(213\) −1.81799e8 −1.28903
\(214\) −1.27493e8 −0.889276
\(215\) −5.54968e7 −0.380832
\(216\) −1.55095e8 −1.04715
\(217\) 2.98865e7 0.198548
\(218\) −6.71011e7 −0.438664
\(219\) 5.39644e7 0.347178
\(220\) 3.79461e7 0.240263
\(221\) 2.73618e7 0.170518
\(222\) 2.40427e7 0.147485
\(223\) −6.00297e7 −0.362493 −0.181246 0.983438i \(-0.558013\pi\)
−0.181246 + 0.983438i \(0.558013\pi\)
\(224\) −6.37538e7 −0.378999
\(225\) −7.78041e7 −0.455369
\(226\) 1.61158e8 0.928696
\(227\) −1.75709e8 −0.997017 −0.498508 0.866885i \(-0.666119\pi\)
−0.498508 + 0.866885i \(0.666119\pi\)
\(228\) 1.84163e7 0.102903
\(229\) −2.35858e8 −1.29785 −0.648927 0.760851i \(-0.724782\pi\)
−0.648927 + 0.760851i \(0.724782\pi\)
\(230\) −3.81660e7 −0.206838
\(231\) 1.37749e7 0.0735270
\(232\) 1.98959e7 0.104606
\(233\) 3.05100e8 1.58014 0.790071 0.613015i \(-0.210044\pi\)
0.790071 + 0.613015i \(0.210044\pi\)
\(234\) 1.76190e7 0.0898928
\(235\) −8.42321e7 −0.423389
\(236\) −3.06284e7 −0.151682
\(237\) −1.10497e8 −0.539175
\(238\) 2.97114e7 0.142858
\(239\) 1.73701e8 0.823019 0.411509 0.911406i \(-0.365002\pi\)
0.411509 + 0.911406i \(0.365002\pi\)
\(240\) 1.81360e6 0.00846842
\(241\) −4.92419e7 −0.226608 −0.113304 0.993560i \(-0.536143\pi\)
−0.113304 + 0.993560i \(0.536143\pi\)
\(242\) −1.24690e8 −0.565559
\(243\) −2.04919e8 −0.916137
\(244\) 1.56724e8 0.690672
\(245\) −4.48926e7 −0.195026
\(246\) 1.00976e8 0.432460
\(247\) −1.58027e7 −0.0667255
\(248\) 1.25827e8 0.523833
\(249\) 4.49847e7 0.184658
\(250\) 2.82549e7 0.114368
\(251\) −4.76583e7 −0.190231 −0.0951153 0.995466i \(-0.530322\pi\)
−0.0951153 + 0.995466i \(0.530322\pi\)
\(252\) −3.14899e7 −0.123957
\(253\) 1.79602e7 0.0697253
\(254\) 2.18938e8 0.838306
\(255\) −1.52812e8 −0.577121
\(256\) −2.66907e8 −0.994307
\(257\) 3.83220e8 1.40826 0.704129 0.710072i \(-0.251338\pi\)
0.704129 + 0.710072i \(0.251338\pi\)
\(258\) −3.25278e7 −0.117920
\(259\) 3.68726e7 0.131873
\(260\) −6.67513e7 −0.235534
\(261\) 1.58856e7 0.0553048
\(262\) −1.15205e8 −0.395748
\(263\) −5.15431e8 −1.74713 −0.873566 0.486707i \(-0.838198\pi\)
−0.873566 + 0.486707i \(0.838198\pi\)
\(264\) 5.79946e7 0.193987
\(265\) 1.97171e8 0.650853
\(266\) −1.71598e7 −0.0559018
\(267\) −5.72799e7 −0.184167
\(268\) 6.79005e7 0.215477
\(269\) −5.22355e7 −0.163619 −0.0818093 0.996648i \(-0.526070\pi\)
−0.0818093 + 0.996648i \(0.526070\pi\)
\(270\) −2.85041e8 −0.881322
\(271\) −5.46494e8 −1.66799 −0.833994 0.551774i \(-0.813951\pi\)
−0.833994 + 0.551774i \(0.813951\pi\)
\(272\) −1.84082e6 −0.00554652
\(273\) −2.42316e7 −0.0720796
\(274\) 6.53549e7 0.191934
\(275\) 8.42762e7 0.244366
\(276\) 3.68193e7 0.105413
\(277\) 4.84372e7 0.136931 0.0684653 0.997654i \(-0.478190\pi\)
0.0684653 + 0.997654i \(0.478190\pi\)
\(278\) 4.30249e8 1.20105
\(279\) 1.00465e8 0.276950
\(280\) −1.89005e8 −0.514541
\(281\) −2.46686e8 −0.663244 −0.331622 0.943412i \(-0.607596\pi\)
−0.331622 + 0.943412i \(0.607596\pi\)
\(282\) −4.93701e7 −0.131097
\(283\) 4.15186e8 1.08890 0.544452 0.838792i \(-0.316738\pi\)
0.544452 + 0.838792i \(0.316738\pi\)
\(284\) −4.50171e8 −1.16617
\(285\) 8.82562e7 0.225833
\(286\) −1.90846e7 −0.0482394
\(287\) 1.54860e8 0.386681
\(288\) −2.14312e8 −0.528655
\(289\) −2.55233e8 −0.622006
\(290\) 3.65656e7 0.0880399
\(291\) 1.94704e8 0.463180
\(292\) 1.33627e8 0.314089
\(293\) 4.53635e8 1.05359 0.526793 0.849994i \(-0.323394\pi\)
0.526793 + 0.849994i \(0.323394\pi\)
\(294\) −2.63125e7 −0.0603873
\(295\) −1.46780e8 −0.332883
\(296\) 1.55240e8 0.347922
\(297\) 1.34135e8 0.297095
\(298\) 1.49642e8 0.327565
\(299\) −3.15940e7 −0.0683527
\(300\) 1.72770e8 0.369440
\(301\) −4.98857e7 −0.105437
\(302\) 2.73758e8 0.571929
\(303\) 4.15881e8 0.858854
\(304\) 1.06316e6 0.00217041
\(305\) 7.51069e8 1.51576
\(306\) 9.98767e7 0.199269
\(307\) 8.47339e7 0.167137 0.0835686 0.996502i \(-0.473368\pi\)
0.0835686 + 0.996502i \(0.473368\pi\)
\(308\) 3.41094e7 0.0665192
\(309\) 3.82990e8 0.738470
