Properties

Label 91.10.a.b.1.2
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $13$
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] = [91,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4945 x^{11} - 8694 x^{10} + 9009530 x^{9} + 27431200 x^{8} - 7320118704 x^{7} - 28566940352 x^{6} + 2456696387328 x^{5} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-34.7365\) 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)
 
\(f(q)\) \(=\) \(q-36.7365 q^{2} -181.146 q^{3} +837.568 q^{4} +2263.64 q^{5} +6654.66 q^{6} -2401.00 q^{7} -11960.2 q^{8} +13130.9 q^{9} +O(q^{10})\) \(q-36.7365 q^{2} -181.146 q^{3} +837.568 q^{4} +2263.64 q^{5} +6654.66 q^{6} -2401.00 q^{7} -11960.2 q^{8} +13130.9 q^{9} -83158.3 q^{10} +4940.24 q^{11} -151722. q^{12} -28561.0 q^{13} +88204.2 q^{14} -410050. q^{15} +10541.0 q^{16} -178661. q^{17} -482381. q^{18} -471966. q^{19} +1.89595e6 q^{20} +434931. q^{21} -181487. q^{22} +2.11688e6 q^{23} +2.16654e6 q^{24} +3.17096e6 q^{25} +1.04923e6 q^{26} +1.18689e6 q^{27} -2.01100e6 q^{28} -4.05803e6 q^{29} +1.50638e7 q^{30} +1.98399e6 q^{31} +5.73639e6 q^{32} -894905. q^{33} +6.56338e6 q^{34} -5.43501e6 q^{35} +1.09980e7 q^{36} -9.92848e6 q^{37} +1.73384e7 q^{38} +5.17371e6 q^{39} -2.70736e7 q^{40} +8.06137e6 q^{41} -1.59778e7 q^{42} -3.04626e7 q^{43} +4.13779e6 q^{44} +2.97236e7 q^{45} -7.77669e7 q^{46} +5.28351e7 q^{47} -1.90947e6 q^{48} +5.76480e6 q^{49} -1.16490e8 q^{50} +3.23637e7 q^{51} -2.39218e7 q^{52} -6.65323e6 q^{53} -4.36023e7 q^{54} +1.11829e7 q^{55} +2.87165e7 q^{56} +8.54947e7 q^{57} +1.49078e8 q^{58} +1.63029e7 q^{59} -3.43445e8 q^{60} -1.19476e8 q^{61} -7.28849e7 q^{62} -3.15272e7 q^{63} -2.16132e8 q^{64} -6.46519e7 q^{65} +3.28756e7 q^{66} +2.03127e8 q^{67} -1.49641e8 q^{68} -3.83465e8 q^{69} +1.99663e8 q^{70} +4.61351e7 q^{71} -1.57048e8 q^{72} +1.31899e8 q^{73} +3.64737e8 q^{74} -5.74406e8 q^{75} -3.95303e8 q^{76} -1.18615e7 q^{77} -1.90064e8 q^{78} +4.76417e8 q^{79} +2.38611e7 q^{80} -4.73456e8 q^{81} -2.96146e8 q^{82} -9.05932e7 q^{83} +3.64285e8 q^{84} -4.04425e8 q^{85} +1.11909e9 q^{86} +7.35096e8 q^{87} -5.90863e7 q^{88} -4.07571e7 q^{89} -1.09194e9 q^{90} +6.85750e7 q^{91} +1.77303e9 q^{92} -3.59393e8 q^{93} -1.94097e9 q^{94} -1.06836e9 q^{95} -1.03912e9 q^{96} -6.02654e8 q^{97} -2.11778e8 q^{98} +6.48696e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 26 q^{2} + 163 q^{3} + 3286 q^{4} - 2640 q^{5} - 995 q^{6} - 31213 q^{7} - 6738 q^{8} + 94756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 26 q^{2} + 163 q^{3} + 3286 q^{4} - 2640 q^{5} - 995 q^{6} - 31213 q^{7} - 6738 q^{8} + 94756 q^{9} + 42588 q^{10} - 107493 q^{11} + 157399 q^{12} - 371293 q^{13} + 62426 q^{14} - 469556 q^{15} + 1033802 q^{16} + 50812 q^{17} - 2994615 q^{18} + 479470 q^{19} - 1834962 q^{20} - 391363 q^{21} - 5474013 q^{22} - 984639 q^{23} - 12496965 q^{24} + 4519039 q^{25} + 742586 q^{26} + 5965117 q^{27} - 7889686 q^{28} - 3441800 q^{29} + 25168012 q^{30} - 2185751 q^{31} - 2746342 q^{32} + 34793355 q^{33} - 966694 q^{34} + 6338640 q^{35} + 23974587 q^{36} - 31532363 q^{37} - 51039796 q^{38} - 4655443 q^{39} + 27446642 q^{40} - 38029287 q^{41} + 2388995 q^{42} - 65479740 q^{43} - 64795239 q^{44} - 190647152 q^{45} - 68737615 q^{46} + 18884785 q^{47} - 43918333 q^{48} + 74942413 q^{49} - 295918964 q^{50} - 97799092 q^{51} - 93851446 q^{52} - 37670088 q^{53} - 420784337 q^{54} - 11739604 q^{55} + 16177938 q^{56} - 119447794 q^{57} - 351819004 q^{58} - 86030686 q^{59} - 1421949708 q^{60} - 413609773 q^{61} + 21747651 q^{62} - 227509156 q^{63} - 611561502 q^{64} + 75401040 q^{65} - 154290083 q^{66} + 121596783 q^{67} - 613335382 q^{68} - 1089108303 q^{69} - 102253788 q^{70} - 900222116 q^{71} - 1897573017 q^{72} - 586910355 q^{73} - 688661251 q^{74} - 1466887131 q^{75} - 180912510 q^{76} + 258090693 q^{77} + 28418195 q^{78} - 590012173 q^{79} - 1724662122 q^{80} - 58178363 q^{81} + 145984865 q^{82} + 94283256 q^{83} - 377914999 q^{84} - 1689818164 q^{85} + 13901738 q^{86} + 1073171888 q^{87} - 1814132379 q^{88} - 1154652750 q^{89} + 2671175016 q^{90} + 891474493 q^{91} + 670826733 q^{92} - 5057835587 q^{93} - 2961146369 q^{94} - 3377803464 q^{95} - 4898921405 q^{96} - 2173622401 q^{97} - 149884826 q^{98} - 4653424330 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −36.7365 −1.62354 −0.811769 0.583979i \(-0.801495\pi\)
−0.811769 + 0.583979i \(0.801495\pi\)
\(3\) −181.146 −1.29117 −0.645584 0.763689i \(-0.723386\pi\)
−0.645584 + 0.763689i \(0.723386\pi\)
\(4\) 837.568 1.63587
\(5\) 2263.64 1.61973 0.809866 0.586615i \(-0.199540\pi\)
0.809866 + 0.586615i \(0.199540\pi\)
\(6\) 6654.66 2.09626
\(7\) −2401.00 −0.377964
\(8\) −11960.2 −1.03237
\(9\) 13130.9 0.667117
\(10\) −83158.3 −2.62969
