Properties

Label 91.8.a.b.1.4
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} - 22500802 x^{2} + 176026849 x + 176334338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-5.28525\) 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)
 
\(f(q)\) \(=\) \(q-6.28525 q^{2} +79.3185 q^{3} -88.4957 q^{4} -293.470 q^{5} -498.536 q^{6} -343.000 q^{7} +1360.73 q^{8} +4104.43 q^{9} +O(q^{10})\) \(q-6.28525 q^{2} +79.3185 q^{3} -88.4957 q^{4} -293.470 q^{5} -498.536 q^{6} -343.000 q^{7} +1360.73 q^{8} +4104.43 q^{9} +1844.53 q^{10} +5225.54 q^{11} -7019.35 q^{12} +2197.00 q^{13} +2155.84 q^{14} -23277.6 q^{15} +2774.93 q^{16} -35943.3 q^{17} -25797.4 q^{18} -15900.9 q^{19} +25970.9 q^{20} -27206.3 q^{21} -32843.8 q^{22} -54319.2 q^{23} +107931. q^{24} +7999.90 q^{25} -13808.7 q^{26} +152088. q^{27} +30354.0 q^{28} -11069.5 q^{29} +146306. q^{30} -196401. q^{31} -191614. q^{32} +414482. q^{33} +225912. q^{34} +100660. q^{35} -363224. q^{36} -569532. q^{37} +99941.3 q^{38} +174263. q^{39} -399334. q^{40} +18430.8 q^{41} +170998. q^{42} -23342.0 q^{43} -462438. q^{44} -1.20453e6 q^{45} +341410. q^{46} +956732. q^{47} +220104. q^{48} +117649. q^{49} -50281.4 q^{50} -2.85097e6 q^{51} -194425. q^{52} -933786. q^{53} -955909. q^{54} -1.53354e6 q^{55} -466730. q^{56} -1.26124e6 q^{57} +69574.3 q^{58} -1.11153e6 q^{59} +2.05997e6 q^{60} -609277. q^{61} +1.23443e6 q^{62} -1.40782e6 q^{63} +849152. q^{64} -644755. q^{65} -2.60512e6 q^{66} +121784. q^{67} +3.18083e6 q^{68} -4.30852e6 q^{69} -632675. q^{70} +1.39633e6 q^{71} +5.58502e6 q^{72} +2.85451e6 q^{73} +3.57965e6 q^{74} +634541. q^{75} +1.40716e6 q^{76} -1.79236e6 q^{77} -1.09528e6 q^{78} -477949. q^{79} -814361. q^{80} +3.08699e6 q^{81} -115842. q^{82} +7.53259e6 q^{83} +2.40764e6 q^{84} +1.05483e7 q^{85} +146710. q^{86} -878014. q^{87} +7.11054e6 q^{88} -9.23609e6 q^{89} +7.57076e6 q^{90} -753571. q^{91} +4.80701e6 q^{92} -1.55782e7 q^{93} -6.01329e6 q^{94} +4.66646e6 q^{95} -1.51986e7 q^{96} +1.58162e7 q^{97} -739453. q^{98} +2.14479e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9} - 5124 q^{10} - 9826 q^{11} - 20919 q^{12} + 19773 q^{13} + 1715 q^{14} - 20346 q^{15} + 31113 q^{16} - 22766 q^{17} - 12978 q^{18} - 17769 q^{19} - 44204 q^{20} + 8918 q^{21} - 203553 q^{22} - 49103 q^{23} + 52737 q^{24} + 227466 q^{25} - 10985 q^{26} + 103624 q^{27} - 134799 q^{28} - 487455 q^{29} - 287992 q^{30} - 63843 q^{31} - 587099 q^{32} - 314392 q^{33} - 576240 q^{34} + 62083 q^{35} - 1514926 q^{36} - 796926 q^{37} - 766702 q^{38} - 57122 q^{39} - 2887296 q^{40} - 1567546 q^{41} - 241129 q^{42} - 277899 q^{43} - 1281195 q^{44} - 1650593 q^{45} - 1907445 q^{46} + 1077367 q^{47} - 1110835 q^{48} + 1058841 q^{49} - 267459 q^{50} - 3054368 q^{51} + 863421 q^{52} - 7322659 q^{53} - 3355387 q^{54} - 2613324 q^{55} - 410571 q^{56} - 3751946 q^{57} - 2992332 q^{58} - 169804 q^{59} - 2754416 q^{60} - 6352284 q^{61} + 6001087 q^{62} - 1101373 q^{63} + 1657017 q^{64} - 397657 q^{65} - 5962713 q^{66} + 921120 q^{67} + 5615224 q^{68} - 5202780 q^{69} + 1757532 q^{70} + 3786654 q^{71} + 2229758 q^{72} + 5792889 q^{73} - 1991961 q^{74} + 145628 q^{75} - 2806026 q^{76} + 3370318 q^{77} + 1544491 q^{78} + 3464037 q^{79} + 15422512 q^{80} - 5010363 q^{81} - 12539943 q^{82} + 6834945 q^{83} + 7175217 q^{84} + 3880662 q^{85} - 7977524 q^{86} + 3727078 q^{87} + 7013709 q^{88} - 20408371 q^{89} + 34910060 q^{90} - 6782139 q^{91} - 3544371 q^{92} + 3121742 q^{93} + 61343967 q^{94} + 3360807 q^{95} + 23547905 q^{96} + 41644125 q^{97} - 588245 q^{98} + 50754068 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.28525 −0.555542 −0.277771 0.960647i \(-0.589596\pi\)
−0.277771 + 0.960647i \(0.589596\pi\)
\(3\) 79.3185 1.69610 0.848048 0.529920i \(-0.177778\pi\)
0.848048 + 0.529920i \(0.177778\pi\)
\(4\) −88.4957 −0.691373
\(5\) −293.470 −1.04995 −0.524976 0.851117i \(-0.675926\pi\)
−0.524976 + 0.851117i \(0.675926\pi\)
\(6\) −498.536 −0.942253
\(7\) −343.000 −0.377964
\(8\) 1360.73 0.939629
\(9\) 4104.43 1.87674
\(10\) 1844.53 0.583293
\(11\) 5225.54 1.18374 0.591870 0.806033i \(-0.298390\pi\)
0.591870 + 0.806033i \(0.298390\pi\)
\(12\) −7019.35 −1.17263
\(13\) 2197.00 0.277350
\(14\) 2155.84 0.209975
\(15\) −23277.6 −1.78082
\(16\) 2774.93 0.169369
\(17\) −35943.3 −1.77438 −0.887190 0.461405i \(-0.847346\pi\)
−0.887190 + 0.461405i \(0.847346\pi\)
\(18\) −25797.4 −1.04261
\(19\) −15900.9 −0.531845 −0.265923 0.963994i \(-0.585677\pi\)
−0.265923 + 0.963994i \(0.585677\pi\)
\(20\) 25970.9 0.725908
\(21\) −27206.3 −0.641064
\(22\) −32843.8 −0.657618
\(23\) −54319.2 −0.930906 −0.465453 0.885073i \(-0.654109\pi\)
−0.465453 + 0.885073i \(0.654109\pi\)
\(24\) 107931. 1.59370
\(25\) 7999.90 0.102399
\(26\) −13808.7 −0.154080
\(27\) 152088. 1.48703
\(28\) 30354.0 0.261314
\(29\) −11069.5 −0.0842818 −0.0421409 0.999112i \(-0.513418\pi\)
−0.0421409 + 0.999112i \(0.513418\pi\)
\(30\) 146306. 0.989320
\(31\) −196401. −1.18407 −0.592035 0.805912i \(-0.701675\pi\)
−0.592035 + 0.805912i \(0.701675\pi\)
\(32\) −191614. −1.03372
\(33\) 414482. 2.00774
\(34\) 225912. 0.985743
\(35\) 100660. 0.396844
\(36\) −363224. −1.29753
\(37\) −569532. −1.84847 −0.924234 0.381826i \(-0.875295\pi\)
−0.924234 + 0.381826i \(0.875295\pi\)
\(38\) 99941.3 0.295463
\(39\) 174263. 0.470412
\(40\) −399334. −0.986565
\(41\) 18430.8 0.0417640 0.0208820 0.999782i \(-0.493353\pi\)
0.0208820 + 0.999782i \(0.493353\pi\)
\(42\) 170998. 0.356138
\(43\) −23342.0 −0.0447711 −0.0223855 0.999749i \(-0.507126\pi\)