\(310\) 2.31251e8 0.440877
\(311\) −1.23785e8 −0.233349 −0.116674 0.993170i \(-0.537223\pi\)
−0.116674 + 0.993170i \(0.537223\pi\)
\(312\) −1.02019e8 −0.190169
\(313\) −9.31573e8 −1.71716 −0.858582 0.512677i \(-0.828654\pi\)
−0.858582 + 0.512677i \(0.828654\pi\)
\(314\) 4.31553e8 0.786647
\(315\) −1.50909e8 −0.272037
\(316\) −2.73612e8 −0.487786
\(317\) 6.40816e8 1.12986 0.564932 0.825137i \(-0.308902\pi\)
0.564932 + 0.825137i \(0.308902\pi\)
\(318\) 1.15566e8 0.201528
\(319\) −1.72071e7 −0.0296784
\(320\) −4.86084e8 −0.829251
\(321\) 5.89422e8 0.994624
\(322\) −3.43071e7 −0.0572650
\(323\) −8.95808e7 −0.147913
\(324\) 7.42005e7 0.121199
\(325\) −1.48251e8 −0.239556
\(326\) 3.90829e8 0.624777
\(327\) 3.10221e8 0.490630
\(328\) 6.51986e8 1.02019
\(329\) −7.57155e7 −0.117219
\(330\) 1.06585e8 0.163267
\(331\) −4.01542e8 −0.608601 −0.304300 0.952576i \(-0.598423\pi\)
−0.304300 + 0.952576i \(0.598423\pi\)
\(332\) 1.11391e8 0.167058
\(333\) 1.23949e8 0.183946
\(334\) 1.48238e8 0.217695
\(335\) 3.25399e8 0.472890
\(336\) 1.63023e6 0.00234456
\(337\) 1.41278e8 0.201080 0.100540 0.994933i \(-0.467943\pi\)
0.100540 + 0.994933i \(0.467943\pi\)
\(338\) 3.35719e7 0.0472898
\(339\) −7.45066e8 −1.03871
\(340\) −3.78393e8 −0.522116
\(341\) −1.08822e8 −0.148620
\(342\) −5.76835e7 −0.0779759
\(343\) −4.03536e7 −0.0539949
\(344\) −2.10027e8 −0.278176
\(345\) 1.76449e8 0.231341
\(346\) −3.13837e8 −0.407322
\(347\) −3.94593e8 −0.506987 −0.253493 0.967337i \(-0.581580\pi\)
−0.253493 + 0.967337i \(0.581580\pi\)
\(348\) −3.52753e7 −0.0448687
\(349\) 1.12883e9 1.42148 0.710739 0.703456i \(-0.248361\pi\)
0.710739 + 0.703456i \(0.248361\pi\)
\(350\) −1.60982e8 −0.200696
\(351\) −2.35959e8 −0.291247
\(352\) 2.32139e8 0.283694
\(353\) −1.16667e9 −1.41168 −0.705839 0.708372i \(-0.749430\pi\)
−0.705839 + 0.708372i \(0.749430\pi\)
\(354\) −8.60310e7 −0.103073
\(355\) −2.15735e9 −2.55930
\(356\) −1.41836e8 −0.166615
\(357\) −1.37362e8 −0.159782
\(358\) 5.90865e8 0.680609
\(359\) 1.52650e9 1.74127 0.870635 0.491929i \(-0.163708\pi\)
0.870635 + 0.491929i \(0.163708\pi\)
\(360\) −6.35351e8 −0.717720
\(361\) −8.42135e8 −0.942120
\(362\) 4.80588e8 0.532467
\(363\) 5.76466e8 0.632558
\(364\) −6.00022e7 −0.0652097
\(365\) 6.40378e8 0.689305
\(366\) 4.40216e8 0.469334
\(367\) −1.56116e9 −1.64860 −0.824302 0.566150i \(-0.808432\pi\)
−0.824302 + 0.566150i \(0.808432\pi\)
\(368\) 2.12556e6 0.00222334
\(369\) 5.20571e8 0.539371
\(370\) 2.85307e8 0.292823
\(371\) 1.77236e8 0.180195
\(372\) −2.23091e8 −0.224689
\(373\) 6.30670e8 0.629247 0.314623 0.949217i \(-0.398122\pi\)
0.314623 + 0.949217i \(0.398122\pi\)
\(374\) −1.08185e8 −0.106934
\(375\) −1.30628e8 −0.127916
\(376\) −3.18775e8 −0.309262
\(377\) 3.02692e7 0.0290941
\(378\) −2.56221e8 −0.244002
\(379\) −1.03258e9 −0.974283 −0.487141 0.873323i \(-0.661960\pi\)
−0.487141 + 0.873323i \(0.661960\pi\)
\(380\) 2.18540e8 0.204309
\(381\) −1.01219e9 −0.937616
\(382\) 7.30680e7 0.0670664
\(383\) 2.21286e8 0.201261 0.100630 0.994924i \(-0.467914\pi\)
0.100630 + 0.994924i \(0.467914\pi\)
\(384\) 4.80127e8 0.432710
\(385\) 1.63462e8 0.145984
\(386\) −1.91061e8 −0.169090
\(387\) −1.67693e8 −0.147071
\(388\) 4.82127e8 0.419035
\(389\) 1.89442e9 1.63175 0.815873 0.578231i \(-0.196257\pi\)
0.815873 + 0.578231i \(0.196257\pi\)
\(390\) −1.87495e8 −0.160053
\(391\) −1.79097e8 −0.151520
\(392\) −1.69895e8 −0.142456
\(393\) 5.32617e8 0.442630
\(394\) 6.26325e8 0.515897
\(395\) −1.31123e9 −1.07050
\(396\) 1.14661e8 0.0927858
\(397\) −4.13478e8 −0.331654 −0.165827 0.986155i \(-0.553029\pi\)
−0.165827 + 0.986155i \(0.553029\pi\)
\(398\) 1.26957e8 0.100940
\(399\) 7.93328e7 0.0625241
\(400\) 9.97392e6 0.00779212
\(401\) 1.29463e9 1.00263 0.501314 0.865265i \(-0.332850\pi\)
0.501314 + 0.865265i \(0.332850\pi\)
\(402\) 1.90723e8 0.146424
\(403\) 1.91430e8 0.145695
\(404\) 1.02980e9 0.776997
\(405\) 3.55590e8 0.265985
\(406\) 3.28685e7 0.0243747
\(407\) −1.34260e8 −0.0987112