\(11\) 4940.24 0.101737 0.0508687 0.998705i \(-0.483801\pi\)
0.0508687 + 0.998705i \(0.483801\pi\)
\(12\) −151722. −2.11219
\(13\) −28561.0 −0.277350
\(14\) 88204.2 0.613640
\(15\) −410050. −2.09135
\(16\) 10541.0 0.0402109
\(17\) −178661. −0.518812 −0.259406 0.965768i \(-0.583527\pi\)
−0.259406 + 0.965768i \(0.583527\pi\)
\(18\) −482381. −1.08309
\(19\) −471966. −0.830844 −0.415422 0.909629i \(-0.636366\pi\)
−0.415422 + 0.909629i \(0.636366\pi\)
\(20\) 1.89595e6 2.64968
\(21\) 434931. 0.488016
\(22\) −181487. −0.165175
\(23\) 2.11688e6 1.57733 0.788664 0.614825i \(-0.210773\pi\)
0.788664 + 0.614825i \(0.210773\pi\)
\(24\) 2.16654e6 1.33296
\(25\) 3.17096e6 1.62353
\(26\) 1.04923e6 0.450288
\(27\) 1.18689e6 0.429809
\(28\) −2.01100e6 −0.618302
\(29\) −4.05803e6 −1.06543 −0.532715 0.846295i \(-0.678828\pi\)
−0.532715 + 0.846295i \(0.678828\pi\)
\(30\) 1.50638e7 3.39538
\(31\) 1.98399e6 0.385845 0.192922 0.981214i \(-0.438203\pi\)
0.192922 + 0.981214i \(0.438203\pi\)
\(32\) 5.73639e6 0.967082
\(33\) −894905. −0.131360
\(34\) 6.56338e6 0.842311
\(35\) −5.43501e6 −0.612201
\(36\) 1.09980e7 1.09132
\(37\) −9.92848e6 −0.870913 −0.435457 0.900210i \(-0.643413\pi\)
−0.435457 + 0.900210i \(0.643413\pi\)
\(38\) 1.73384e7 1.34891
\(39\) 5.17371e6 0.358106
\(40\) −2.70736e7 −1.67216
\(41\) 8.06137e6 0.445535 0.222767 0.974872i \(-0.428491\pi\)
0.222767 + 0.974872i \(0.428491\pi\)
\(42\) −1.59778e7 −0.792312
\(43\) −3.04626e7 −1.35881 −0.679405 0.733763i \(-0.737762\pi\)
−0.679405 + 0.733763i \(0.737762\pi\)
\(44\) 4.13779e6 0.166430
\(45\) 2.97236e7 1.08055
\(46\) −7.77669e7 −2.56085
\(47\) 5.28351e7 1.57936 0.789681 0.613517i \(-0.210246\pi\)
0.789681 + 0.613517i \(0.210246\pi\)
\(48\) −1.90947e6 −0.0519190
\(49\) 5.76480e6 0.142857
\(50\) −1.16490e8 −2.63586
\(51\) 3.23637e7 0.669874
\(52\) −2.39218e7 −0.453710
\(53\) −6.65323e6 −0.115822 −0.0579110 0.998322i \(-0.518444\pi\)
−0.0579110 + 0.998322i \(0.518444\pi\)
\(54\) −4.36023e7 −0.697810
\(55\) 1.11829e7 0.164787
\(56\) 2.87165e7 0.390198
\(57\) 8.54947e7 1.07276
\(58\) 1.49078e8 1.72976
\(59\) 1.63029e7 0.175158 0.0875790 0.996158i \(-0.472087\pi\)
0.0875790 + 0.996158i \(0.472087\pi\)
\(60\) −3.43445e8 −3.42118
\(61\) −1.19476e8 −1.10484 −0.552418 0.833567i \(-0.686295\pi\)
−0.552418 + 0.833567i \(0.686295\pi\)
\(62\) −7.28849e7 −0.626434
\(63\) −3.15272e7 −0.252146
\(64\) −2.16132e8 −1.61031
\(65\) −6.46519e7 −0.449233
\(66\) 3.28756e7 0.213268
\(67\) 2.03127e8 1.23149 0.615746 0.787944i \(-0.288855\pi\)
0.615746 + 0.787944i \(0.288855\pi\)
\(68\) −1.49641e8 −0.848711
\(69\) −3.83465e8 −2.03660
\(70\) 1.99663e8 0.993931
\(71\) 4.61351e7 0.215461 0.107731 0.994180i \(-0.465642\pi\)
0.107731 + 0.994180i \(0.465642\pi\)
\(72\) −1.57048e8 −0.688709
\(73\) 1.31899e8 0.543610 0.271805 0.962352i \(-0.412379\pi\)
0.271805 + 0.962352i \(0.412379\pi\)
\(74\) 3.64737e8 1.41396
\(75\) −5.74406e8 −2.09625
\(76\) −3.95303e8 −1.35916
\(77\) −1.18615e7 −0.0384532
\(78\) −1.90064e8 −0.581398
\(79\) 4.76417e8 1.37615 0.688074 0.725640i \(-0.258456\pi\)
0.688074 + 0.725640i \(0.258456\pi\)
\(80\) 2.38611e7 0.0651308
\(81\) −4.73456e8 −1.22207
\(82\) −2.96146e8 −0.723342
\(83\) −9.05932e7 −0.209529 −0.104764 0.994497i \(-0.533409\pi\)
−0.104764 + 0.994497i \(0.533409\pi\)
\(84\) 3.64285e8 0.798333
\(85\) −4.04425e8 −0.840336
\(86\) 1.11909e9 2.20608
\(87\) 7.35096e8 1.37565
\(88\) −5.90863e7 −0.105030
\(89\) −4.07571e7 −0.0688570 −0.0344285 0.999407i \(-0.510961\pi\)
−0.0344285 + 0.999407i \(0.510961\pi\)
\(90\) −1.09194e9 −1.75431
\(91\) 6.85750e7 0.104828
\(92\) 1.77303e9 2.58031
\(93\) −3.59393e8 −0.498191
\(94\) −1.94097e9 −2.56415
\(95\) −1.06836e9 −1.34574
\(96\) −1.03912e9 −1.24867
\(97\) −6.02654e8 −0.691187 −0.345593 0.938384i \(-0.612322\pi\)
−0.345593 + 0.938384i \(0.612322\pi\)
\(98\) −2.11778e8 −0.231934
\(99\) 6.48696e7 0.0678708
\(100\) 2.65589e9 2.65589
\(101\) −1.25464e9 −1.19970 −0.599849 0.800114i \(-0.704773\pi\)
−0.599849 + 0.800114i \(0.704773\pi\)
\(102\) −1.18893e9 −1.08757
\(103\) −4.08373e8 −0.357511 −0.178756 0.983894i \(-0.557207\pi\)
−0.178756 + 0.983894i \(0.557207\pi\)
\(104\) 3.41595e8 0.286327
\(105\) 9.84530e8 0.790455
\(106\) 2.44416e8 0.188041
\(107\) −1.88302e9 −1.38876 −0.694380 0.719609i \(-0.744321\pi\)
−0.694380 + 0.719609i \(0.744321\pi\)
\(108\) 9.94104e8 0.703113
\(109\) 1.46345e9 0.993022 0.496511 0.868030i \(-0.334614\pi\)
0.496511 + 0.868030i \(0.334614\pi\)
\(110\) −4.10822e8 −0.267539
\(111\) 1.79850e9 1.12450
\(112\) −2.53090e7 −0.0151983
\(113\) 2.31179e9 1.33381 0.666907 0.745141i \(-0.267618\pi\)
0.666907 + 0.745141i \(0.267618\pi\)
\(114\) −3.14077e9 −1.74167
\(115\) 4.79187e9 2.55485
\(116\) −3.39888e9 −1.74291
\(117\) −3.75030e8 −0.185025
\(118\) −5.98910e8 −0.284376
\(119\) 4.28965e8 0.196092