−0.0223855 + 0.999749i \(0.507126\pi\)
\(44\) −462438. −0.818406
\(45\) −1.20453e6 −1.97049
\(46\) 341410. 0.517158
\(47\) 956732. 1.34415 0.672075 0.740483i \(-0.265403\pi\)
0.672075 + 0.740483i \(0.265403\pi\)
\(48\) 220104. 0.287265
\(49\) 117649. 0.142857
\(50\) −50281.4 −0.0568869
\(51\) −2.85097e6 −3.00952
\(52\) −194425. −0.191752
\(53\) −933786. −0.861553 −0.430776 0.902459i \(-0.641760\pi\)
−0.430776 + 0.902459i \(0.641760\pi\)
\(54\) −955909. −0.826111
\(55\) −1.53354e6 −1.24287
\(56\) −466730. −0.355146
\(57\) −1.26124e6 −0.902060
\(58\) 69574.3 0.0468221
\(59\) −1.11153e6 −0.704595 −0.352298 0.935888i \(-0.614600\pi\)
−0.352298 + 0.935888i \(0.614600\pi\)
\(60\) 2.05997e6 1.23121
\(61\) −609277. −0.343685 −0.171842 0.985124i \(-0.554972\pi\)
−0.171842 + 0.985124i \(0.554972\pi\)
\(62\) 1.23443e6 0.657802
\(63\) −1.40782e6 −0.709341
\(64\) 849152. 0.404907
\(65\) −644755. −0.291204
\(66\) −2.60512e6 −1.11538
\(67\) 121784. 0.0494687 0.0247343 0.999694i \(-0.492126\pi\)
0.0247343 + 0.999694i \(0.492126\pi\)
\(68\) 3.18083e6 1.22676
\(69\) −4.30852e6 −1.57891
\(70\) −632675. −0.220464
\(71\) 1.39633e6 0.463004 0.231502 0.972834i \(-0.425636\pi\)
0.231502 + 0.972834i \(0.425636\pi\)
\(72\) 5.58502e6 1.76344
\(73\) 2.85451e6 0.858818 0.429409 0.903110i \(-0.358722\pi\)
0.429409 + 0.903110i \(0.358722\pi\)
\(74\) 3.57965e6 1.02690
\(75\) 634541. 0.173678
\(76\) 1.40716e6 0.367703
\(77\) −1.79236e6 −0.447412
\(78\) −1.09528e6 −0.261334
\(79\) −477949. −0.109065 −0.0545326 0.998512i \(-0.517367\pi\)
−0.0545326 + 0.998512i \(0.517367\pi\)
\(80\) −814361. −0.177829
\(81\) 3.08699e6 0.645413
\(82\) −115842. −0.0232017
\(83\) 7.53259e6 1.44601 0.723005 0.690843i \(-0.242760\pi\)
0.723005 + 0.690843i \(0.242760\pi\)
\(84\) 2.40764e6 0.443214
\(85\) 1.05483e7 1.86301
\(86\) 146710. 0.0248722
\(87\) −878014. −0.142950
\(88\) 7.11054e6 1.11228
\(89\) −9.23609e6 −1.38875 −0.694374 0.719615i \(-0.744319\pi\)
−0.694374 + 0.719615i \(0.744319\pi\)
\(90\) 7.57076e6 1.09469
\(91\) −753571. −0.104828
\(92\) 4.80701e6 0.643603
\(93\) −1.55782e7 −2.00830
\(94\) −6.01329e6 −0.746732
\(95\) 4.66646e6 0.558412
\(96\) −1.51986e7 −1.75329
\(97\) 1.58162e7 1.75955 0.879776 0.475389i \(-0.157693\pi\)
0.879776 + 0.475389i \(0.157693\pi\)
\(98\) −739453. −0.0793632
\(99\) 2.14479e7 2.22157
\(100\) −707957. −0.0707957
\(101\) −1.94673e7 −1.88010 −0.940048 0.341042i \(-0.889220\pi\)
−0.940048 + 0.341042i \(0.889220\pi\)
\(102\) 1.79190e7 1.67191
\(103\) −9.38945e6 −0.846661 −0.423331 0.905975i \(-0.639139\pi\)
−0.423331 + 0.905975i \(0.639139\pi\)
\(104\) 2.98952e6 0.260606
\(105\) 7.98423e6 0.673086
\(106\) 5.86907e6 0.478629
\(107\) −1.46471e7 −1.15587 −0.577935 0.816083i \(-0.696141\pi\)
−0.577935 + 0.816083i \(0.696141\pi\)
\(108\) −1.34591e7 −1.02809
\(109\) −641790. −0.0474679 −0.0237339 0.999718i \(-0.507555\pi\)
−0.0237339 + 0.999718i \(0.507555\pi\)
\(110\) 9.63868e6 0.690468
\(111\) −4.51744e7 −3.13518
\(112\) −951802. −0.0640153
\(113\) 1.24089e6 0.0809019 0.0404509 0.999182i \(-0.487121\pi\)
0.0404509 + 0.999182i \(0.487121\pi\)
\(114\) 7.92720e6 0.501133
\(115\) 1.59411e7 0.977407
\(116\) 979600. 0.0582701
\(117\) 9.01743e6 0.520514
\(118\) 6.98625e6 0.391433
\(119\) 1.23285e7 0.670652
\(120\) −3.16746e7 −1.67331
\(121\) 7.81909e6 0.401243
\(122\) 3.82945e6 0.190931
\(123\) 1.46191e6 0.0708357
\(124\) 1.73806e7 0.818634
\(125\) 2.05796e7 0.942438
\(126\) 8.84849e6 0.394069
\(127\) −3.18169e7 −1.37831 −0.689153 0.724616i \(-0.742017\pi\)
−0.689153 + 0.724616i \(0.742017\pi\)
\(128\) 1.91895e7 0.808778
\(129\) −1.85145e6 −0.0759361
\(130\) 4.05244e6 0.161776
\(131\) −1.22654e7 −0.476685 −0.238343 0.971181i \(-0.576604\pi\)
−0.238343 + 0.971181i \(0.576604\pi\)
\(132\) −3.66799e7 −1.38809
\(133\) 5.45402e6 0.201019
\(134\) −765445. −0.0274819
\(135\) −4.46333e7 −1.56131
\(136\) −4.89091e7 −1.66726
\(137\) 2.63318e7 0.874902 0.437451 0.899242i \(-0.355881\pi\)
0.437451 + 0.899242i \(0.355881\pi\)
\(138\) 2.70801e7 0.877149
\(139\) 5.57051e7 1.75931 0.879657 0.475609i \(-0.157772\pi\)
0.879657 + 0.475609i \(0.157772\pi\)
\(140\) −8.90801e6 −0.274367
\(141\) 7.58866e7 2.27981
\(142\) −8.77629e6 −0.257218
\(143\) 1.14805e7 0.328311
\(144\) 1.13895e7 0.317861
\(145\) 3.24856e6 0.0884918
\(146\) −1.79413e7 −0.477110
\(147\) 9.33175e6 0.242299
\(148\) 5.04011e7 1.27798
\(149\) 3.77273e7 0.934337 0.467168 0.884168i \(-0.345274\pi\)
0.467168 + 0.884168i \(0.345274\pi\)
\(150\) −3.98824e6 −0.0964856
\(151\) 3.00307e6 0.0709816 0.0354908 0.999370i \(-0.488701\pi\)
0.0354908 + 0.999370i \(0.488701\pi\)
\(152\) −2.16369e7 −0.499737
\(153\) −1.47527e8 −3.33005
\(154\) 1.12654e7 0.248556
\(155\) 5.76379e7 1.24322
\(156\) −1.54215e7 −0.325230
\(157\) −9.46951e6 −0.195290 −0.0976448 0.995221i \(-0.531131\pi\)
−0.0976448 + 0.995221i \(0.531131\pi\)
\(158\) 3.00402e6 0.0605904
\(159\) −7.40665e7 −1.46128
\(160\) 5.62332e7 1.08536
\(161\) 1.86315e7 0.351849
\(162\) −1.94025e7 −0.358554
\(163\) 1.72034e6 0.0311142 0.0155571 0.999879i \(-0.495048\pi\)
0.0155571 + 0.999879i \(0.495048\pi\)
\(164\) −1.63105e6 −0.0288745
\(165\) −1.21638e8 −2.10803
\(166\) −4.73442e7 −0.803320
\(167\) −1.00841e8 −1.67544 −0.837718 0.546103i \(-0.816111\pi\)
−0.837718 + 0.546103i \(0.816111\pi\)
\(168\) −3.70203e7 −0.602362
\(169\) 4.82681e6 0.0769231
\(170\) −6.62986e7 −1.03498
\(171\) −6.52643e7 −0.998135
\(172\) 2.06566e6 0.0309535
\(173\) 1.04816e8 1.53910 0.769550 0.638586i \(-0.220480\pi\)
0.769550 + 0.638586i \(0.220480\pi\)