\(408\) −5.78315e8 −0.421554
\(409\) 1.53435e8 0.110890 0.0554449 0.998462i \(-0.482342\pi\)
0.0554449 + 0.998462i \(0.482342\pi\)
\(410\) 1.19825e9 0.858626
\(411\) −3.02148e8 −0.214671
\(412\) 9.48360e8 0.668087
\(413\) −1.31940e8 −0.0921617
\(414\) −1.15325e8 −0.0798774
\(415\) 5.33819e8 0.366628
\(416\) −4.08359e8 −0.278109
\(417\) −1.98912e9 −1.34334
\(418\) 6.24819e7 0.0418444
\(419\) −1.34262e9 −0.891667 −0.445833 0.895116i \(-0.647093\pi\)
−0.445833 + 0.895116i \(0.647093\pi\)
\(420\) 3.35105e8 0.220703
\(421\) −1.80068e9 −1.17611 −0.588055 0.808821i \(-0.700106\pi\)
−0.588055 + 0.808821i \(0.700106\pi\)
\(422\) 8.09454e8 0.524323
\(423\) −2.54522e8 −0.163506
\(424\) 7.46191e8 0.475411
\(425\) −8.40392e8 −0.531032
\(426\) −1.26447e9 −0.792453
\(427\) 6.75130e8 0.419652
\(428\) 1.45953e9 0.899827
\(429\) 8.82317e7 0.0539541
\(430\) −3.85997e8 −0.234123
\(431\) −2.86118e9 −1.72137 −0.860686 0.509137i \(-0.829965\pi\)
−0.860686 + 0.509137i \(0.829965\pi\)
\(432\) 1.58746e7 0.00947350
\(433\) −2.21570e9 −1.31160 −0.655802 0.754933i \(-0.727669\pi\)
−0.655802 + 0.754933i \(0.727669\pi\)
\(434\) 2.07869e8 0.122061
\(435\) −1.69050e8 −0.0984695
\(436\) 7.68170e8 0.443869
\(437\) 1.03437e8 0.0592913
\(438\) 3.75338e8 0.213434
\(439\) −6.24237e8 −0.352147 −0.176073 0.984377i \(-0.556340\pi\)
−0.176073 + 0.984377i \(0.556340\pi\)
\(440\) 6.88203e8 0.385152
\(441\) −1.35651e8 −0.0753161
\(442\) 1.90309e8 0.104829
\(443\) −1.35040e9 −0.737987 −0.368993 0.929432i \(-0.620298\pi\)
−0.368993 + 0.929432i \(0.620298\pi\)
\(444\) −2.75239e8 −0.149235
\(445\) −6.79722e8 −0.365655
\(446\) −4.17525e8 −0.222849
\(447\) −6.91824e8 −0.366369
\(448\) −4.36937e8 −0.229586
\(449\) 8.40873e7 0.0438397 0.0219199 0.999760i \(-0.493022\pi\)
0.0219199 + 0.999760i \(0.493022\pi\)
\(450\) −5.41150e8 −0.279946
\(451\) −5.63875e8 −0.289444
\(452\) −1.84493e9 −0.939715
\(453\) −1.26563e9 −0.639682
\(454\) −1.22211e9 −0.612933
\(455\) −2.87548e8 −0.143110
\(456\) 3.34004e8 0.164959
\(457\) −5.67963e8 −0.278364 −0.139182 0.990267i \(-0.544447\pi\)
−0.139182 + 0.990267i \(0.544447\pi\)
\(458\) −1.64046e9 −0.797878
\(459\) −1.33758e9 −0.645617
\(460\) 4.36923e8 0.209292
\(461\) −8.22112e8 −0.390821 −0.195410 0.980722i \(-0.562604\pi\)
−0.195410 + 0.980722i \(0.562604\pi\)
\(462\) 9.58086e7 0.0452020
\(463\) −1.87940e9 −0.880006 −0.440003 0.897996i \(-0.645023\pi\)
−0.440003 + 0.897996i \(0.645023\pi\)
\(464\) −2.03642e6 −0.000946357 0
\(465\) −1.06912e9 −0.493105
\(466\) 2.12206e9 0.971420
\(467\) 3.08803e9 1.40305 0.701524 0.712646i \(-0.252503\pi\)
0.701524 + 0.712646i \(0.252503\pi\)
\(468\) −2.01701e8 −0.0909593
\(469\) 2.92499e8 0.130924
\(470\) −5.85859e8 −0.260286
\(471\) −1.99515e9 −0.879837
\(472\) −5.55488e8 −0.243152
\(473\) 1.81643e8 0.0789232
\(474\) −7.68536e8 −0.331467
\(475\) 4.85366e8 0.207798
\(476\) −3.40135e8 −0.144553
\(477\) 5.95788e8 0.251349
\(478\) 1.20814e9 0.505965
\(479\) 4.41054e9 1.83366 0.916828 0.399282i \(-0.130741\pi\)
0.916828 + 0.399282i \(0.130741\pi\)
\(480\) 2.28063e9 0.941263
\(481\) 2.36178e8 0.0967681
\(482\) −3.42492e8 −0.139311
\(483\) 1.58608e8 0.0640489
\(484\) 1.42744e9 0.572269
\(485\) 2.31049e9 0.919620
\(486\) −1.42527e9 −0.563211
\(487\) −8.06410e8 −0.316377 −0.158188 0.987409i \(-0.550565\pi\)
−0.158188 + 0.987409i \(0.550565\pi\)
\(488\) 2.84241e9 1.10718
\(489\) −1.80688e9 −0.698792
\(490\) −3.12241e8 −0.119896
\(491\) 1.88312e9 0.717948 0.358974 0.933348i \(-0.383127\pi\)
0.358974 + 0.933348i \(0.383127\pi\)
\(492\) −1.15597e9 −0.437591
\(493\) 1.71587e8 0.0644941
\(494\) −1.09913e8 −0.0410207
\(495\) 5.49488e8 0.203629
\(496\) −1.28789e7 −0.00473907
\(497\) −1.93922e9 −0.708567
\(498\) 3.12882e8 0.113521
\(499\) 4.27934e9 1.54179 0.770895 0.636962i \(-0.219809\pi\)
0.770895 + 0.636962i \(0.219809\pi\)
\(500\) −3.23460e8 −0.115725
\(501\) −6.85334e8 −0.243484
\(502\) −3.31477e8 −0.116947