\(120\) 4.90428e9 2.15904
\(121\) −2.33354e9 −0.989649
\(122\) 4.38914e9 1.79374
\(123\) −1.46028e9 −0.575261
\(124\) 1.66173e9 0.631194
\(125\) 2.75674e9 1.00995
\(126\) 1.15820e9 0.409369
\(127\) 4.98338e9 1.69984 0.849918 0.526915i \(-0.176651\pi\)
0.849918 + 0.526915i \(0.176651\pi\)
\(128\) 5.00288e9 1.64731
\(129\) 5.51818e9 1.75445
\(130\) 2.37508e9 0.729346
\(131\) −4.28792e9 −1.27211 −0.636056 0.771643i \(-0.719435\pi\)
−0.636056 + 0.771643i \(0.719435\pi\)
\(132\) −7.49543e8 −0.214889
\(133\) 1.13319e9 0.314029
\(134\) −7.46218e9 −1.99937
\(135\) 2.68671e9 0.696174
\(136\) 2.13682e9 0.535604
\(137\) −2.17817e9 −0.528262 −0.264131 0.964487i \(-0.585085\pi\)
−0.264131 + 0.964487i \(0.585085\pi\)
\(138\) 1.40872e10 3.30649
\(139\) −3.00478e9 −0.682726 −0.341363 0.939932i \(-0.610889\pi\)
−0.341363 + 0.939932i \(0.610889\pi\)
\(140\) −4.55219e9 −1.00148
\(141\) −9.57086e9 −2.03922
\(142\) −1.69484e9 −0.349809
\(143\) −1.41098e8 −0.0282169
\(144\) 1.38413e8 0.0268253
\(145\) −9.18594e9 −1.72571
\(146\) −4.84549e9 −0.882572
\(147\) −1.04427e9 −0.184453
\(148\) −8.31577e9 −1.42470
\(149\) −1.13565e10 −1.88758 −0.943792 0.330539i \(-0.892770\pi\)
−0.943792 + 0.330539i \(0.892770\pi\)
\(150\) 2.11016e10 3.40334
\(151\) −9.47221e9 −1.48271 −0.741353 0.671115i \(-0.765816\pi\)
−0.741353 + 0.671115i \(0.765816\pi\)
\(152\) 5.64481e9 0.857735
\(153\) −2.34597e9 −0.346108
\(154\) 4.35750e8 0.0624301
\(155\) 4.49106e9 0.624965
\(156\) 4.33333e9 0.585816
\(157\) 6.85342e9 0.900241 0.450120 0.892968i \(-0.351381\pi\)
0.450120 + 0.892968i \(0.351381\pi\)
\(158\) −1.75019e10 −2.23423
\(159\) 1.20521e9 0.149546
\(160\) 1.29851e10 1.56641
\(161\) −5.08264e9 −0.596174
\(162\) 1.73931e10 1.98408
\(163\) −5.81211e9 −0.644896 −0.322448 0.946587i \(-0.604506\pi\)
−0.322448 + 0.946587i \(0.604506\pi\)
\(164\) 6.75195e9 0.728839
\(165\) −2.02574e9 −0.212768
\(166\) 3.32807e9 0.340178
\(167\) −1.81398e9 −0.180471 −0.0902356 0.995920i \(-0.528762\pi\)
−0.0902356 + 0.995920i \(0.528762\pi\)
\(168\) −5.20187e9 −0.503811
\(169\) 8.15731e8 0.0769231
\(170\) 1.48571e10 1.36432
\(171\) −6.19731e9 −0.554270
\(172\) −2.55145e10 −2.22284
\(173\) −1.40074e10 −1.18891 −0.594456 0.804128i \(-0.702633\pi\)
−0.594456 + 0.804128i \(0.702633\pi\)
\(174\) −2.70048e10 −2.23342
\(175\) −7.61347e9 −0.613637
\(176\) 5.20752e7 0.00409095
\(177\) −2.95320e9 −0.226158
\(178\) 1.49727e9 0.111792
\(179\) −9.19182e9 −0.669210 −0.334605 0.942358i \(-0.608603\pi\)
−0.334605 + 0.942358i \(0.608603\pi\)
\(180\) 2.48955e10 1.76764
\(181\) −2.50761e10 −1.73663 −0.868313 0.496017i \(-0.834795\pi\)
−0.868313 + 0.496017i \(0.834795\pi\)
\(182\) −2.51920e9 −0.170193
\(183\) 2.16427e10 1.42653
\(184\) −2.53184e10 −1.62838
\(185\) −2.24745e10 −1.41065
\(186\) 1.32028e10 0.808832
\(187\) −8.82629e8 −0.0527826
\(188\) 4.42529e10 2.58364
\(189\) −2.84973e9 −0.162452
\(190\) 3.92478e10 2.18487
\(191\) 2.60438e10 1.41597 0.707985 0.706227i \(-0.249604\pi\)
0.707985 + 0.706227i \(0.249604\pi\)
\(192\) 3.91514e10 2.07918
\(193\) 3.40811e10 1.76810 0.884049 0.467394i \(-0.154807\pi\)
0.884049 + 0.467394i \(0.154807\pi\)
\(194\) 2.21394e10 1.12217
\(195\) 1.17114e10 0.580035
\(196\) 4.82841e9 0.233696
\(197\) −3.69881e10 −1.74970 −0.874851 0.484392i \(-0.839041\pi\)
−0.874851 + 0.484392i \(0.839041\pi\)
\(198\) −2.38308e9 −0.110191
\(199\) −2.61934e10 −1.18400 −0.592001 0.805937i \(-0.701662\pi\)
−0.592001 + 0.805937i \(0.701662\pi\)
\(200\) −3.79253e10 −1.67608
\(201\) −3.67957e10 −1.59006
\(202\) 4.60909e10 1.94775
\(203\) 9.74334e9 0.402694
\(204\) 2.71068e10 1.09583
\(205\) 1.82481e10 0.721647
\(206\) 1.50022e10 0.580433
\(207\) 2.77965e10 1.05226
\(208\) −3.01062e8 −0.0111525
\(209\) −2.33162e9 −0.0845279
\(210\) −3.61681e10 −1.28333
\(211\) −1.41870e10 −0.492743 −0.246371 0.969176i \(-0.579238\pi\)
−0.246371 + 0.969176i \(0.579238\pi\)
\(212\) −5.57253e9 −0.189470
\(213\) −8.35718e9 −0.278197
\(214\) 6.91754e10 2.25470
\(215\) −6.89565e10 −2.20091
\(216\) −1.41955e10 −0.443720
\(217\) −4.76357e9 −0.145836
\(218\) −5.37620e10 −1.61221
\(219\) −2.38929e10 −0.701893
\(220\) 9.36647e9 0.269572
\(221\) 5.10274e9 0.143893
\(222\) −6.60707e10 −1.82566
\(223\) 4.58529e10 1.24164 0.620819 0.783954i \(-0.286800\pi\)
0.620819 + 0.783954i \(0.286800\pi\)
\(224\) −1.37731e10 −0.365523
\(225\) 4.16374e10 1.08308
\(226\) −8.49270e10 −2.16550
\(227\) 3.66125e10 0.915194 0.457597 0.889160i \(-0.348710\pi\)
0.457597 + 0.889160i \(0.348710\pi\)
\(228\) 7.16076e10 1.75490
\(229\) −4.61194e10 −1.10821 −0.554107 0.832445i \(-0.686940\pi\)
−0.554107 + 0.832445i \(0.686940\pi\)
\(230\) −1.76036e11 −4.14789
\(231\) 2.14867e9 0.0496495
\(232\) 4.85349e10 1.09991
\(233\) −7.71305e10 −1.71445 −0.857224 0.514943i \(-0.827813\pi\)