\(174\) 5.51853e6 0.0794148
\(175\) −2.74397e6 −0.0387031
\(176\) 1.45005e7 0.200488
\(177\) −8.81651e7 −1.19506
\(178\) 5.80511e7 0.771508
\(179\) −5.81700e7 −0.758077 −0.379038 0.925381i \(-0.623745\pi\)
−0.379038 + 0.925381i \(0.623745\pi\)
\(180\) 1.06596e8 1.36234
\(181\) 1.19705e8 1.50051 0.750255 0.661149i \(-0.229931\pi\)
0.750255 + 0.661149i \(0.229931\pi\)
\(182\) 4.73638e6 0.0582367
\(183\) −4.83269e7 −0.582922
\(184\) −7.39137e7 −0.874707
\(185\) 1.67141e8 1.94080
\(186\) 9.79130e7 1.11569
\(187\) −1.87823e8 −2.10041
\(188\) −8.46666e7 −0.929308
\(189\) −5.21661e7 −0.562046
\(190\) −2.93298e7 −0.310222
\(191\) −1.09441e8 −1.13648 −0.568240 0.822863i \(-0.692375\pi\)
−0.568240 + 0.822863i \(0.692375\pi\)
\(192\) 6.73535e7 0.686761
\(193\) 5.00461e7 0.501094 0.250547 0.968104i \(-0.419390\pi\)
0.250547 + 0.968104i \(0.419390\pi\)
\(194\) −9.94089e7 −0.977506
\(195\) −5.11410e7 −0.493910
\(196\) −1.04114e7 −0.0987675
\(197\) 1.05766e7 0.0985627 0.0492814 0.998785i \(-0.484307\pi\)
0.0492814 + 0.998785i \(0.484307\pi\)
\(198\) −1.34805e8 −1.23418
\(199\) 1.76419e8 1.58693 0.793466 0.608614i \(-0.208274\pi\)
0.793466 + 0.608614i \(0.208274\pi\)
\(200\) 1.08857e7 0.0962169
\(201\) 9.65976e6 0.0839036
\(202\) 1.22357e8 1.04447
\(203\) 3.79683e6 0.0318555
\(204\) 2.52298e8 2.08070
\(205\) −5.40891e6 −0.0438501
\(206\) 5.90150e7 0.470356
\(207\) −2.22949e8 −1.74707
\(208\) 6.09653e6 0.0469744
\(209\) −8.30910e7 −0.629567
\(210\) −5.01829e7 −0.373928
\(211\) −2.13989e8 −1.56820 −0.784102 0.620632i \(-0.786876\pi\)
−0.784102 + 0.620632i \(0.786876\pi\)
\(212\) 8.26360e7 0.595654
\(213\) 1.10755e8 0.785299
\(214\) 9.20607e7 0.642135
\(215\) 6.85018e6 0.0470075
\(216\) 2.06950e8 1.39726
\(217\) 6.73655e7 0.447537
\(218\) 4.03381e6 0.0263704
\(219\) 2.26415e8 1.45664
\(220\) 1.35712e8 0.859287
\(221\) −7.89674e7 −0.492124
\(222\) 2.83932e8 1.74172
\(223\) −1.09968e8 −0.664048 −0.332024 0.943271i \(-0.607731\pi\)
−0.332024 + 0.943271i \(0.607731\pi\)
\(224\) 6.57237e7 0.390710
\(225\) 3.28350e7 0.192176
\(226\) −7.79929e6 −0.0449444
\(227\) 1.55040e8 0.879735 0.439868 0.898063i \(-0.355025\pi\)
0.439868 + 0.898063i \(0.355025\pi\)
\(228\) 1.11614e8 0.623660
\(229\) 3.45066e8 1.89879 0.949397 0.314078i \(-0.101695\pi\)
0.949397 + 0.314078i \(0.101695\pi\)
\(230\) −1.00194e8 −0.542991
\(231\) −1.42167e8 −0.758854
\(232\) −1.50625e7 −0.0791936
\(233\) −6.20447e7 −0.321336 −0.160668 0.987009i \(-0.551365\pi\)
−0.160668 + 0.987009i \(0.551365\pi\)
\(234\) −5.66768e7 −0.289168
\(235\) −2.80773e8 −1.41129
\(236\) 9.83658e7 0.487138
\(237\) −3.79102e7 −0.184985
\(238\) −7.74879e7 −0.372576
\(239\) −2.61194e8 −1.23757 −0.618787 0.785559i \(-0.712376\pi\)
−0.618787 + 0.785559i \(0.712376\pi\)
\(240\) −6.45939e7 −0.301615
\(241\) −1.02075e8 −0.469741 −0.234870 0.972027i \(-0.575467\pi\)
−0.234870 + 0.972027i \(0.575467\pi\)
\(242\) −4.91449e7 −0.222907
\(243\) −8.77605e7 −0.392353
\(244\) 5.39184e7 0.237614
\(245\) −3.45265e7 −0.149993
\(246\) −9.18845e6 −0.0393522
\(247\) −3.49344e7 −0.147507
\(248\) −2.67248e8 −1.11259
\(249\) 5.97474e8 2.45257
\(250\) −1.29348e8 −0.523564
\(251\) −2.93524e7 −0.117162 −0.0585809 0.998283i \(-0.518658\pi\)
−0.0585809 + 0.998283i \(0.518658\pi\)
\(252\) 1.24586e8 0.490419
\(253\) −2.83847e8 −1.10195
\(254\) 1.99977e8 0.765707
\(255\) 8.36675e8 3.15985
\(256\) −2.29302e8 −0.854218
\(257\) −3.72786e8 −1.36992 −0.684958 0.728583i \(-0.740179\pi\)
−0.684958 + 0.728583i \(0.740179\pi\)
\(258\) 1.16368e7 0.0421857
\(259\) 1.95349e8 0.698655
\(260\) 5.70580e7 0.201331
\(261\) −4.54339e7 −0.158175
\(262\) 7.70910e7 0.264819
\(263\) −2.46625e8 −0.835972 −0.417986 0.908453i \(-0.637264\pi\)
−0.417986 + 0.908453i \(0.637264\pi\)
\(264\) 5.63998e8 1.88653
\(265\) 2.74039e8 0.904589
\(266\) −3.42799e7 −0.111674
\(267\) −7.32593e8 −2.35545
\(268\) −1.07774e7 −0.0342013
\(269\) −4.40801e8 −1.38073 −0.690367 0.723459i \(-0.742551\pi\)
−0.690367 + 0.723459i \(0.742551\pi\)
\(270\) 2.80531e8 0.867377
\(271\) 4.61850e8 1.40964 0.704821 0.709386i \(-0.251028\pi\)
0.704821 + 0.709386i \(0.251028\pi\)
\(272\) −9.97403e7 −0.300524
\(273\) −5.97721e7 −0.177799
\(274\) −1.65502e8 −0.486045
\(275\) 4.18038e7 0.121214
\(276\) 3.81285e8 1.09161
\(277\) 5.21353e8 1.47385 0.736924 0.675976i \(-0.236278\pi\)
0.736924 + 0.675976i \(0.236278\pi\)
\(278\) −3.50120e8 −0.977373
\(279\) −8.06114e8 −2.22219
\(280\) 1.36971e8 0.372887
\(281\) 2.46808e8 0.663571 0.331785 0.943355i \(-0.392349\pi\)
0.331785 + 0.943355i \(0.392349\pi\)
\(282\) −4.76966e8 −1.26653
\(283\) 5.37098e8 1.40864 0.704321 0.709881i \(-0.251251\pi\)
0.704321 + 0.709881i \(0.251251\pi\)
\(284\) −1.23569e8 −0.320108
\(285\) 3.70137e8 0.947120
\(286\) −7.21578e7 −0.182391
\(287\) −6.32178e6 −0.0157853
\(288\) −7.86468e8 −1.94002
\(289\) 8.81581e8 2.14842
\(290\) −2.04180e7 −0.0491610
\(291\) 1.25452e9 2.98437
\(292\) −2.52612e8 −0.593763
\(293\) −7.71723e8 −1.79236 −0.896179 0.443693i \(-0.853668\pi\)
−0.896179 + 0.443693i \(0.853668\pi\)
\(294\) −5.86523e7 −0.134608
\(295\) 3.26202e8 0.739791
\(296\) −7.74978e8 −1.73687
\(297\) 7.94740e8 1.76026
\(298\) −2.37125e8 −0.519064
\(299\) −1.19339e8 −0.258187
\(300\) −5.61541e7 −0.120076
\(301\) 8.00629e6 0.0169219
\(302\) −1.88750e7 −0.0394333
\(303\) −1.54411e9 −3.18882
\(304\) −4.41241e7 −0.0900779
\(305\) 1.78805e8 0.360852
\(306\) 9.27242e8 1.84998
\(307\) 1.30784e8 0.257971 0.128986 0.991646i \(-0.458828\pi\)
0.128986 + 0.991646i \(0.458828\pi\)
\(308\) 1.58616e8 0.309328