\(503\) 2.89029e9 1.01263 0.506317 0.862347i \(-0.331006\pi\)
0.506317 + 0.862347i \(0.331006\pi\)
\(504\) −5.71112e8 −0.198708
\(505\) 4.93512e9 1.70521
\(506\) 1.24919e8 0.0428648
\(507\) −1.55209e8 −0.0528920
\(508\) −2.50639e9 −0.848253
\(509\) 4.52661e9 1.52146 0.760731 0.649067i \(-0.224840\pi\)
0.760731 + 0.649067i \(0.224840\pi\)
\(510\) −1.06285e9 −0.354795
\(511\) 5.75631e8 0.190841
\(512\) 5.47945e7 0.0180423
\(513\) 7.72515e8 0.252637
\(514\) 2.66541e9 0.865751
\(515\) 4.54482e9 1.46619
\(516\) 3.72376e8 0.119319
\(517\) 2.75694e8 0.0877427
\(518\) 2.56460e8 0.0810710
\(519\) 1.45093e9 0.455575
\(520\) −1.21062e9 −0.377570
\(521\) 4.03795e9 1.25092 0.625459 0.780257i \(-0.284912\pi\)
0.625459 + 0.780257i \(0.284912\pi\)
\(522\) 1.10489e8 0.0339996
\(523\) −5.83872e9 −1.78469 −0.892343 0.451358i \(-0.850940\pi\)
−0.892343 + 0.451358i \(0.850940\pi\)
\(524\) 1.31887e9 0.400443
\(525\) 7.44251e8 0.224472
\(526\) −3.58498e9 −1.07408
\(527\) 1.08516e9 0.322967
\(528\) −5.93598e6 −0.00175499
\(529\) −3.19803e9 −0.939263
\(530\) 1.37138e9 0.400123
\(531\) −4.43523e8 −0.128554
\(532\) 1.96444e8 0.0565650
\(533\) 9.91917e8 0.283746
\(534\) −3.98399e8 −0.113220
\(535\) 6.99448e9 1.97477
\(536\) 1.23147e9 0.345419
\(537\) −2.73168e9 −0.761237
\(538\) −3.63313e8 −0.100587
\(539\) 1.46935e8 0.0404171
\(540\) 3.26314e9 0.891779
\(541\) 3.67872e8 0.0998863 0.0499432 0.998752i \(-0.484096\pi\)
0.0499432 + 0.998752i \(0.484096\pi\)
\(542\) −3.80103e9 −1.02542
\(543\) −2.22185e9 −0.595546
\(544\) −2.31486e9 −0.616495
\(545\) 3.68129e9 0.974121
\(546\) −1.68538e8 −0.0443122
\(547\) 1.47383e9 0.385028 0.192514 0.981294i \(-0.438336\pi\)
0.192514 + 0.981294i \(0.438336\pi\)
\(548\) −7.48179e8 −0.194211
\(549\) 2.26949e9 0.585362
\(550\) 5.86166e8 0.150228
\(551\) −9.90995e7 −0.0252372
\(552\) 6.67767e8 0.168981
\(553\) −1.17865e9 −0.296379
\(554\) 3.36895e8 0.0841804
\(555\) −1.31903e9 −0.327513
\(556\) −4.92546e9 −1.21530
\(557\) 3.10977e9 0.762491 0.381246 0.924474i \(-0.375495\pi\)
0.381246 + 0.924474i \(0.375495\pi\)
\(558\) 6.98765e8 0.170260
\(559\) −3.19530e8 −0.0773696
\(560\) 1.93454e7 0.00465501
\(561\) 5.00159e8 0.119602
\(562\) −1.71578e9 −0.407741
\(563\) 4.68729e9 1.10699 0.553493 0.832854i \(-0.313294\pi\)
0.553493 + 0.832854i \(0.313294\pi\)
\(564\) 5.65186e8 0.132652
\(565\) −8.84146e9 −2.06231
\(566\) 2.88774e9 0.669423
\(567\) 3.19637e8 0.0736405
\(568\) −8.16445e9 −1.86942
\(569\) −4.16238e9 −0.947216 −0.473608 0.880736i \(-0.657049\pi\)
−0.473608 + 0.880736i \(0.657049\pi\)
\(570\) 6.13848e8 0.138835
\(571\) 1.08668e9 0.244273 0.122137 0.992513i \(-0.461025\pi\)
0.122137 + 0.992513i \(0.461025\pi\)
\(572\) 2.18479e8 0.0488117
\(573\) −3.37807e8 −0.0750115
\(574\) 1.07710e9 0.237719
\(575\) 9.70382e8 0.212865
\(576\) −1.46879e9 −0.320244
\(577\) 4.80319e9 1.04091 0.520456 0.853888i \(-0.325762\pi\)
0.520456 + 0.853888i \(0.325762\pi\)
\(578\) −1.77522e9 −0.382389
\(579\) 8.83312e8 0.189121
\(580\) −4.18601e8 −0.0890844
\(581\) 4.79846e8 0.101504
\(582\) 1.35423e9 0.284748
\(583\) −6.45348e8 −0.134882
\(584\) 2.42350e9 0.503498
\(585\) −9.66609e8 −0.199621
\(586\) 3.15516e9 0.647710
\(587\) 5.36662e9 1.09513 0.547567 0.836762i \(-0.315554\pi\)
0.547567 + 0.836762i \(0.315554\pi\)
\(588\) 3.01223e8 0.0611038
\(589\) −6.26733e8 −0.126380
\(590\) −1.02090e9 −0.204645
\(591\) −2.89562e9 −0.577012
\(592\) −1.58894e7 −0.00314762
\(593\) 4.81236e9 0.947691 0.473845 0.880608i \(-0.342866\pi\)
0.473845 + 0.880608i \(0.342866\pi\)
\(594\) 9.32950e8 0.182644
\(595\) −1.63003e9 −0.317238
\(596\) −1.71309e9 −0.331451
\(597\) −5.86945e8 −0.112898
\(598\) −2.19746e8 −0.0420210
\(599\) 1.29580e9 0.246345 0.123173 0.992385i \(-0.460693\pi\)
0.123173 + 0.992385i \(0.460693\pi\)
\(600\) 3.13342e9 0.592228
\(601\) −6.01433e9 −1.13012 −0.565062 0.825048i \(-0.691148\pi\)
−0.565062 + 0.825048i \(0.691148\pi\)
\(602\) −3.46970e8 −0.0648192
\(603\) 9.83251e8 0.182622
\(604\) −3.13396e9 −0.578715