−0.857224 + 0.514943i \(0.827813\pi\)
\(234\) 1.37773e10 0.300395
\(235\) 1.19600e11 2.55814
\(236\) 1.36548e10 0.286536
\(237\) −8.63010e10 −1.77684
\(238\) −1.57587e10 −0.318363
\(239\) −6.91404e10 −1.37070 −0.685348 0.728216i \(-0.740350\pi\)
−0.685348 + 0.728216i \(0.740350\pi\)
\(240\) −4.32235e9 −0.0840948
\(241\) 4.18522e10 0.799174 0.399587 0.916695i \(-0.369154\pi\)
0.399587 + 0.916695i \(0.369154\pi\)
\(242\) 8.57261e10 1.60673
\(243\) 6.24030e10 1.14809
\(244\) −1.00070e11 −1.80737
\(245\) 1.30495e10 0.231390
\(246\) 5.36457e10 0.933957
\(247\) 1.34798e10 0.230435
\(248\) −2.37290e10 −0.398333
\(249\) 1.64106e10 0.270537
\(250\) −1.01273e11 −1.63969
\(251\) 8.40440e10 1.33652 0.668260 0.743928i \(-0.267040\pi\)
0.668260 + 0.743928i \(0.267040\pi\)
\(252\) −2.64062e10 −0.412480
\(253\) 1.04579e10 0.160473
\(254\) −1.83072e11 −2.75975
\(255\) 7.32600e10 1.08502
\(256\) −7.31287e10 −1.06416
\(257\) −1.15453e11 −1.65084 −0.825419 0.564521i \(-0.809061\pi\)
−0.825419 + 0.564521i \(0.809061\pi\)
\(258\) −2.02718e11 −2.84842
\(259\) 2.38383e10 0.329174
\(260\) −5.41504e10 −0.734888
\(261\) −5.32855e10 −0.710766
\(262\) 1.57523e11 2.06532
\(263\) −5.72660e10 −0.738068 −0.369034 0.929416i \(-0.620311\pi\)
−0.369034 + 0.929416i \(0.620311\pi\)
\(264\) 1.07032e10 0.135612
\(265\) −1.50605e10 −0.187601
\(266\) −4.16294e10 −0.509838
\(267\) 7.38298e9 0.0889060
\(268\) 1.70133e11 2.01457
\(269\) 5.92578e10 0.690018 0.345009 0.938599i \(-0.387876\pi\)
0.345009 + 0.938599i \(0.387876\pi\)
\(270\) −9.87001e10 −1.13027
\(271\) −1.29346e10 −0.145677 −0.0728386 0.997344i \(-0.523206\pi\)
−0.0728386 + 0.997344i \(0.523206\pi\)
\(272\) −1.88327e9 −0.0208619
\(273\) −1.24221e10 −0.135351
\(274\) 8.00183e10 0.857654
\(275\) 1.56653e10 0.165174
\(276\) −3.21178e11 −3.33162
\(277\) 8.86317e10 0.904545 0.452272 0.891880i \(-0.350614\pi\)
0.452272 + 0.891880i \(0.350614\pi\)
\(278\) 1.10385e11 1.10843
\(279\) 2.60515e10 0.257404
\(280\) 6.50038e10 0.632016
\(281\) 4.17541e10 0.399503 0.199752 0.979847i \(-0.435986\pi\)
0.199752 + 0.979847i \(0.435986\pi\)
\(282\) 3.51599e11 3.31076
\(283\) −8.52161e10 −0.789737 −0.394869 0.918738i \(-0.629210\pi\)
−0.394869 + 0.918738i \(0.629210\pi\)
\(284\) 3.86413e10 0.352467
\(285\) 1.93529e11 1.73758
\(286\) 5.18345e9 0.0458112
\(287\) −1.93554e10 −0.168396
\(288\) 7.53237e10 0.645157
\(289\) −8.66681e10 −0.730834
\(290\) 3.37459e11 2.80175
\(291\) 1.09168e11 0.892439
\(292\) 1.10474e11 0.889278
\(293\) −1.32163e10 −0.104763 −0.0523813 0.998627i \(-0.516681\pi\)
−0.0523813 + 0.998627i \(0.516681\pi\)
\(294\) 3.83628e10 0.299466
\(295\) 3.69039e10 0.283709
\(296\) 1.18747e11 0.899101
\(297\) 5.86354e9 0.0437276
\(298\) 4.17198e11 3.06456
\(299\) −6.04603e10 −0.437472
\(300\) −4.81104e11 −3.42920
\(301\) 7.31407e10 0.513582
\(302\) 3.47975e11 2.40723
\(303\) 2.27272e11 1.54901
\(304\) −4.97501e9 −0.0334089
\(305\) −2.70452e11 −1.78954
\(306\) 8.61828e10 0.561919
\(307\) −1.28224e11 −0.823849 −0.411925 0.911218i \(-0.635143\pi\)
−0.411925 + 0.911218i \(0.635143\pi\)
\(308\) −9.93482e9 −0.0629045
\(309\) 7.39751e10 0.461607
\(310\) −1.64985e11 −1.01465
\(311\) 1.32478e11 0.803011 0.401505 0.915857i \(-0.368487\pi\)
0.401505 + 0.915857i \(0.368487\pi\)
\(312\) −6.18786e10 −0.369696
\(313\) 7.23487e10 0.426070 0.213035 0.977045i \(-0.431665\pi\)
0.213035 + 0.977045i \(0.431665\pi\)
\(314\) −2.51770e11 −1.46157
\(315\) −7.13663e10 −0.408409
\(316\) 3.99031e11 2.25121
\(317\) −1.28788e11 −0.716320 −0.358160 0.933660i \(-0.616596\pi\)
−0.358160 + 0.933660i \(0.616596\pi\)
\(318\) −4.42750e10 −0.242793
\(319\) −2.00477e10 −0.108394
\(320\) −4.89245e11 −2.60826
\(321\) 3.41101e11 1.79312
\(322\) 1.86718e11 0.967911
\(323\) 8.43219e10 0.431052
\(324\) −3.96551e11 −1.99916
\(325\) −9.05657e10 −0.450286
\(326\) 2.13516e11 1.04701
\(327\) −2.65098e11 −1.28216
\(328\) −9.64157e10 −0.459955
\(329\) −1.26857e11 −0.596943
\(330\) 7.44187e10 0.345437
\(331\) 8.71662e10 0.399137 0.199569 0.979884i \(-0.436046\pi\)
0.199569 + 0.979884i \(0.436046\pi\)
\(332\) −7.58779e10 −0.342763
\(333\) −1.30369e11 −0.581001
\(334\) 6.66391e10 0.293002
\(335\) 4.59808e11 1.99469
\(336\) 4.58463e9 0.0196235
\(337\) 2.73441e10 0.115486 0.0577430 0.998331i \(-0.481610\pi\)
0.0577430 + 0.998331i \(0.481610\pi\)
\(338\) −2.99671e10 −0.124888
\(339\) −4.18771e11 −1.72218
\(340\) −3.38733e11 −1.37468
\(341\) 9.80141e9 0.0392549
\(342\) 2.27667e11 0.899878
\(343\) −1.38413e10 −0.0539949
\(344\) 3.64339e11 1.40279
\(345\) −8.68028e11 −3.29874
\(346\) 5.14582e11 1.93024
\(347\) 4.21270e11 1.55983 0.779916 0.625884i \(-0.215262\pi\)
0.779916 + 0.625884i \(0.215262\pi\)
\(348\) 6.15693e11 2.25039
\(349\) 6.84953e10 0.247142 0.123571 0.992336i \(-0.460565\pi\)