\(309\) −7.44757e8 −1.43602
\(310\) −3.62268e8 −0.690660
\(311\) −5.05719e8 −0.953340 −0.476670 0.879082i \(-0.658156\pi\)
−0.476670 + 0.879082i \(0.658156\pi\)
\(312\) 2.37124e8 0.442013
\(313\) 3.25829e8 0.600599 0.300299 0.953845i \(-0.402913\pi\)
0.300299 + 0.953845i \(0.402913\pi\)
\(314\) 5.95182e7 0.108492
\(315\) 4.13153e8 0.744774
\(316\) 4.22964e7 0.0754047
\(317\) 2.83996e8 0.500732 0.250366 0.968151i \(-0.419449\pi\)
0.250366 + 0.968151i \(0.419449\pi\)
\(318\) 4.65526e8 0.811801
\(319\) −5.78439e7 −0.0997678
\(320\) −2.49201e8 −0.425133
\(321\) −1.16179e9 −1.96047
\(322\) −1.17103e8 −0.195467
\(323\) 5.71532e8 0.943695
\(324\) −2.73185e8 −0.446221
\(325\) 1.75758e7 0.0284003
\(326\) −1.08128e7 −0.0172852
\(327\) −5.09058e7 −0.0805101
\(328\) 2.50794e7 0.0392426
\(329\) −3.28159e8 −0.508041
\(330\) 7.64526e8 1.17110
\(331\) −9.22687e8 −1.39848 −0.699241 0.714886i \(-0.746478\pi\)
−0.699241 + 0.714886i \(0.746478\pi\)
\(332\) −6.66602e8 −0.999731
\(333\) −2.33760e9 −3.46909
\(334\) 6.33808e8 0.930776
\(335\) −3.57401e7 −0.0519397
\(336\) −7.54956e7 −0.108576
\(337\) 6.01648e7 0.0856323 0.0428161 0.999083i \(-0.486367\pi\)
0.0428161 + 0.999083i \(0.486367\pi\)
\(338\) −3.03377e7 −0.0427340
\(339\) 9.84255e7 0.137217
\(340\) −9.33478e8 −1.28804
\(341\) −1.02630e9 −1.40163
\(342\) 4.10202e8 0.554506
\(343\) −4.03536e7 −0.0539949
\(344\) −3.17621e7 −0.0420682
\(345\) 1.26442e9 1.65777
\(346\) −6.58795e8 −0.855036
\(347\) 1.08011e9 1.38776 0.693881 0.720090i \(-0.255900\pi\)
0.693881 + 0.720090i \(0.255900\pi\)
\(348\) 7.77005e7 0.0988317
\(349\) −6.96677e8 −0.877288 −0.438644 0.898661i \(-0.644541\pi\)
−0.438644 + 0.898661i \(0.644541\pi\)
\(350\) 1.72465e7 0.0215012
\(351\) 3.34137e8 0.412429
\(352\) −1.00129e9 −1.22366
\(353\) −2.15539e7 −0.0260804 −0.0130402 0.999915i \(-0.504151\pi\)
−0.0130402 + 0.999915i \(0.504151\pi\)
\(354\) 5.54139e8 0.663907
\(355\) −4.09782e8 −0.486132
\(356\) 8.17354e8 0.960142
\(357\) 9.77882e8 1.13749
\(358\) 3.65613e8 0.421144
\(359\) −1.02429e9 −1.16840 −0.584200 0.811609i \(-0.698592\pi\)
−0.584200 + 0.811609i \(0.698592\pi\)
\(360\) −1.63904e9 −1.85153
\(361\) −6.41032e8 −0.717141
\(362\) −7.52378e8 −0.833597
\(363\) 6.20198e8 0.680546
\(364\) 6.66878e7 0.0724755
\(365\) −8.37714e8 −0.901717
\(366\) 3.03747e8 0.323838
\(367\) −746813. −0.000788643 0 −0.000394321 1.00000i \(-0.500126\pi\)
−0.000394321 1.00000i \(0.500126\pi\)
\(368\) −1.50732e8 −0.157666
\(369\) 7.56481e7 0.0783801
\(370\) −1.05052e9 −1.07820
\(371\) 3.20289e8 0.325636
\(372\) 1.37861e9 1.38848
\(373\) 1.07550e9 1.07307 0.536535 0.843878i \(-0.319733\pi\)
0.536535 + 0.843878i \(0.319733\pi\)
\(374\) 1.18051e9 1.16686
\(375\) 1.63235e9 1.59846
\(376\) 1.30185e9 1.26300
\(377\) −2.43196e7 −0.0233756
\(378\) 3.27877e8 0.312241
\(379\) 1.61102e9 1.52007 0.760037 0.649880i \(-0.225181\pi\)
0.760037 + 0.649880i \(0.225181\pi\)
\(380\) −4.12961e8 −0.386071
\(381\) −2.52367e9 −2.33774
\(382\) 6.87861e8 0.631362
\(383\) 7.52684e8 0.684568 0.342284 0.939596i \(-0.388799\pi\)
0.342284 + 0.939596i \(0.388799\pi\)
\(384\) 1.52208e9 1.37176
\(385\) 5.26005e8 0.469761
\(386\) −3.14552e8 −0.278379
\(387\) −9.58054e7 −0.0840237
\(388\) −1.39967e9 −1.21651
\(389\) 1.42901e9 1.23087 0.615434 0.788188i \(-0.288981\pi\)
0.615434 + 0.788188i \(0.288981\pi\)
\(390\) 3.21434e8 0.274388
\(391\) 1.95241e9 1.65178
\(392\) 1.60088e8 0.134233
\(393\) −9.72872e8 −0.808504
\(394\) −6.64763e7 −0.0547558
\(395\) 1.40264e8 0.114513
\(396\) −1.89804e9 −1.53594
\(397\) 1.04601e9 0.839017 0.419509 0.907751i \(-0.362202\pi\)
0.419509 + 0.907751i \(0.362202\pi\)
\(398\) −1.10883e9 −0.881609
\(399\) 4.32605e8 0.340947
\(400\) 2.21992e7 0.0173431
\(401\) 6.70774e7 0.0519483 0.0259741 0.999663i \(-0.491731\pi\)
0.0259741 + 0.999663i \(0.491731\pi\)
\(402\) −6.07140e7 −0.0466120
\(403\) −4.31493e8 −0.328402
\(404\) 1.72277e9 1.29985
\(405\) −9.05940e8 −0.677652
\(406\) −2.38640e7 −0.0176971
\(407\) −2.97611e9 −2.18811
\(408\) −3.87939e9 −2.82783
\(409\) −1.12516e9 −0.813173 −0.406587 0.913612i \(-0.633281\pi\)
−0.406587 + 0.913612i \(0.633281\pi\)
\(410\) 3.39963e7 0.0243606
\(411\) 2.08860e9 1.48392
\(412\) 8.30926e8 0.585358
\(413\) 3.81255e8 0.266312
\(414\) 1.40129e9 0.970571
\(415\) −2.21059e9 −1.51824
\(416\) −4.20977e8 −0.286703
\(417\) 4.41845e9 2.98396
\(418\) 5.22247e8 0.349751
\(419\) 2.20420e9 1.46387 0.731933 0.681377i \(-0.238619\pi\)
0.731933 + 0.681377i \(0.238619\pi\)
\(420\) −7.06570e8 −0.465353
\(421\) −1.40920e9 −0.920418 −0.460209 0.887811i \(-0.652226\pi\)
−0.460209 + 0.887811i \(0.652226\pi\)
\(422\) 1.34497e9 0.871204
\(423\) 3.92684e9 2.52262
\(424\) −1.27063e9 −0.809540
\(425\) −2.87543e8 −0.181694
\(426\) −6.96123e8 −0.436267
\(427\) 2.08982e8 0.129901
\(428\) 1.29621e9 0.799137
\(429\) 9.10617e8 0.556846
\(430\) −4.30550e7 −0.0261147
\(431\) 2.63366e9 1.58449 0.792245 0.610204i \(-0.208912\pi\)
0.792245 + 0.610204i \(0.208912\pi\)
\(432\) 4.22033e8 0.251857
\(433\) 2.83906e9 1.68061 0.840306 0.542112i \(-0.182375\pi\)
0.840306 + 0.542112i \(0.182375\pi\)
\(434\) −4.23409e8 −0.248626
\(435\) 2.57671e8 0.150091
\(436\) 5.67956e7 0.0328180
\(437\) 8.63726e8 0.495098
\(438\) −1.42308e9 −0.809224
\(439\) −1.92343e9 −1.08505 −0.542527 0.840039i \(-0.682532\pi\)
−0.542527 + 0.840039i \(0.682532\pi\)
\(440\) −2.08673e9 −1.16784
\(441\) 4.82882e8 0.268106
\(442\) 4.96330e8 0.273396
\(443\) −3.37999e9 −1.84715 −0.923575 0.383417i \(-0.874747\pi\)
−0.923575 + 0.383417i \(0.874747\pi\)