\(605\) 6.84073e9 1.25591
\(606\) 2.89257e9 0.527995
\(607\) −4.04454e9 −0.734021 −0.367010 0.930217i \(-0.619619\pi\)
−0.367010 + 0.930217i \(0.619619\pi\)
\(608\) 1.33694e9 0.241241
\(609\) −1.51957e8 −0.0272622
\(610\) 5.22390e9 0.931839
\(611\) −4.84977e8 −0.0860155
\(612\) −1.14338e9 −0.201633
\(613\) −1.01120e9 −0.177308 −0.0886538 0.996062i \(-0.528256\pi\)
−0.0886538 + 0.996062i \(0.528256\pi\)
\(614\) 5.89350e8 0.102750
\(615\) −5.53974e9 −0.960343
\(616\) 6.18620e8 0.106633
\(617\) −9.03950e9 −1.54934 −0.774669 0.632367i \(-0.782083\pi\)
−0.774669 + 0.632367i \(0.782083\pi\)
\(618\) 2.66381e9 0.453987
\(619\) 1.24718e9 0.211355 0.105678 0.994400i \(-0.466299\pi\)
0.105678 + 0.994400i \(0.466299\pi\)
\(620\) −2.64735e9 −0.446108
\(621\) 1.54447e9 0.258797
\(622\) −8.60959e8 −0.143455
\(623\) −6.10997e8 −0.101235
\(624\) 1.04420e7 0.00172044
\(625\) −6.82190e9 −1.11770
\(626\) −6.47937e9 −1.05566
\(627\) −2.88866e8 −0.0468015
\(628\) −4.94039e9 −0.795981
\(629\) 1.33882e9 0.214509
\(630\) −1.04962e9 −0.167239
\(631\) −1.02219e10 −1.61968 −0.809839 0.586652i \(-0.800446\pi\)
−0.809839 + 0.586652i \(0.800446\pi\)
\(632\) −4.96231e9 −0.781942
\(633\) −3.74226e9 −0.586437
\(634\) 4.45707e9 0.694603
\(635\) −1.20113e10 −1.86159
\(636\) −1.32299e9 −0.203919
\(637\) −2.58475e8 −0.0396214
\(638\) −1.19680e8 −0.0182453
\(639\) −6.51881e9 −0.988361
\(640\) 5.69752e9 0.859124
\(641\) −8.10582e9 −1.21561 −0.607804 0.794087i \(-0.707949\pi\)
−0.607804 + 0.794087i \(0.707949\pi\)
\(642\) 4.09961e9 0.611462
\(643\) −1.03980e10 −1.54245 −0.771225 0.636563i \(-0.780356\pi\)
−0.771225 + 0.636563i \(0.780356\pi\)
\(644\) 3.92746e8 0.0579444
\(645\) 1.78454e9 0.261858
\(646\) −6.23061e8 −0.0909321
\(647\) 5.54098e9 0.804306 0.402153 0.915572i \(-0.368262\pi\)
0.402153 + 0.915572i \(0.368262\pi\)
\(648\) 1.34573e9 0.194287
\(649\) 4.80418e8 0.0689862
\(650\) −1.03113e9 −0.147271
\(651\) −9.61020e8 −0.136521
\(652\) −4.47419e9 −0.632190
\(653\) −5.05858e9 −0.710939 −0.355469 0.934688i \(-0.615679\pi\)
−0.355469 + 0.934688i \(0.615679\pi\)
\(654\) 2.15768e9 0.301623
\(655\) 6.32039e9 0.878819
\(656\) −6.67334e7 −0.00922954
\(657\) 1.93501e9 0.266198
\(658\) −5.26624e8 −0.0720626
\(659\) −8.78170e9 −1.19531 −0.597653 0.801755i \(-0.703900\pi\)
−0.597653 + 0.801755i \(0.703900\pi\)
\(660\) −1.22018e9 −0.165204
\(661\) 1.14547e10 1.54269 0.771347 0.636415i \(-0.219583\pi\)
0.771347 + 0.636415i \(0.219583\pi\)
\(662\) −2.79284e9 −0.374148
\(663\) −8.79835e8 −0.117248
\(664\) 2.02023e9 0.267801
\(665\) 9.41417e8 0.124138
\(666\) 8.62105e8 0.113084
\(667\) −1.98128e8 −0.0258526
\(668\) −1.69702e9 −0.220277
\(669\) 1.93029e9 0.249248
\(670\) 2.26325e9 0.290717
\(671\) −2.45827e9 −0.314124
\(672\) 2.05004e9 0.260598
\(673\) 1.16861e10 1.47780 0.738902 0.673813i \(-0.235345\pi\)
0.738902 + 0.673813i \(0.235345\pi\)
\(674\) 9.82630e8 0.123618
\(675\) 7.24725e9 0.907006
\(676\) −3.84329e8 −0.0478509
\(677\) 9.29654e9 1.15149 0.575746 0.817629i \(-0.304712\pi\)
0.575746 + 0.817629i \(0.304712\pi\)
\(678\) −5.18216e9 −0.638567
\(679\) 2.07688e9 0.254606
\(680\) −6.86267e9 −0.836974
\(681\) 5.65002e9 0.685544
\(682\) −7.56892e8 −0.0913668
\(683\) 3.85605e9 0.463095 0.231547 0.972824i \(-0.425621\pi\)
0.231547 + 0.972824i \(0.425621\pi\)
\(684\) 6.60357e8 0.0789010
\(685\) −3.58549e9 −0.426218
\(686\) −2.80671e8 −0.0331943
\(687\) 7.58415e9 0.892398
\(688\) 2.14971e7 0.00251663
\(689\) 1.13524e9 0.132227
\(690\) 1.22725e9 0.142221
\(691\) 1.52193e10 1.75477 0.877386 0.479785i \(-0.159286\pi\)
0.877386 + 0.479785i \(0.159286\pi\)
\(692\) 3.59278e9 0.412154
\(693\) 4.93930e8 0.0563767
\(694\) −2.74451e9 −0.311679
\(695\) −2.36043e10 −2.66713
\(696\) −6.39765e8 −0.0719264
\(697\) 5.62289e9 0.628991
\(698\) 7.85136e9 0.873878
\(699\) −9.81068e9 −1.08650
\(700\) 1.84291e9 0.203078
\(701\) 1.74595e10 1.91434 0.957170 0.289527i \(-0.0934980\pi\)
0.957170 + 0.289527i \(0.0934980\pi\)