0.123571 + 0.992336i \(0.460565\pi\)
\(350\) 2.79692e11 0.996262
\(351\) −3.38989e10 −0.119207
\(352\) 2.83391e10 0.0983885
\(353\) −3.33099e11 −1.14179 −0.570897 0.821022i \(-0.693404\pi\)
−0.570897 + 0.821022i \(0.693404\pi\)
\(354\) 1.08490e11 0.367177
\(355\) 1.04433e11 0.348989
\(356\) −3.41368e10 −0.112641
\(357\) −7.77053e10 −0.253188
\(358\) 3.37675e11 1.08649
\(359\) −4.87654e11 −1.54948 −0.774741 0.632279i \(-0.782120\pi\)
−0.774741 + 0.632279i \(0.782120\pi\)
\(360\) −3.55500e11 −1.11552
\(361\) −9.99360e10 −0.309699
\(362\) 9.21207e11 2.81948
\(363\) 4.22712e11 1.27780
\(364\) 5.74362e10 0.171486
\(365\) 2.98572e11 0.880503
\(366\) −7.95075e11 −2.31603
\(367\) −3.97139e11 −1.14273 −0.571367 0.820694i \(-0.693587\pi\)
−0.571367 + 0.820694i \(0.693587\pi\)
\(368\) 2.23142e10 0.0634257
\(369\) 1.05853e11 0.297224
\(370\) 8.25635e11 2.29024
\(371\) 1.59744e10 0.0437766
\(372\) −3.01016e11 −0.814978
\(373\) −2.70965e11 −0.724810 −0.362405 0.932021i \(-0.618044\pi\)
−0.362405 + 0.932021i \(0.618044\pi\)
\(374\) 3.24247e10 0.0856946
\(375\) −4.99372e11 −1.30402
\(376\) −6.31918e11 −1.63048
\(377\) 1.15901e11 0.295497
\(378\) 1.04689e11 0.263748
\(379\) −3.62277e11 −0.901913 −0.450957 0.892546i \(-0.648917\pi\)
−0.450957 + 0.892546i \(0.648917\pi\)
\(380\) −8.94826e11 −2.20147
\(381\) −9.02719e11 −2.19478
\(382\) −9.56757e11 −2.29888
\(383\) −1.63326e11 −0.387848 −0.193924 0.981017i \(-0.562121\pi\)
−0.193924 + 0.981017i \(0.562121\pi\)
\(384\) −9.06251e11 −2.12695
\(385\) −2.68502e10 −0.0622838
\(386\) −1.25202e12 −2.87057
\(387\) −4.00000e11 −0.906485
\(388\) −5.04764e11 −1.13069
\(389\) 4.37595e11 0.968945 0.484473 0.874806i \(-0.339012\pi\)
0.484473 + 0.874806i \(0.339012\pi\)
\(390\) −4.30237e11 −0.941709
\(391\) −3.78205e11 −0.818336
\(392\) −6.89482e10 −0.147481
\(393\) 7.76739e11 1.64251
\(394\) 1.35881e12 2.84071
\(395\) 1.07844e12 2.22899
\(396\) 5.43327e10 0.111028
\(397\) −1.67915e11 −0.339261 −0.169630 0.985508i \(-0.554257\pi\)
−0.169630 + 0.985508i \(0.554257\pi\)
\(398\) 9.62251e11 1.92227
\(399\) −2.05273e11 −0.405465
\(400\) 3.34252e10 0.0652835
\(401\) −4.53747e11 −0.876322 −0.438161 0.898897i \(-0.644370\pi\)
−0.438161 + 0.898897i \(0.644370\pi\)
\(402\) 1.35174e12 2.58153
\(403\) −5.66649e10 −0.107014
\(404\) −1.05084e12 −1.96255
\(405\) −1.07174e12 −1.97943
\(406\) −3.57936e11 −0.653790
\(407\) −4.90491e10 −0.0886045
\(408\) −3.87077e11 −0.691555
\(409\) 5.07750e11 0.897212 0.448606 0.893730i \(-0.351921\pi\)
0.448606 + 0.893730i \(0.351921\pi\)
\(410\) −6.70370e11 −1.17162
\(411\) 3.94567e11 0.682076
\(412\) −3.42040e11 −0.584843
\(413\) −3.91432e10 −0.0662035
\(414\) −1.02115e12 −1.70839
\(415\) −2.05071e11 −0.339381
\(416\) −1.63837e11 −0.268220
\(417\) 5.44304e11 0.881514
\(418\) 8.56556e10 0.137234
\(419\) 9.94618e11 1.57650 0.788249 0.615357i \(-0.210988\pi\)
0.788249 + 0.615357i \(0.210988\pi\)
\(420\) 8.24610e11 1.29308
\(421\) −4.38374e11 −0.680104 −0.340052 0.940407i \(-0.610445\pi\)
−0.340052 + 0.940407i \(0.610445\pi\)
\(422\) 5.21181e11 0.799986
\(423\) 6.93770e11 1.05362
\(424\) 7.95740e10 0.119571
\(425\) −5.66527e11 −0.842307
\(426\) 3.07013e11 0.451663
\(427\) 2.86863e11 0.417589
\(428\) −1.57715e12 −2.27184
\(429\) 2.55594e10 0.0364328
\(430\) 2.53322e12 3.57326
\(431\) −1.19861e12 −1.67313 −0.836566 0.547866i \(-0.815440\pi\)
−0.836566 + 0.547866i \(0.815440\pi\)
\(432\) 1.25111e10 0.0172830
\(433\) −6.29247e11 −0.860251 −0.430126 0.902769i \(-0.641531\pi\)
−0.430126 + 0.902769i \(0.641531\pi\)
\(434\) 1.74997e11 0.236770
\(435\) 1.66400e12 2.22818
\(436\) 1.22574e12 1.62446
\(437\) −9.99097e11 −1.31051
\(438\) 8.77741e11 1.13955
\(439\) 8.66984e11 1.11409 0.557046 0.830482i \(-0.311935\pi\)
0.557046 + 0.830482i \(0.311935\pi\)
\(440\) −1.33750e11 −0.170121
\(441\) 7.56968e10 0.0953024
\(442\) −1.87457e11 −0.233615
\(443\) −2.63092e11 −0.324557 −0.162278 0.986745i \(-0.551884\pi\)
−0.162278 + 0.986745i \(0.551884\pi\)
\(444\) 1.50637e12 1.83953
\(445\) −9.22595e10 −0.111530
\(446\) −1.68447e12 −2.01585
\(447\) 2.05718e12 2.43719
\(448\) 5.18932e11 0.608638
\(449\) −5.64159e10 −0.0655078 −0.0327539 0.999463i \(-0.510428\pi\)
−0.0327539 + 0.999463i \(0.510428\pi\)
\(450\) −1.52961e12 −1.75843
\(451\) 3.98251e10 0.0453276
\(452\) 1.93628e12 2.18195
\(453\) 1.71585e12 1.91442
\(454\) −1.34501e12 −1.48585
\(455\) 1.55229e11 0.169794
\(456\) −1.02253e12 −1.10748
\(457\) 1.12328e12 1.20466 0.602331 0.798246i \(-0.294239\pi\)
0.602331 + 0.798246i \(0.294239\pi\)
\(458\) 1.69426e12 1.79923
\(459\) −2.12052e11 −0.222990
\(460\) 4.01352e12 4.17941
\(461\) 2.95941e11 0.305177 0.152588 0.988290i \(-0.451239\pi\)
0.152588 + 0.988290i \(0.451239\pi\)
\(462\) −7.89344e10 −0.0806078
\(463\) −1.64043e12 −1.65899 −0.829494 0.558516i \(-0.811371\pi\)