\(444\) 3.99774e9 2.16758
\(445\) 2.71052e9 1.45812
\(446\) 6.91176e8 0.368907
\(447\) 2.99247e9 1.58472
\(448\) −2.91259e8 −0.153041
\(449\) −7.47706e8 −0.389824 −0.194912 0.980821i \(-0.562442\pi\)
−0.194912 + 0.980821i \(0.562442\pi\)
\(450\) −2.06376e8 −0.106762
\(451\) 9.63111e7 0.0494377
\(452\) −1.09813e8 −0.0559333
\(453\) 2.38199e8 0.120392
\(454\) −9.74462e8 −0.488730
\(455\) 2.21151e8 0.110065
\(456\) −1.71620e9 −0.847602
\(457\) 1.37231e9 0.672582 0.336291 0.941758i \(-0.390827\pi\)
0.336291 + 0.941758i \(0.390827\pi\)
\(458\) −2.16882e9 −1.05486
\(459\) −5.46653e9 −2.63856
\(460\) −1.41072e9 −0.675752
\(461\) −2.64939e9 −1.25948 −0.629741 0.776805i \(-0.716839\pi\)
−0.629741 + 0.776805i \(0.716839\pi\)
\(462\) 8.93557e8 0.421575
\(463\) −2.32553e9 −1.08890 −0.544451 0.838793i \(-0.683262\pi\)
−0.544451 + 0.838793i \(0.683262\pi\)
\(464\) −3.07170e7 −0.0142747
\(465\) 4.57175e9 2.10862
\(466\) 3.89966e8 0.178516
\(467\) −2.60290e9 −1.18263 −0.591315 0.806440i \(-0.701391\pi\)
−0.591315 + 0.806440i \(0.701391\pi\)
\(468\) −7.98004e8 −0.359869
\(469\) −4.17721e7 −0.0186974
\(470\) 1.76472e9 0.784033
\(471\) −7.51108e8 −0.331230
\(472\) −1.51249e9 −0.662058
\(473\) −1.21974e8 −0.0529974
\(474\) 2.38275e8 0.102767
\(475\) −1.27206e8 −0.0544603
\(476\) −1.09102e9 −0.463671
\(477\) −3.83266e9 −1.61691
\(478\) 1.64167e9 0.687525
\(479\) 9.63575e8 0.400600 0.200300 0.979735i \(-0.435808\pi\)
0.200300 + 0.979735i \(0.435808\pi\)
\(480\) 4.46033e9 1.84087
\(481\) −1.25126e9 −0.512673
\(482\) 6.41564e8 0.260961
\(483\) 1.47782e9 0.596770
\(484\) −6.91955e8 −0.277408
\(485\) −4.64160e9 −1.84744
\(486\) 5.51596e8 0.217969
\(487\) −3.27359e9 −1.28432 −0.642159 0.766572i \(-0.721961\pi\)
−0.642159 + 0.766572i \(0.721961\pi\)
\(488\) −8.29060e8 −0.322936
\(489\) 1.36455e8 0.0527726
\(490\) 2.17008e8 0.0833275
\(491\) 1.73467e9 0.661351 0.330675 0.943745i \(-0.392723\pi\)
0.330675 + 0.943745i \(0.392723\pi\)
\(492\) −1.29372e8 −0.0489738
\(493\) 3.97873e8 0.149548
\(494\) 2.19571e8 0.0819466
\(495\) −6.29431e9 −2.33255
\(496\) −5.45000e8 −0.200544
\(497\) −4.78942e8 −0.174999
\(498\) −3.75527e9 −1.36251
\(499\) −1.21566e9 −0.437987 −0.218994 0.975726i \(-0.570277\pi\)
−0.218994 + 0.975726i \(0.570277\pi\)
\(500\) −1.82121e9 −0.651576
\(501\) −7.99853e9 −2.84170
\(502\) 1.84487e8 0.0650884
\(503\) −1.95400e9 −0.684600 −0.342300 0.939591i \(-0.611206\pi\)
−0.342300 + 0.939591i \(0.611206\pi\)
\(504\) −1.91566e9 −0.666518
\(505\) 5.71307e9 1.97401
\(506\) 1.78405e9 0.612181
\(507\) 3.82855e8 0.130469
\(508\) 2.81566e9 0.952923
\(509\) 8.35731e8 0.280902 0.140451 0.990088i \(-0.455145\pi\)
0.140451 + 0.990088i \(0.455145\pi\)
\(510\) −5.25871e9 −1.75543
\(511\) −9.79096e8 −0.324603
\(512\) −1.01504e9 −0.334223
\(513\) −2.41834e9 −0.790872
\(514\) 2.34305e9 0.761046
\(515\) 2.75553e9 0.888954
\(516\) 1.63845e8 0.0525001
\(517\) 4.99944e9 1.59113
\(518\) −1.22782e9 −0.388133
\(519\) 8.31387e9 2.61046
\(520\) −8.77336e8 −0.273624
\(521\) −3.99708e8 −0.123826 −0.0619128 0.998082i \(-0.519720\pi\)
−0.0619128 + 0.998082i \(0.519720\pi\)
\(522\) 2.85563e8 0.0878729
\(523\) 1.45373e9 0.444354 0.222177 0.975006i \(-0.428684\pi\)
0.222177 + 0.975006i \(0.428684\pi\)
\(524\) 1.08543e9 0.329567
\(525\) −2.17647e8 −0.0656441
\(526\) 1.55010e9 0.464418
\(527\) 7.05930e9 2.10099
\(528\) 1.15016e9 0.340048
\(529\) −4.54250e8 −0.133414
\(530\) −1.72240e9 −0.502537
\(531\) −4.56220e9 −1.32234
\(532\) −4.82658e8 −0.138979
\(533\) 4.04926e7 0.0115832
\(534\) 4.60453e9 1.30855
\(535\) 4.29850e9 1.21361
\(536\) 1.65716e8 0.0464822
\(537\) −4.61396e9 −1.28577
\(538\) 2.77054e9 0.767056
\(539\) 6.14779e8 0.169106
\(540\) 3.94985e9 1.07945
\(541\) −1.35141e9 −0.366940 −0.183470 0.983025i \(-0.558733\pi\)
−0.183470 + 0.983025i \(0.558733\pi\)
\(542\) −2.90284e9 −0.783115
\(543\) 9.49486e9 2.54501
\(544\) 6.88725e9 1.83421
\(545\) 1.88346e8 0.0498390
\(546\) 3.75683e8 0.0987750
\(547\) 3.27852e9 0.856491 0.428245 0.903662i \(-0.359132\pi\)
0.428245 + 0.903662i \(0.359132\pi\)
\(548\) −2.33025e9 −0.604883
\(549\) −2.50073e9 −0.645007
\(550\) −2.62747e8 −0.0673393
\(551\) 1.76015e8 0.0448249
\(552\) −5.86273e9 −1.48359
\(553\) 1.63936e8 0.0412228
\(554\) −3.27683e9 −0.818785
\(555\) 1.32574e10 3.29179
\(556\) −4.92966e9 −1.21634
\(557\) −4.62141e9 −1.13313 −0.566567 0.824016i \(-0.691729\pi\)
−0.566567 + 0.824016i \(0.691729\pi\)
\(558\) 5.06662e9 1.23452
\(559\) −5.12823e7 −0.0124173
\(560\) 2.79326e8 0.0672130
\(561\) −1.48978e10 −3.56249
\(562\) −1.55125e9 −0.368642
\(563\) 4.93762e9 1.16611 0.583054 0.812434i \(-0.301858\pi\)
0.583054 + 0.812434i \(0.301858\pi\)
\(564\) −6.71563e9 −1.57620
\(565\) −3.64164e8 −0.0849430
\(566\) −3.37579e9 −0.782561
\(567\) −1.05884e9 −0.243943
\(568\) 1.90003e9 0.435052
\(569\) −6.57550e7 −0.0149636 −0.00748180 0.999972i \(-0.502382\pi\)
−0.00748180 + 0.999972i \(0.502382\pi\)
\(570\) −2.32640e9 −0.526165
\(571\) 2.78142e9 0.625231 0.312615 0.949880i \(-0.398795\pi\)
0.312615 + 0.949880i \(0.398795\pi\)
\(572\) −1.01598e9 −0.226985
\(573\) −8.68066e9 −1.92758
\(574\) 3.97339e7 0.00876940
\(575\) −4.34548e8 −0.0953236
\(576\) 3.48529e9 0.759906
\(577\) 1.91932e9 0.415940 0.207970 0.978135i \(-0.433314\pi\)
0.207970 + 0.978135i \(0.433314\pi\)
\(578\) −5.54095e9 −1.19354
\(579\) 3.96958e9 0.849903
\(580\) −2.87484e8 −0.0611808
\(581\) −2.58368e9 −0.546540
\(582\) −7.88497e9 −1.65794
\(583\) −4.87954e9 −1.01986
\(584\) 3.88421e9 0.806971
\(585\) −2.64635e9 −0.546515