\(702\) −1.64116e9 −0.179049
\(703\) −7.73234e8 −0.0839397
\(704\) 1.59097e9 0.171853
\(705\) 2.70854e9 0.291121
\(706\) −8.11452e9 −0.867854
\(707\) 4.43614e9 0.472104
\(708\) 9.84877e8 0.104296
\(709\) 1.06434e10 1.12155 0.560777 0.827967i \(-0.310502\pi\)
0.560777 + 0.827967i \(0.310502\pi\)
\(710\) −1.50050e10 −1.57337
\(711\) −3.96210e9 −0.413411
\(712\) −2.57240e9 −0.267090
\(713\) −1.25301e9 −0.129462
\(714\) −9.55391e8 −0.0982284
\(715\) 1.04702e9 0.107123
\(716\) −6.76419e9 −0.688684
\(717\) −5.58547e9 −0.565904
\(718\) 1.06173e10 1.07048
\(719\) −3.93718e9 −0.395034 −0.197517 0.980299i \(-0.563288\pi\)
−0.197517 + 0.980299i \(0.563288\pi\)
\(720\) 6.50307e7 0.00649314
\(721\) 4.08530e9 0.405930
\(722\) −5.85730e9 −0.579185
\(723\) 1.58341e9 0.155815
\(724\) −5.50174e9 −0.538785
\(725\) −9.29690e8 −0.0906056
\(726\) 4.00949e9 0.388876
\(727\) 1.76816e8 0.0170668 0.00853339 0.999964i \(-0.497284\pi\)
0.00853339 + 0.999964i \(0.497284\pi\)
\(728\) −1.08822e9 −0.104534
\(729\) 8.62735e9 0.824766
\(730\) 4.45402e9 0.423762
\(731\) −1.81132e9 −0.171508
\(732\) −5.03957e9 −0.474903
\(733\) 5.88295e9 0.551736 0.275868 0.961195i \(-0.411035\pi\)
0.275868 + 0.961195i \(0.411035\pi\)
\(734\) −1.08583e10 −1.01351
\(735\) 1.44355e9 0.134099
\(736\) 2.67292e9 0.247124
\(737\) −1.06504e9 −0.0980011
\(738\) 3.62073e9 0.331588
\(739\) 1.68753e10 1.53814 0.769069 0.639166i \(-0.220720\pi\)
0.769069 + 0.639166i \(0.220720\pi\)
\(740\) −3.26617e9 −0.296298
\(741\) 5.08146e8 0.0458802
\(742\) 1.23273e9 0.110778
\(743\) −5.61136e9 −0.501888 −0.250944 0.968002i \(-0.580741\pi\)
−0.250944 + 0.968002i \(0.580741\pi\)
\(744\) −4.04605e9 −0.360185
\(745\) −8.20965e9 −0.727407
\(746\) 4.38649e9 0.386840
\(747\) 1.61303e9 0.141586
\(748\) 1.23849e9 0.108203
\(749\) 6.28729e9 0.546735
\(750\) −9.08555e8 −0.0786387
\(751\) −9.05348e9 −0.779966 −0.389983 0.920822i \(-0.627519\pi\)
−0.389983 + 0.920822i \(0.627519\pi\)
\(752\) 3.26279e7 0.00279786
\(753\) 1.53248e9 0.130802
\(754\) 2.10531e8 0.0178861
\(755\) −1.50189e10 −1.27006
\(756\) 2.93321e9 0.246897
\(757\) 1.51419e10 1.26866 0.634329 0.773063i \(-0.281277\pi\)
0.634329 + 0.773063i \(0.281277\pi\)
\(758\) −7.18188e9 −0.598957
\(759\) −5.77523e8 −0.0479428
\(760\) 3.96352e9 0.327516
\(761\) 1.28555e10 1.05741 0.528703 0.848807i \(-0.322679\pi\)
0.528703 + 0.848807i \(0.322679\pi\)
\(762\) −7.04009e9 −0.576416
\(763\) 3.30909e9 0.269695
\(764\) −8.36478e8 −0.0678622
\(765\) −5.47942e9 −0.442507
\(766\) 1.53911e9 0.123728
\(767\) −8.45107e8 −0.0676282
\(768\) 8.58258e9 0.683681
\(769\) 2.32284e10 1.84194 0.920972 0.389629i \(-0.127397\pi\)
0.920972 + 0.389629i \(0.127397\pi\)
\(770\) 1.13693e9 0.0897462
\(771\) −1.23227e10 −0.968311
\(772\) 2.18726e9 0.171096
\(773\) 1.44242e10 1.12321 0.561607 0.827404i \(-0.310183\pi\)
0.561607 + 0.827404i \(0.310183\pi\)
\(774\) −1.16636e9 −0.0904146
\(775\) −5.87961e9 −0.453725
\(776\) 8.74401e9 0.671730
\(777\) −1.18566e9 −0.0906750
\(778\) 1.31763e10 1.00314
\(779\) −3.24748e9 −0.246131
\(780\) 2.14643e9 0.161952
\(781\) 7.06108e9 0.530387
\(782\) −1.24567e9 −0.0931495
\(783\) −1.47971e9 −0.110156
\(784\) 1.73895e7 0.00128878
\(785\) −2.36758e10 −1.74687
\(786\) 3.70451e9 0.272115
\(787\) −1.09086e10 −0.797735 −0.398867 0.917009i \(-0.630597\pi\)
−0.398867 + 0.917009i \(0.630597\pi\)
\(788\) −7.17013e9 −0.522018
\(789\) 1.65740e10 1.20132
\(790\) −9.11997e9 −0.658110
\(791\) −7.94752e9 −0.570971
\(792\) 2.07953e9 0.148740
\(793\) 4.32437e9 0.307941
\(794\) −2.87586e9 −0.203890
\(795\) −6.34017e9 −0.447524
\(796\) −1.45339e9 −0.102138
\(797\) 1.69316e9 0.118466 0.0592330 0.998244i \(-0.481135\pi\)
0.0592330 + 0.998244i \(0.481135\pi\)
\(798\) 5.51783e8 0.0384378
\(799\) −2.74919e9 −0.190674
\(800\) 1.25424e10 0.866093
\(801\) −2.05390e9 −0.141210
\(802\) 9.00453e9 0.616383
\(803\) −2.09598e9 −0.142851
\(804\) −2.18339e9 −0.148161
\(805\) 1.88216e9 0.127166