−0.829494 + 0.558516i \(0.811371\pi\)
\(464\) −4.27759e10 −0.0428418
\(465\) −8.13537e11 −0.806935
\(466\) 2.83350e12 2.78347
\(467\) −1.15569e12 −1.12438 −0.562192 0.827007i \(-0.690042\pi\)
−0.562192 + 0.827007i \(0.690042\pi\)
\(468\) −3.14113e11 −0.302677
\(469\) −4.87709e11 −0.465460
\(470\) −4.39367e12 −4.15324
\(471\) −1.24147e12 −1.16236
\(472\) −1.94986e11 −0.180827
\(473\) −1.50493e11 −0.138242
\(474\) 3.17039e12 2.88477
\(475\) −1.49658e12 −1.34890
\(476\) 3.59288e11 0.320783
\(477\) −8.73626e10 −0.0772668
\(478\) 2.53997e12 2.22538
\(479\) −1.87129e12 −1.62417 −0.812083 0.583542i \(-0.801666\pi\)
−0.812083 + 0.583542i \(0.801666\pi\)
\(480\) −2.35220e12 −2.02250
\(481\) 2.83567e11 0.241548
\(482\) −1.53750e12 −1.29749
\(483\) 9.20700e11 0.769761
\(484\) −1.95450e12 −1.61894
\(485\) −1.36419e12 −1.11954
\(486\) −2.29246e12 −1.86397
\(487\) −9.21564e11 −0.742412 −0.371206 0.928551i \(-0.621056\pi\)
−0.371206 + 0.928551i \(0.621056\pi\)
\(488\) 1.42896e12 1.14060
\(489\) 1.05284e12 0.832670
\(490\) −4.79391e11 −0.375671
\(491\) −1.40709e12 −1.09259 −0.546294 0.837593i \(-0.683962\pi\)
−0.546294 + 0.837593i \(0.683962\pi\)
\(492\) −1.22309e12 −0.941054
\(493\) 7.25013e11 0.552757
\(494\) −4.95201e11 −0.374119
\(495\) 1.46842e11 0.109932
\(496\) 2.09133e10 0.0155152
\(497\) −1.10770e11 −0.0814366
\(498\) −6.02867e11 −0.439227
\(499\) 4.85195e11 0.350319 0.175160 0.984540i \(-0.443956\pi\)
0.175160 + 0.984540i \(0.443956\pi\)
\(500\) 2.30895e12 1.65215
\(501\) 3.28595e11 0.233019
\(502\) −3.08748e12 −2.16989
\(503\) 2.20742e12 1.53755 0.768776 0.639518i \(-0.220866\pi\)
0.768776 + 0.639518i \(0.220866\pi\)
\(504\) 3.77072e11 0.260307
\(505\) −2.84005e12 −1.94319
\(506\) −3.84187e11 −0.260535
\(507\) −1.47766e11 −0.0993207
\(508\) 4.17392e12 2.78072
\(509\) −8.18353e11 −0.540394 −0.270197 0.962805i \(-0.587089\pi\)
−0.270197 + 0.962805i \(0.587089\pi\)
\(510\) −2.69131e12 −1.76156
\(511\) −3.16689e11 −0.205465
\(512\) 1.25017e11 0.0803995
\(513\) −5.60173e11 −0.357104
\(514\) 4.24132e12 2.68020
\(515\) −9.24411e11 −0.579072
\(516\) 4.62185e12 2.87007
\(517\) 2.61018e11 0.160680
\(518\) −8.75734e11 −0.534427
\(519\) 2.53738e12 1.53509
\(520\) 7.73250e11 0.463773
\(521\) 7.25289e10 0.0431262 0.0215631 0.999767i \(-0.493136\pi\)
0.0215631 + 0.999767i \(0.493136\pi\)
\(522\) 1.95752e12 1.15395
\(523\) 5.08349e11 0.297101 0.148551 0.988905i \(-0.452539\pi\)
0.148551 + 0.988905i \(0.452539\pi\)
\(524\) −3.59142e12 −2.08102
\(525\) 1.37915e12 0.792308
\(526\) 2.10375e12 1.19828
\(527\) −3.54463e11 −0.200181
\(528\) −9.43322e9 −0.00528211
\(529\) 2.68005e12 1.48796
\(530\) 5.53271e11 0.304577
\(531\) 2.14071e11 0.116851
\(532\) 9.49123e11 0.513713
\(533\) −2.30241e11 −0.123569
\(534\) −2.71225e11 −0.144342
\(535\) −4.26248e12 −2.24942
\(536\) −2.42945e12 −1.27135
\(537\) 1.66506e12 0.864064
\(538\) −2.17692e12 −1.12027
\(539\) 2.84795e10 0.0145339
\(540\) 2.25030e12 1.13885
\(541\) 1.12944e12 0.566860 0.283430 0.958993i \(-0.408528\pi\)
0.283430 + 0.958993i \(0.408528\pi\)
\(542\) 4.75172e11 0.236512
\(543\) 4.54243e12 2.24228
\(544\) −1.02487e12 −0.501734
\(545\) 3.31273e12 1.60843
\(546\) 4.56343e11 0.219748
\(547\) −1.88876e12 −0.902056 −0.451028 0.892510i \(-0.648942\pi\)
−0.451028 + 0.892510i \(0.648942\pi\)
\(548\) −1.82437e12 −0.864171
\(549\) −1.56883e12 −0.737055
\(550\) −5.75487e11 −0.268166
\(551\) 1.91525e12 0.885205
\(552\) 4.58632e12 2.10251
\(553\) −1.14388e12 −0.520135
\(554\) −3.25601e12 −1.46856
\(555\) 4.07117e12 1.82138
\(556\) −2.51671e12 −1.11685
\(557\) 4.55848e11 0.200665 0.100333 0.994954i \(-0.468009\pi\)
0.100333 + 0.994954i \(0.468009\pi\)
\(558\) −9.57042e11 −0.417904
\(559\) 8.70042e11 0.376866
\(560\) −5.72906e10 −0.0246171
\(561\) 1.59885e11 0.0681513
\(562\) −1.53390e12 −0.648609
\(563\) 4.13409e12 1.73417 0.867085 0.498160i \(-0.165991\pi\)
0.867085 + 0.498160i \(0.165991\pi\)
\(564\) −8.01624e12 −3.33591
\(565\) 5.23307e12 2.16042
\(566\) 3.13054e12 1.28217
\(567\) 1.13677e12 0.461900
\(568\) −5.51785e11 −0.222435
\(569\) 4.08880e12 1.63528 0.817638 0.575733i \(-0.195283\pi\)
0.817638 + 0.575733i \(0.195283\pi\)
\(570\) −7.10959e12 −2.82103
\(571\) −4.29824e12 −1.69211 −0.846055 0.533096i \(-0.821028\pi\)
−0.846055 + 0.533096i \(0.821028\pi\)
\(572\) −1.18179e11 −0.0461593
\(573\) −4.71773e12 −1.82826
\(574\) 7.11047e11 0.273398
\(575\) 6.71255e12 2.56084
\(576\) −2.83799e12 −1.07426
\(577\) −1.59653e12 −0.599634 −0.299817 0.953997i \(-0.596926\pi\)
−0.299817 + 0.953997i \(0.596926\pi\)
\(578\) 3.18388e12 1.18654
\(579\) −6.17366e12 −2.28291
\(580\) −7.69385e12 −2.82304
\(581\) 2.17514e11 0.0791945
\(582\) −4.01046e12 −1.44891
\(583\) −3.28685e10 −0.0117834
\(584\) −1.57754e12 −0.561205