\(586\) 4.85047e9 0.995731
\(587\) −3.65483e9 −0.745820 −0.372910 0.927868i \(-0.621640\pi\)
−0.372910 + 0.927868i \(0.621640\pi\)
\(588\) −8.25819e8 −0.167519
\(589\) 3.12296e9 0.629742
\(590\) −2.05026e9 −0.410985
\(591\) 8.38917e8 0.167172
\(592\) −1.58041e9 −0.313072
\(593\) −5.54651e9 −1.09227 −0.546133 0.837698i \(-0.683901\pi\)
−0.546133 + 0.837698i \(0.683901\pi\)
\(594\) −4.99514e9 −0.977901
\(595\) −3.61806e9 −0.704153
\(596\) −3.33870e9 −0.645975
\(597\) 1.39933e10 2.69159
\(598\) 7.50077e8 0.143434
\(599\) −4.44815e9 −0.845641 −0.422820 0.906214i \(-0.638960\pi\)
−0.422820 + 0.906214i \(0.638960\pi\)
\(600\) 8.63438e8 0.163193
\(601\) −2.18830e9 −0.411194 −0.205597 0.978637i \(-0.565914\pi\)
−0.205597 + 0.978637i \(0.565914\pi\)
\(602\) −5.03215e7 −0.00940083
\(603\) 4.99856e8 0.0928398
\(604\) −2.65759e8 −0.0490748
\(605\) −2.29467e9 −0.421286
\(606\) 9.70514e9 1.77153
\(607\) 3.10528e9 0.563561 0.281780 0.959479i \(-0.409075\pi\)
0.281780 + 0.959479i \(0.409075\pi\)
\(608\) 3.04685e9 0.549779
\(609\) 3.01159e8 0.0540300
\(610\) −1.12383e9 −0.200469
\(611\) 2.10194e9 0.372800
\(612\) 1.30555e10 2.30230
\(613\) −4.84819e9 −0.850096 −0.425048 0.905171i \(-0.639743\pi\)
−0.425048 + 0.905171i \(0.639743\pi\)
\(614\) −8.22012e8 −0.143314
\(615\) −4.29027e8 −0.0743740
\(616\) −2.43892e9 −0.420401
\(617\) 4.98963e9 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(618\) 4.68098e9 0.797769
\(619\) −3.67596e9 −0.622951 −0.311475 0.950254i \(-0.600823\pi\)
−0.311475 + 0.950254i \(0.600823\pi\)
\(620\) −5.10070e9 −0.859526
\(621\) −8.26128e9 −1.38429
\(622\) 3.17857e9 0.529621
\(623\) 3.16798e9 0.524897
\(624\) 4.83568e8 0.0796730
\(625\) −6.66451e9 −1.09191
\(626\) −2.04791e9 −0.333658
\(627\) −6.59066e9 −1.06781
\(628\) 8.38011e8 0.135018
\(629\) 2.04708e10 3.27988
\(630\) −2.59677e9 −0.413754
\(631\) −1.78403e9 −0.282682 −0.141341 0.989961i \(-0.545141\pi\)
−0.141341 + 0.989961i \(0.545141\pi\)
\(632\) −6.50358e8 −0.102481
\(633\) −1.69733e10 −2.65982
\(634\) −1.78499e9 −0.278178
\(635\) 9.33733e9 1.44715
\(636\) 6.55457e9 1.01029
\(637\) 2.58475e8 0.0396214
\(638\) 3.63563e8 0.0554253
\(639\) 5.73115e9 0.868938
\(640\) −5.63156e9 −0.849177
\(641\) 5.61163e9 0.841562 0.420781 0.907162i \(-0.361756\pi\)
0.420781 + 0.907162i \(0.361756\pi\)
\(642\) 7.30212e9 1.08912
\(643\) 2.13249e9 0.316337 0.158168 0.987412i \(-0.449441\pi\)
0.158168 + 0.987412i \(0.449441\pi\)
\(644\) −1.64881e9 −0.243259
\(645\) 5.43346e8 0.0797292
\(646\) −3.59222e9 −0.524263
\(647\) −6.37987e9 −0.926077 −0.463038 0.886338i \(-0.653241\pi\)
−0.463038 + 0.886338i \(0.653241\pi\)
\(648\) 4.20055e9 0.606449
\(649\) −5.80835e9 −0.834058
\(650\) −1.10468e8 −0.0157776
\(651\) 5.34333e9 0.759065
\(652\) −1.52243e8 −0.0215115
\(653\) −4.54878e9 −0.639292 −0.319646 0.947537i \(-0.603564\pi\)
−0.319646 + 0.947537i \(0.603564\pi\)
\(654\) 3.19956e8 0.0447268
\(655\) 3.59953e9 0.500497
\(656\) 5.11444e7 0.00707350
\(657\) 1.17161e10 1.61178
\(658\) 2.06256e9 0.282238
\(659\) 3.40787e9 0.463857 0.231928 0.972733i \(-0.425497\pi\)
0.231928 + 0.972733i \(0.425497\pi\)
\(660\) 1.07645e10 1.45743
\(661\) 2.52665e9 0.340283 0.170142 0.985420i \(-0.445577\pi\)
0.170142 + 0.985420i \(0.445577\pi\)
\(662\) 5.79932e9 0.776916
\(663\) −6.26358e9 −0.834690
\(664\) 1.02498e10 1.35871
\(665\) −1.60059e9 −0.211060
\(666\) 1.46924e10 1.92723
\(667\) 6.01285e8 0.0784585
\(668\) 8.92396e9 1.15835
\(669\) −8.72250e9 −1.12629
\(670\) 2.24636e8 0.0288547
\(671\) −3.18380e9 −0.406834
\(672\) 5.21311e9 0.662681
\(673\) 2.92213e9 0.369528 0.184764 0.982783i \(-0.440848\pi\)
0.184764 + 0.982783i \(0.440848\pi\)
\(674\) −3.78150e8 −0.0475724
\(675\) 1.21669e9 0.152270
\(676\) −4.27152e8 −0.0531825
\(677\) −1.57979e10 −1.95676 −0.978382 0.206807i \(-0.933693\pi\)
−0.978382 + 0.206807i \(0.933693\pi\)
\(678\) −6.18629e8 −0.0762300
\(679\) −5.42497e9 −0.665048
\(680\) 1.43534e10 1.75054
\(681\) 1.22975e10 1.49212
\(682\) 6.45055e9 0.778667
\(683\) 1.12352e10 1.34930 0.674648 0.738140i \(-0.264296\pi\)
0.674648 + 0.738140i \(0.264296\pi\)
\(684\) 5.77561e9 0.690083
\(685\) −7.72762e9 −0.918604
\(686\) 2.53632e8 0.0299965
\(687\) 2.73701e10 3.22054
\(688\) −6.47724e7 −0.00758282
\(689\) −2.05153e9 −0.238952
\(690\) −7.94721e9 −0.920964
\(691\) 3.86818e9 0.445999 0.223000 0.974819i \(-0.428415\pi\)
0.223000 + 0.974819i \(0.428415\pi\)
\(692\) −9.27578e9 −1.06409
\(693\) −7.35662e9 −0.839676
\(694\) −6.78876e9 −0.770960
\(695\) −1.63478e10 −1.84719
\(696\) −1.19474e9 −0.134320
\(697\) −6.62465e8 −0.0741051
\(698\) 4.37878e9 0.487371
\(699\) −4.92130e9 −0.545016
\(700\) 2.42829e8 0.0267583
\(701\) −1.21215e10 −1.32906 −0.664529 0.747263i \(-0.731368\pi\)
−0.664529 + 0.747263i \(0.731368\pi\)
\(702\) −2.10013e9 −0.229122
\(703\) 9.05609e9 0.983099
\(704\) 4.43728e9 0.479305
\(705\) −2.22705e10 −2.39369
\(706\) 1.35471e8 0.0144888
\(707\) 6.67727e9 0.710610
\(708\) 7.80223e9 0.826232
\(709\) 1.05985e10 1.11682 0.558410 0.829565i \(-0.311412\pi\)
0.558410 + 0.829565i \(0.311412\pi\)
\(710\) 2.57558e9 0.270067
\(711\) −1.96171e9 −0.204687
\(712\) −1.25678e10 −1.30491
\(713\) 1.06683e10 1.10226
\(714\) −6.14623e9 −0.631924
\(715\) −3.36919e9 −0.344710
\(716\) 5.14779e9 0.524114
\(717\) −2.07176e10 −2.09904
\(718\) 6.43791e9 0.649096
\(719\) −1.13284e10 −1.13663 −0.568313 0.822813i \(-0.692404\pi\)
−0.568313 + 0.822813i \(0.692404\pi\)
\(720\) −3.34249e9 −0.333738
\(721\) 3.22058e9 0.320008
\(722\) 4.02904e9 0.398402
\(723\) −8.09641e9 −0.796725