\(806\) 1.33146e9 0.0895682
\(807\) 1.67967e9 0.112503
\(808\) 1.86769e10 1.24556
\(809\) −1.36657e10 −0.907429 −0.453715 0.891147i \(-0.649901\pi\)
−0.453715 + 0.891147i \(0.649901\pi\)
\(810\) 2.47324e9 0.163519
\(811\) 1.83163e10 1.20577 0.602885 0.797828i \(-0.294018\pi\)
0.602885 + 0.797828i \(0.294018\pi\)
\(812\) −3.76277e8 −0.0246639
\(813\) 1.75729e10 1.14690
\(814\) −9.33819e8 −0.0606844
\(815\) −2.14416e10 −1.38741
\(816\) 5.91928e7 0.00381376
\(817\) 1.04612e9 0.0671129
\(818\) 1.06718e9 0.0681714
\(819\) −8.68878e8 −0.0552669
\(820\) −1.37175e10 −0.868814
\(821\) 1.34510e10 0.848307 0.424153 0.905590i \(-0.360572\pi\)
0.424153 + 0.905590i \(0.360572\pi\)
\(822\) −2.10153e9 −0.131973
\(823\) 1.06821e9 0.0667968 0.0333984 0.999442i \(-0.489367\pi\)
0.0333984 + 0.999442i \(0.489367\pi\)
\(824\) 1.71998e10 1.07097
\(825\) −2.70996e9 −0.168025
\(826\) −9.17680e8 −0.0566580
\(827\) −1.47731e9 −0.0908245 −0.0454123 0.998968i \(-0.514460\pi\)
−0.0454123 + 0.998968i \(0.514460\pi\)
\(828\) 1.32024e9 0.0808251
\(829\) −5.54603e9 −0.338097 −0.169048 0.985608i \(-0.554069\pi\)
−0.169048 + 0.985608i \(0.554069\pi\)
\(830\) 3.71287e9 0.225391
\(831\) −1.55753e9 −0.0941528
\(832\) −2.79869e9 −0.168470
\(833\) −1.46522e9 −0.0878303
\(834\) −1.38349e10 −0.825839
\(835\) −8.13263e9 −0.483424
\(836\) −7.15289e8 −0.0423409
\(837\) −9.35808e9 −0.551630
\(838\) −9.33829e9 −0.548168
\(839\) 2.04818e10 1.19729 0.598646 0.801013i \(-0.295705\pi\)
0.598646 + 0.801013i \(0.295705\pi\)
\(840\) 6.07758e9 0.353796
\(841\) −1.70601e10 −0.988996
\(842\) −1.25242e10 −0.723034
\(843\) 7.93237e9 0.456043
\(844\) −9.26659e9 −0.530544
\(845\) −1.84182e9 −0.105014
\(846\) −1.77028e9 −0.100518
\(847\) 6.14908e9 0.347711
\(848\) −7.63756e7 −0.00430100
\(849\) −1.33506e10 −0.748726
\(850\) −5.84517e9 −0.326461
\(851\) −1.54591e9 −0.0859867
\(852\) 1.44755e10 0.801855
\(853\) 2.15373e10 1.18814 0.594072 0.804412i \(-0.297519\pi\)
0.594072 + 0.804412i \(0.297519\pi\)
\(854\) 4.69573e9 0.257989
\(855\) 3.16463e9 0.173157
\(856\) 2.64705e10 1.44246
\(857\) 1.35756e10 0.736760 0.368380 0.929675i \(-0.379913\pi\)
0.368380 + 0.929675i \(0.379913\pi\)
\(858\) 6.13678e8 0.0331692
\(859\) −1.21787e10 −0.655581 −0.327790 0.944750i \(-0.606304\pi\)
−0.327790 + 0.944750i \(0.606304\pi\)
\(860\) 4.41887e9 0.236901
\(861\) −4.97963e9 −0.265880
\(862\) −1.99004e10 −1.05824
\(863\) −6.53280e9 −0.345988 −0.172994 0.984923i \(-0.555344\pi\)
−0.172994 + 0.984923i \(0.555344\pi\)
\(864\) 1.99626e10 1.05298
\(865\) 1.72177e10 0.904520
\(866\) −1.54108e10 −0.806331
\(867\) 8.20719e9 0.427688
\(868\) −2.37968e9 −0.123509
\(869\) 4.29169e9 0.221850
\(870\) −1.17579e9 −0.0605358
\(871\) 1.87353e9 0.0960720
\(872\) 1.39318e10 0.711540
\(873\) 6.98156e9 0.355142
\(874\) 7.19435e8 0.0364504
\(875\) −1.39339e9 −0.0703143
\(876\) −4.29685e9 −0.215966
\(877\) −1.56638e10 −0.784151 −0.392075 0.919933i \(-0.628243\pi\)
−0.392075 + 0.919933i \(0.628243\pi\)
\(878\) −4.34175e9 −0.216488
\(879\) −1.45869e10 −0.724440
\(880\) −7.04403e7 −0.00348443
\(881\) 3.73448e9 0.183998 0.0919992 0.995759i \(-0.470674\pi\)
0.0919992 + 0.995759i \(0.470674\pi\)
\(882\) −9.43492e8 −0.0463018
\(883\) 9.21741e9 0.450554 0.225277 0.974295i \(-0.427671\pi\)
0.225277 + 0.974295i \(0.427671\pi\)
\(884\) −2.17865e9 −0.106073
\(885\) 4.71982e9 0.228889
\(886\) −9.39242e9 −0.453690
\(887\) −3.72046e10 −1.79004 −0.895022 0.446023i \(-0.852840\pi\)
−0.895022 + 0.446023i \(0.852840\pi\)
\(888\) −4.99183e9 −0.239229
\(889\) −1.07969e10 −0.515398
\(890\) −4.72767e9 −0.224793
\(891\) −1.16386e9 −0.0551225
\(892\) 4.77980e9 0.225493
\(893\) 1.58779e9 0.0746126
\(894\) −4.81184e9 −0.225232
\(895\) −3.24160e10 −1.51140
\(896\) 5.12145e9 0.237857
\(897\) 1.01593e9 0.0469990
\(898\) 5.84852e8 0.0269512
\(899\) 1.20047e9 0.0551052
\(900\) 6.19506e9 0.283267
\(901\) 6.43533e9 0.293112
\(902\) −3.92192e9 −0.177941
\(903\) 1.60411e9 0.0724980