\(585\) −8.48935e11 −0.299691
\(586\) 4.85521e11 0.170086
\(587\) −3.18350e11 −0.110671 −0.0553354 0.998468i \(-0.517623\pi\)
−0.0553354 + 0.998468i \(0.517623\pi\)
\(588\) −8.74647e11 −0.301741
\(589\) −9.36377e11 −0.320577
\(590\) −1.35572e12 −0.460612
\(591\) 6.70025e12 2.25916
\(592\) −1.04656e11 −0.0350202
\(593\) 5.02708e12 1.66944 0.834718 0.550678i \(-0.185631\pi\)
0.834718 + 0.550678i \(0.185631\pi\)
\(594\) −2.15406e11 −0.0709935
\(595\) 9.71025e11 0.317617
\(596\) −9.51184e12 −3.08785
\(597\) 4.74482e12 1.52875
\(598\) 2.22110e12 0.710252
\(599\) −3.37361e11 −0.107072 −0.0535358 0.998566i \(-0.517049\pi\)
−0.0535358 + 0.998566i \(0.517049\pi\)
\(600\) 6.87002e12 2.16410
\(601\) −1.88701e12 −0.589984 −0.294992 0.955500i \(-0.595317\pi\)
−0.294992 + 0.955500i \(0.595317\pi\)
\(602\) −2.68693e12 −0.833820
\(603\) 2.66724e12 0.821549
\(604\) −7.93362e12 −2.42552
\(605\) −5.28231e12 −1.60297
\(606\) −8.34918e12 −2.51488
\(607\) −6.92739e11 −0.207119 −0.103560 0.994623i \(-0.533023\pi\)
−0.103560 + 0.994623i \(0.533023\pi\)
\(608\) −2.70738e12 −0.803494
\(609\) −1.76497e12 −0.519946
\(610\) 9.93545e12 2.90538
\(611\) −1.50902e12 −0.438036
\(612\) −1.96491e12 −0.566189
\(613\) 3.82623e12 1.09446 0.547229 0.836983i \(-0.315683\pi\)
0.547229 + 0.836983i \(0.315683\pi\)
\(614\) 4.71051e12 1.33755
\(615\) −3.30556e12 −0.931768
\(616\) 1.41866e11 0.0396977
\(617\) −4.74746e12 −1.31880 −0.659399 0.751793i \(-0.729189\pi\)
−0.659399 + 0.751793i \(0.729189\pi\)
\(618\) −2.71759e12 −0.749437
\(619\) 4.54492e12 1.24428 0.622140 0.782906i \(-0.286264\pi\)
0.622140 + 0.782906i \(0.286264\pi\)
\(620\) 3.76156e12 1.02236
\(621\) 2.51252e12 0.677949
\(622\) −4.86677e12 −1.30372
\(623\) 9.78578e10 0.0260255
\(624\) 5.45362e10 0.0143997
\(625\) 4.69957e10 0.0123197
\(626\) −2.65784e12 −0.691741
\(627\) 4.22364e11 0.109140
\(628\) 5.74020e12 1.47268
\(629\) 1.77383e12 0.451840
\(630\) 2.62175e12 0.663068
\(631\) 2.30717e12 0.579358 0.289679 0.957124i \(-0.406452\pi\)
0.289679 + 0.957124i \(0.406452\pi\)
\(632\) −5.69805e12 −1.42069
\(633\) 2.56992e12 0.636214
\(634\) 4.73120e12 1.16297
\(635\) 1.12806e13 2.75328
\(636\) 1.00944e12 0.244638
\(637\) −1.64648e11 −0.0396214
\(638\) 7.36480e11 0.175982
\(639\) 6.05793e11 0.143738
\(640\) 1.13247e13 2.66820
\(641\) −4.20974e12 −0.984904 −0.492452 0.870340i \(-0.663899\pi\)
−0.492452 + 0.870340i \(0.663899\pi\)
\(642\) −1.25308e13 −2.91120
\(643\) −5.26829e12 −1.21540 −0.607702 0.794165i \(-0.707908\pi\)
−0.607702 + 0.794165i \(0.707908\pi\)
\(644\) −4.25706e12 −0.975266
\(645\) 1.24912e13 2.84174
\(646\) −3.09769e12 −0.699828
\(647\) 4.94062e12 1.10844 0.554220 0.832370i \(-0.313017\pi\)
0.554220 + 0.832370i \(0.313017\pi\)
\(648\) 5.66263e12 1.26163
\(649\) 8.05401e10 0.0178201
\(650\) 3.32706e12 0.731057
\(651\) 8.62901e11 0.188298
\(652\) −4.86804e12 −1.05497
\(653\) −1.05701e12 −0.227494 −0.113747 0.993510i \(-0.536285\pi\)
−0.113747 + 0.993510i \(0.536285\pi\)
\(654\) 9.73877e12 2.08163
\(655\) −9.70631e12 −2.06048
\(656\) 8.49752e10 0.0179153
\(657\) 1.73194e12 0.362652
\(658\) 4.66028e12 0.969159
\(659\) −6.19800e12 −1.28017 −0.640084 0.768305i \(-0.721100\pi\)
−0.640084 + 0.768305i \(0.721100\pi\)
\(660\) −1.69670e12 −0.348062
\(661\) −3.53191e12 −0.719619 −0.359810 0.933026i \(-0.617158\pi\)
−0.359810 + 0.933026i \(0.617158\pi\)
\(662\) −3.20218e12 −0.648014
\(663\) −9.24341e11 −0.185790
\(664\) 1.08351e12 0.216311
\(665\) 2.56514e12 0.508643
\(666\) 4.78931e12 0.943277
\(667\) −8.59039e12 −1.68053
\(668\) −1.51933e12 −0.295228
\(669\) −8.30607e12 −1.60316
\(670\) −1.68917e13 −3.23845
\(671\) −5.90242e11 −0.112403
\(672\) 2.49493e12 0.471952
\(673\) −1.03450e12 −0.194384 −0.0971922 0.995266i \(-0.530986\pi\)
−0.0971922 + 0.995266i \(0.530986\pi\)
\(674\) −1.00453e12 −0.187496
\(675\) 3.76359e12 0.697807
\(676\) 6.83230e11 0.125836
\(677\) 3.37289e12 0.617097 0.308548 0.951209i \(-0.400157\pi\)
0.308548 + 0.951209i \(0.400157\pi\)
\(678\) 1.53842e13 2.79602
\(679\) 1.44697e12 0.261244
\(680\) 4.83701e12 0.867534
\(681\) −6.63221e12 −1.18167
\(682\) −3.60069e11 −0.0637318
\(683\) −8.33511e12 −1.46561 −0.732804 0.680439i \(-0.761789\pi\)
−0.732804 + 0.680439i \(0.761789\pi\)
\(684\) −5.19067e12 −0.906715
\(685\) −4.93060e12 −0.855643
\(686\) 5.08480e11 0.0876628
\(687\) 8.35434e12 1.43089
\(688\) −3.21107e11 −0.0546389
\(689\) 1.90023e11 0.0321232
\(690\) 3.18883e13 5.35563
\(691\) −7.68704e12 −1.28265 −0.641325 0.767270i \(-0.721615\pi\)
−0.641325 + 0.767270i \(0.721615\pi\)
\(692\) −1.17321e13 −1.94491
\(693\) −1.55752e11 −0.0256527
\(694\) −1.54760e13 −2.53245
\(695\) −6.80176e12 −1.10583
\(696\) −8.79191e12 −1.42017
\(697\) −1.44025e12 −0.231149
\(698\) −2.51628e12 −0.401244
\(699\) 1.39719e13 2.21364