\(724\) −1.05934e10 −1.03741
\(725\) −8.85547e7 −0.00863035
\(726\) −3.89810e9 −0.378072
\(727\) 1.01385e9 0.0978594 0.0489297 0.998802i \(-0.484419\pi\)
0.0489297 + 0.998802i \(0.484419\pi\)
\(728\) −1.02541e9 −0.0984999
\(729\) −1.37123e10 −1.31088
\(730\) 5.26524e9 0.500942
\(731\) 8.38987e8 0.0794409
\(732\) 4.27673e9 0.403016
\(733\) 6.74736e9 0.632805 0.316403 0.948625i \(-0.397525\pi\)
0.316403 + 0.948625i \(0.397525\pi\)
\(734\) 4.69390e6 0.000438125 0
\(735\) −2.73859e9 −0.254403
\(736\) 1.04083e10 0.962297
\(737\) 6.36389e8 0.0585581
\(738\) −4.75467e8 −0.0435435
\(739\) −1.61384e10 −1.47097 −0.735486 0.677540i \(-0.763046\pi\)
−0.735486 + 0.677540i \(0.763046\pi\)
\(740\) −1.47912e10 −1.34182
\(741\) −2.77094e9 −0.250187
\(742\) −2.01309e9 −0.180905
\(743\) −1.55945e10 −1.39479 −0.697397 0.716685i \(-0.745659\pi\)
−0.697397 + 0.716685i \(0.745659\pi\)
\(744\) −2.11978e10 −1.88705
\(745\) −1.10718e10 −0.981008
\(746\) −6.75976e9 −0.596136
\(747\) 3.09170e10 2.71378
\(748\) 1.66215e10 1.45216
\(749\) 5.02396e9 0.436878
\(750\) −1.02597e10 −0.888015
\(751\) −7.60712e9 −0.655361 −0.327680 0.944789i \(-0.606267\pi\)
−0.327680 + 0.944789i \(0.606267\pi\)
\(752\) 2.65487e9 0.227657
\(753\) −2.32819e9 −0.198718
\(754\) 1.52855e8 0.0129861
\(755\) −8.81312e8 −0.0745273
\(756\) 4.61647e9 0.388583
\(757\) 1.29750e10 1.08711 0.543554 0.839374i \(-0.317078\pi\)
0.543554 + 0.839374i \(0.317078\pi\)
\(758\) −1.01257e10 −0.844465
\(759\) −2.25143e10 −1.86902
\(760\) 6.34978e9 0.524700
\(761\) −1.26415e10 −1.03981 −0.519903 0.854225i \(-0.674032\pi\)
−0.519903 + 0.854225i \(0.674032\pi\)
\(762\) 1.58619e10 1.29871
\(763\) 2.20134e8 0.0179412
\(764\) 9.68502e9 0.785731
\(765\) 4.32947e10 3.49639
\(766\) −4.73080e9 −0.380307
\(767\) −2.44203e9 −0.195420
\(768\) −1.81879e10 −1.44883
\(769\) −7.25075e9 −0.574964 −0.287482 0.957786i \(-0.592818\pi\)
−0.287482 + 0.957786i \(0.592818\pi\)
\(770\) −3.30607e9 −0.260972
\(771\) −2.95689e10 −2.32351
\(772\) −4.42886e9 −0.346443
\(773\) 1.23442e10 0.961249 0.480625 0.876926i \(-0.340410\pi\)
0.480625 + 0.876926i \(0.340410\pi\)
\(774\) 6.02161e8 0.0466787
\(775\) −1.57119e9 −0.121247
\(776\) 2.15216e10 1.65333
\(777\) 1.54948e10 1.18499
\(778\) −8.98168e9 −0.683800
\(779\) −2.93068e8 −0.0222120
\(780\) 4.52576e9 0.341476
\(781\) 7.29659e9 0.548077
\(782\) −1.22714e10 −0.917635
\(783\) −1.68353e9 −0.125330
\(784\) 3.26468e8 0.0241955
\(785\) 2.77902e9 0.205045
\(786\) 6.11474e9 0.449158
\(787\) −1.07484e10 −0.786019 −0.393009 0.919534i \(-0.628566\pi\)
−0.393009 + 0.919534i \(0.628566\pi\)
\(788\) −9.35980e8 −0.0681436
\(789\) −1.95619e10 −1.41789
\(790\) −8.81593e8 −0.0636170
\(791\) −4.25625e8 −0.0305780
\(792\) 2.91847e10 2.08746
\(793\) −1.33858e9 −0.0953210
\(794\) −6.57446e9 −0.466110
\(795\) 2.17363e10 1.53427
\(796\) −1.56123e10 −1.09716
\(797\) 9.45263e9 0.661376 0.330688 0.943740i \(-0.392719\pi\)
0.330688 + 0.943740i \(0.392719\pi\)
\(798\) −2.71903e9 −0.189410
\(799\) −3.43881e10 −2.38503
\(800\) −1.53290e9 −0.105852
\(801\) −3.79089e10 −2.60632
\(802\) −4.21598e8 −0.0288595
\(803\) 1.49163e10 1.01662
\(804\) −8.54848e8 −0.0580086
\(805\) −5.46779e9 −0.369425
\(806\) 2.71204e9 0.182441
\(807\) −3.49637e10 −2.34186
\(808\) −2.64897e10 −1.76659
\(809\) −5.07982e9 −0.337309 −0.168655 0.985675i \(-0.553942\pi\)
−0.168655 + 0.985675i \(0.553942\pi\)
\(810\) 5.69406e9 0.376465
\(811\) 7.47602e9 0.492150 0.246075 0.969251i \(-0.420859\pi\)
0.246075 + 0.969251i \(0.420859\pi\)
\(812\) −3.36003e8 −0.0220240
\(813\) 3.66333e10 2.39089
\(814\) 1.87056e10 1.21559
\(815\) −5.04869e8 −0.0326684
\(816\) −7.91125e9 −0.509718
\(817\) 3.71159e8 0.0238113
\(818\) 7.07191e9 0.451752
\(819\) −3.09298e9 −0.196736
\(820\) 4.78665e8 0.0303168
\(821\) −3.77573e9 −0.238122 −0.119061 0.992887i \(-0.537988\pi\)
−0.119061 + 0.992887i \(0.537988\pi\)
\(822\) −1.31274e10 −0.824379
\(823\) −8.24136e8 −0.0515347 −0.0257673 0.999668i \(-0.508203\pi\)
−0.0257673 + 0.999668i \(0.508203\pi\)
\(824\) −1.27765e10 −0.795548
\(825\) 3.31582e9 0.205590
\(826\) −2.39628e9 −0.147948
\(827\) 1.28403e10 0.789418 0.394709 0.918806i \(-0.370845\pi\)
0.394709 + 0.918806i \(0.370845\pi\)
\(828\) 1.97301e10 1.20788
\(829\) −2.31484e10 −1.41117 −0.705587 0.708623i \(-0.749317\pi\)
−0.705587 + 0.708623i \(0.749317\pi\)
\(830\) 1.38941e10 0.843447
\(831\) 4.13530e10 2.49979
\(832\) 1.86559e9 0.112301
\(833\) −4.22869e9 −0.253483
\(834\) −2.77710e10 −1.65772
\(835\) 2.95937e10 1.75913
\(836\) 7.35319e9 0.435265
\(837\) −2.98702e10 −1.76075
\(838\) −1.38539e10 −0.813239
\(839\) 2.21972e9 0.129757 0.0648785 0.997893i \(-0.479334\pi\)
0.0648785 + 0.997893i \(0.479334\pi\)
\(840\) 1.08644e10 0.632451
\(841\) −1.71273e10 −0.992897
\(842\) 8.85717e9 0.511331
\(843\) 1.95765e10 1.12548
\(844\) 1.89371e10 1.08421
\(845\) −1.41653e9 −0.0807655
\(846\) −2.46811e10 −1.40142
\(847\) −2.68195e9 −0.151655
\(848\) −2.59119e9 −0.145920
\(849\) 4.26018e10 2.38919
\(850\) 1.80728e9 0.100939
\(851\) 3.09365e10 1.72075
\(852\) −9.80134e9 −0.542934
\(853\) 2.82906e10 1.56070 0.780351 0.625341i \(-0.215040\pi\)
0.780351 + 0.625341i \(0.215040\pi\)
\(854\) −1.31350e9 −0.0721653
\(855\) 1.91531e10 1.04799
\(856\) −1.99307e10 −1.08609
\(857\) 1.67394e10 0.908461 0.454231 0.890884i \(-0.349914\pi\)
0.454231 + 0.890884i \(0.349914\pi\)
\(858\) −5.72345e9 −0.309352
\(859\) 2.85605e10 1.53741 0.768704 0.639604i \(-0.220902\pi\)
0.768704 + 0.639604i \(0.220902\pi\)
\(860\) −6.06211e8 −0.0324997