\(904\) −3.34603e10 −1.50640
\(905\) −2.63660e10 −1.18243
\(906\) −8.80286e9 −0.393256
\(907\) −2.49092e9 −0.110849 −0.0554247 0.998463i \(-0.517651\pi\)
−0.0554247 + 0.998463i \(0.517651\pi\)
\(908\) 1.39906e10 0.620205
\(909\) 1.49123e10 0.658525
\(910\) −1.99998e9 −0.0879795
\(911\) −2.57216e10 −1.12715 −0.563576 0.826064i \(-0.690575\pi\)
−0.563576 + 0.826064i \(0.690575\pi\)
\(912\) −3.41867e7 −0.00149236
\(913\) −1.74721e9 −0.0759796
\(914\) −3.95035e9 −0.171129
\(915\) −2.41511e10 −1.04223
\(916\) 1.87799e10 0.807344
\(917\) 5.68135e9 0.243309
\(918\) −9.30326e9 −0.396904
\(919\) 5.43149e9 0.230842 0.115421 0.993317i \(-0.463178\pi\)
0.115421 + 0.993317i \(0.463178\pi\)
\(920\) 7.92418e9 0.335503
\(921\) −2.72467e9 −0.114923
\(922\) −5.71803e9 −0.240264
\(923\) −1.24212e10 −0.519946
\(924\) −1.09681e9 −0.0457383
\(925\) −7.25400e9 −0.301357
\(926\) −1.30718e10 −0.540999
\(927\) 1.37330e10 0.566220
\(928\) −2.56084e9 −0.105187
\(929\) 4.77407e10 1.95359 0.976797 0.214169i \(-0.0687044\pi\)
0.976797 + 0.214169i \(0.0687044\pi\)
\(930\) −7.43602e9 −0.303145
\(931\) 8.46232e8 0.0343689
\(932\) −2.42932e10 −0.982945
\(933\) 3.98038e9 0.160450
\(934\) 2.14782e10 0.862548
\(935\) 5.93523e9 0.237463
\(936\) −3.65812e9 −0.145812
\(937\) −2.24030e10 −0.889648 −0.444824 0.895618i \(-0.646734\pi\)
−0.444824 + 0.895618i \(0.646734\pi\)
\(938\) 2.03442e9 0.0804878
\(939\) 2.99553e10 1.18071
\(940\) 6.70688e9 0.263374
\(941\) −1.06797e10 −0.417826 −0.208913 0.977934i \(-0.566993\pi\)
−0.208913 + 0.977934i \(0.566993\pi\)
\(942\) −1.38769e10 −0.540895
\(943\) −6.49262e9 −0.252133
\(944\) 5.68564e7 0.00219977
\(945\) 1.40568e10 0.541845
\(946\) 1.26338e9 0.0485194
\(947\) −3.66166e10 −1.40105 −0.700525 0.713628i \(-0.747051\pi\)
−0.700525 + 0.713628i \(0.747051\pi\)
\(948\) 8.79816e9 0.335400
\(949\) 3.68706e9 0.140039
\(950\) 3.37586e9 0.127747
\(951\) −2.06059e10 −0.776889
\(952\) −6.16880e9 −0.231724
\(953\) 3.67810e10 1.37657 0.688285 0.725440i \(-0.258364\pi\)
0.688285 + 0.725440i \(0.258364\pi\)
\(954\) 4.14388e9 0.154521
\(955\) −4.00865e9 −0.148931
\(956\) −1.38307e10 −0.511968
\(957\) 5.53305e8 0.0204067
\(958\) 3.06766e10 1.12727
\(959\) −3.22297e9 −0.118003
\(960\) 1.56303e10 0.570189
\(961\) −1.99205e10 −0.724050
\(962\) 1.64269e9 0.0594898
\(963\) 2.11351e10 0.762626
\(964\) 3.92083e9 0.140964
\(965\) 1.04820e10 0.375489
\(966\) 1.10317e9 0.0393751
\(967\) −2.03570e10 −0.723971 −0.361986 0.932184i \(-0.617901\pi\)
−0.361986 + 0.932184i \(0.617901\pi\)
\(968\) 2.58886e10 0.917372
\(969\) 2.88053e9 0.101704
\(970\) 1.60702e10 0.565352
\(971\) 9.34911e9 0.327720 0.163860 0.986484i \(-0.447605\pi\)
0.163860 + 0.986484i \(0.447605\pi\)
\(972\) 1.63164e10 0.569893
\(973\) −2.12177e10 −0.738419
\(974\) −5.60882e9 −0.194498
\(975\) 4.76711e9 0.164717
\(976\) −2.90932e8 −0.0100165
\(977\) −4.95354e10 −1.69936 −0.849679 0.527300i \(-0.823204\pi\)
−0.849679 + 0.527300i \(0.823204\pi\)
\(978\) −1.25674e10 −0.429594
\(979\) 2.22475e9 0.0757779
\(980\) 3.57452e9 0.121318
\(981\) 1.11237e10 0.376190
\(982\) 1.30977e10 0.441371
\(983\) 8.29372e9 0.278492 0.139246 0.990258i \(-0.455532\pi\)
0.139246 + 0.990258i \(0.455532\pi\)
\(984\) −2.09650e10 −0.701476
\(985\) −3.43614e10 −1.14563
\(986\) 1.19344e9 0.0396488
\(987\) 2.43468e9 0.0805995
\(988\) 1.25827e9 0.0415074
\(989\) 2.09149e9 0.0687495
\(990\) 3.82185e9 0.125185
\(991\) 3.81642e9 0.124566 0.0622829 0.998059i \(-0.480162\pi\)
0.0622829 + 0.998059i \(0.480162\pi\)
\(992\) −1.61954e10 −0.526747
\(993\) 1.29118e10 0.418471
\(994\) −1.34879e10 −0.435604
\(995\) −6.96508e9 −0.224153
\(996\) −3.58185e9 −0.114868
\(997\) −4.67648e10 −1.49447 −0.747234 0.664562i \(-0.768618\pi\)
−0.747234 + 0.664562i \(0.768618\pi\)
\(998\) 2.97641e10 0.947842
\(999\) −1.15456e10 −0.366384
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 91.8.a.d.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.d.1.8 10 1.1 even 1 trivial