\(861\) −5.01434e8 −0.0267734
\(862\) −1.65532e10 −0.880251
\(863\) 2.80980e10 1.48812 0.744060 0.668113i \(-0.232898\pi\)
0.744060 + 0.668113i \(0.232898\pi\)
\(864\) −2.91422e10 −1.53718
\(865\) −3.07604e10 −1.61598
\(866\) −1.78442e10 −0.933652
\(867\) 6.99257e10 3.64393
\(868\) −5.96156e9 −0.309415
\(869\) −2.49754e9 −0.129105
\(870\) −1.61953e9 −0.0833817
\(871\) 2.67560e8 0.0137201
\(872\) −8.73301e8 −0.0446022
\(873\) 6.49166e10 3.30222
\(874\) −5.42873e9 −0.275048
\(875\) −7.05882e9 −0.356208
\(876\) −2.00368e10 −1.00708
\(877\) −1.76094e10 −0.881550 −0.440775 0.897618i \(-0.645296\pi\)
−0.440775 + 0.897618i \(0.645296\pi\)
\(878\) 1.20892e10 0.602793
\(879\) −6.12119e10 −3.04001
\(880\) −4.25548e9 −0.210503
\(881\) −6.36370e9 −0.313541 −0.156770 0.987635i \(-0.550108\pi\)
−0.156770 + 0.987635i \(0.550108\pi\)
\(882\) −3.03503e9 −0.148944
\(883\) −9.91576e9 −0.484689 −0.242345 0.970190i \(-0.577916\pi\)
−0.242345 + 0.970190i \(0.577916\pi\)
\(884\) 6.98827e9 0.340241
\(885\) 2.58738e10 1.25476
\(886\) 2.12441e10 1.02617
\(887\) 1.93540e10 0.931192 0.465596 0.884998i \(-0.345840\pi\)
0.465596 + 0.884998i \(0.345840\pi\)
\(888\) −6.14701e10 −2.94591
\(889\) 1.09132e10 0.520951
\(890\) −1.70363e10 −0.810047
\(891\) 1.61312e10 0.764001
\(892\) 9.73170e9 0.459104
\(893\) −1.52129e10 −0.714880
\(894\) −1.88084e10 −0.880382
\(895\) 1.70712e10 0.795944
\(896\) −6.58200e9 −0.305689
\(897\) −9.46582e9 −0.437910
\(898\) 4.69952e9 0.216564
\(899\) 2.17405e9 0.0997956
\(900\) −2.90576e9 −0.132865
\(901\) 3.35633e10 1.52872
\(902\) −6.05339e8 −0.0274647
\(903\) 6.35047e8 0.0287011
\(904\) 1.68851e9 0.0760177
\(905\) −3.51300e10 −1.57546
\(906\) −1.49714e9 −0.0668827
\(907\) −3.78862e10 −1.68599 −0.842996 0.537920i \(-0.819210\pi\)
−0.842996 + 0.537920i \(0.819210\pi\)
\(908\) −1.37203e10 −0.608225
\(909\) −7.99020e10 −3.52845
\(910\) −1.38999e9 −0.0611457
\(911\) 2.34375e10 1.02706 0.513531 0.858071i \(-0.328337\pi\)
0.513531 + 0.858071i \(0.328337\pi\)
\(912\) −3.49986e9 −0.152781
\(913\) 3.93619e10 1.71170
\(914\) −8.62530e9 −0.373648
\(915\) 1.41825e10 0.612040
\(916\) −3.05369e10 −1.31277
\(917\) 4.20703e9 0.180170
\(918\) 3.43585e10 1.46583
\(919\) −1.82970e10 −0.777634 −0.388817 0.921315i \(-0.627116\pi\)
−0.388817 + 0.921315i \(0.627116\pi\)
\(920\) 2.16915e10 0.918400
\(921\) 1.03736e10 0.437544
\(922\) 1.66521e10 0.699696
\(923\) 3.06774e9 0.128414
\(924\) 1.25812e10 0.524651
\(925\) −4.55620e9 −0.189281
\(926\) 1.46165e10 0.604931
\(927\) −3.85383e10 −1.58896
\(928\) 2.12107e9 0.0871238
\(929\) 1.45900e10 0.597036 0.298518 0.954404i \(-0.403508\pi\)
0.298518 + 0.954404i \(0.403508\pi\)
\(930\) −2.87346e10 −1.17143
\(931\) −1.87073e9 −0.0759779
\(932\) 5.49069e9 0.222163
\(933\) −4.01129e10 −1.61696
\(934\) 1.63599e10 0.657002
\(935\) 5.51205e10 2.20533
\(936\) 1.22703e10 0.489090
\(937\) 1.07037e10 0.425054 0.212527 0.977155i \(-0.431831\pi\)
0.212527 + 0.977155i \(0.431831\pi\)
\(938\) 2.62548e8 0.0103872
\(939\) 2.58443e10 1.01867
\(940\) 2.48472e10 0.975729
\(941\) 2.17259e10 0.849990 0.424995 0.905196i \(-0.360276\pi\)
0.424995 + 0.905196i \(0.360276\pi\)
\(942\) 4.72090e9 0.184012
\(943\) −1.00115e9 −0.0388783
\(944\) −3.08443e9 −0.119336
\(945\) 1.53092e10 0.590121
\(946\) 7.66639e8 0.0294423
\(947\) −1.79648e10 −0.687379 −0.343690 0.939083i \(-0.611677\pi\)
−0.343690 + 0.939083i \(0.611677\pi\)
\(948\) 3.35489e9 0.127894
\(949\) 6.27135e9 0.238193
\(950\) 7.99521e8 0.0302550
\(951\) 2.25262e10 0.849290
\(952\) 1.67758e10 0.630165
\(953\) 2.28823e10 0.856395 0.428197 0.903685i \(-0.359149\pi\)
0.428197 + 0.903685i \(0.359149\pi\)
\(954\) 2.40892e10 0.898262
\(955\) 3.21176e10 1.19325
\(956\) 2.31146e10 0.855625
\(957\) −4.58810e9 −0.169216
\(958\) −6.05631e9 −0.222551
\(959\) −9.03182e9 −0.330682
\(960\) −1.97663e10 −0.721066
\(961\) 1.10607e10 0.402024
\(962\) 7.86449e9 0.284812
\(963\) −6.01181e10 −2.16927
\(964\) 9.03317e9 0.324766
\(965\) −1.46870e10 −0.526125
\(966\) −9.28848e9 −0.331531
\(967\) 7.85972e9 0.279521 0.139761 0.990185i \(-0.455367\pi\)
0.139761 + 0.990185i \(0.455367\pi\)
\(968\) 1.06397e10 0.377019
\(969\) 4.53331e10 1.60060
\(970\) 2.91736e10 1.02633
\(971\) 1.00666e10 0.352872 0.176436 0.984312i \(-0.443543\pi\)
0.176436 + 0.984312i \(0.443543\pi\)
\(972\) 7.76642e9 0.271262
\(973\) −1.91068e10 −0.664958
\(974\) 2.05753e10 0.713493
\(975\) 1.39409e9 0.0481696
\(976\) −1.69070e9 −0.0582094
\(977\) 1.20709e10 0.414103 0.207051 0.978330i \(-0.433613\pi\)
0.207051 + 0.978330i \(0.433613\pi\)
\(978\) −8.57653e8 −0.0293174
\(979\) −4.82636e10 −1.64392
\(980\) 3.05545e9 0.103701
\(981\) −2.63418e9 −0.0890849
\(982\) −1.09028e10 −0.367409
\(983\) −8.83111e9 −0.296536 −0.148268 0.988947i \(-0.547370\pi\)
−0.148268 + 0.988947i \(0.547370\pi\)
\(984\) 1.98926e9 0.0665593
\(985\) −3.10391e9 −0.103486
\(986\) −2.50073e9 −0.0830802
\(987\) −2.60291e10 −0.861686
\(988\) 3.09154e9 0.101983
\(989\) 1.26792e9 0.0416777
\(990\) 3.95613e10 1.29583
\(991\) 5.59892e10 1.82745 0.913727 0.406329i \(-0.133191\pi\)
0.913727 + 0.406329i \(0.133191\pi\)
\(992\) 3.76332e10 1.22400
\(993\) −7.31862e10 −2.37196
\(994\) 3.01027e9 0.0972194
\(995\) −5.17736e10 −1.66620
\(996\) −5.28739e10 −1.69564
\(997\) 1.14701e9 0.0366549 0.0183275 0.999832i \(-0.494166\pi\)
0.0183275 + 0.999832i \(0.494166\pi\)
\(998\) 7.64074e9 0.243320
\(999\) −8.66188e10 −2.74874
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.b.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.b.1.4 9 1.1 even 1 trivial