\(700\) −6.37680e12 −1.00383
\(701\) −5.32226e12 −0.832463 −0.416231 0.909259i \(-0.636649\pi\)
−0.416231 + 0.909259i \(0.636649\pi\)
\(702\) 1.24533e12 0.193538
\(703\) 4.68590e12 0.723593
\(704\) −1.06774e12 −0.163828
\(705\) −2.16650e13 −3.30299
\(706\) 1.22369e13 1.85374
\(707\) 3.01238e12 0.453443
\(708\) −2.47350e12 −0.369967
\(709\) 2.56286e11 0.0380906 0.0190453 0.999819i \(-0.493937\pi\)
0.0190453 + 0.999819i \(0.493937\pi\)
\(710\) −3.83651e12 −0.566597
\(711\) 6.25576e12 0.918052
\(712\) 4.87463e11 0.0710857
\(713\) 4.19989e12 0.608604
\(714\) 2.85462e12 0.411061
\(715\) −3.19396e11 −0.0457038
\(716\) −7.69877e12 −1.09474
\(717\) 1.25245e13 1.76980
\(718\) 1.79147e13 2.51564
\(719\) −6.49502e11 −0.0906360 −0.0453180 0.998973i \(-0.514430\pi\)
−0.0453180 + 0.998973i \(0.514430\pi\)
\(720\) 3.13317e11 0.0434498
\(721\) 9.80504e11 0.135127
\(722\) 3.67130e12 0.502808
\(723\) −7.58135e12 −1.03187
\(724\) −2.10029e13 −2.84090
\(725\) −1.28679e13 −1.72976
\(726\) −1.55289e13 −2.07456
\(727\) 7.29457e12 0.968490 0.484245 0.874933i \(-0.339094\pi\)
0.484245 + 0.874933i \(0.339094\pi\)
\(728\) −8.20171e11 −0.108221
\(729\) −1.98501e12 −0.260309
\(730\) −1.09685e13 −1.42953
\(731\) 5.44248e12 0.704967
\(732\) 1.81272e13 2.33362
\(733\) −1.04087e13 −1.33176 −0.665882 0.746057i \(-0.731945\pi\)
−0.665882 + 0.746057i \(0.731945\pi\)
\(734\) 1.45895e13 1.85527
\(735\) −2.36386e12 −0.298764
\(736\) 1.21433e13 1.52541
\(737\) 1.00350e12 0.125289
\(738\) −3.88865e12 −0.482554
\(739\) −5.49183e12 −0.677356 −0.338678 0.940902i \(-0.609980\pi\)
−0.338678 + 0.940902i \(0.609980\pi\)
\(740\) −1.88239e13 −2.30764
\(741\) −2.44181e12 −0.297530
\(742\) −5.86843e11 −0.0710730
\(743\) 3.40388e12 0.409755 0.204878 0.978788i \(-0.434320\pi\)
0.204878 + 0.978788i \(0.434320\pi\)
\(744\) 4.29841e12 0.514315
\(745\) −2.57071e13 −3.05738
\(746\) 9.95431e12 1.17676
\(747\) −1.18957e12 −0.139780
\(748\) −7.39261e11 −0.0863457
\(749\) 4.52112e12 0.524902
\(750\) 1.83452e13 2.11712
\(751\) 1.41818e13 1.62687 0.813435 0.581656i \(-0.197595\pi\)
0.813435 + 0.581656i \(0.197595\pi\)
\(752\) 5.56936e11 0.0635075
\(753\) −1.52242e13 −1.72567
\(754\) −4.25781e12 −0.479750
\(755\) −2.14417e13 −2.40159
\(756\) −2.38684e12 −0.265752
\(757\) 1.30639e13 1.44591 0.722953 0.690897i \(-0.242784\pi\)
0.722953 + 0.690897i \(0.242784\pi\)
\(758\) 1.33088e13 1.46429
\(759\) −1.89441e12 −0.207198
\(760\) 1.27778e13 1.38930
\(761\) −3.27918e12 −0.354433 −0.177216 0.984172i \(-0.556709\pi\)
−0.177216 + 0.984172i \(0.556709\pi\)
\(762\) 3.31627e13 3.56330
\(763\) −3.51375e12 −0.375327
\(764\) 2.18134e13 2.31635
\(765\) −5.31045e12 −0.560602
\(766\) 6.00003e12 0.629685
\(767\) −4.65626e11 −0.0485801
\(768\) 1.32470e13 1.37401
\(769\) −1.98573e12 −0.204763 −0.102382 0.994745i \(-0.532646\pi\)
−0.102382 + 0.994745i \(0.532646\pi\)
\(770\) 9.86383e11 0.101120
\(771\) 2.09138e13 2.13151
\(772\) 2.85453e13 2.89239
\(773\) 1.71094e13 1.72357 0.861783 0.507277i \(-0.169348\pi\)
0.861783 + 0.507277i \(0.169348\pi\)
\(774\) 1.46946e13 1.47171
\(775\) 6.29116e12 0.626431
\(776\) 7.20787e12 0.713558
\(777\) −4.31821e12 −0.425020
\(778\) −1.60757e13 −1.57312
\(779\) −3.80469e12 −0.370170
\(780\) 9.80912e12 0.948865
\(781\) 2.27918e11 0.0219205
\(782\) 1.38939e13 1.32860
\(783\) −4.81646e12 −0.457931
\(784\) 6.07670e10 0.00574441
\(785\) 1.55137e13 1.45815
\(786\) −2.85346e13 −2.66668
\(787\) 8.64917e12 0.803689 0.401845 0.915708i \(-0.368369\pi\)
0.401845 + 0.915708i \(0.368369\pi\)
\(788\) −3.09800e13 −2.86229
\(789\) 1.03735e13 0.952970
\(790\) −3.96180e13 −3.61885
\(791\) −5.55061e12 −0.504135
\(792\) −7.75854e11 −0.0700675
\(793\) 3.41237e12 0.306427
\(794\) 6.16862e12 0.550802
\(795\) 2.72816e12 0.242224
\(796\) −2.19387e13 −1.93688
\(797\) 5.72954e12 0.502988 0.251494 0.967859i \(-0.419078\pi\)
0.251494 + 0.967859i \(0.419078\pi\)
\(798\) 7.54099e12 0.658288
\(799\) −9.43957e12 −0.819392
\(800\) 1.81898e13 1.57009
\(801\) −5.35176e11 −0.0459357
\(802\) 1.66690e13 1.42274
\(803\) 6.51611e11 0.0553056
\(804\) −3.08189e13 −2.60115
\(805\) −1.15053e13 −0.965642
\(806\) 2.08167e12 0.173741
\(807\) −1.07343e13 −0.890930
\(808\) 1.50057e13 1.23853
\(809\) −3.18891e12 −0.261743 −0.130871 0.991399i \(-0.541777\pi\)
−0.130871 + 0.991399i \(0.541777\pi\)
\(810\) 3.93718e13 3.21368
\(811\) 7.19763e12 0.584246 0.292123 0.956381i \(-0.405638\pi\)
0.292123 + 0.956381i \(0.405638\pi\)
\(812\) 8.16071e12 0.658758
\(813\) 2.34305e12 0.188094
\(814\) 1.80189e12 0.143853
\(815\) −1.31566e13 −1.04456
\(816\) 3.41147e11 0.0269362
\(817\) 1.43773e13 1.12896
\(818\) −1.86529e13 −1.45666
\(819\) 9.00448e11 0.0699328
\(820\) 1.52840e13 1.18052
\(821\) 1.43670e13 1.10362 0.551812 0.833968i \(-0.313937\pi\)
0.551812 + 0.833968i \(0.313937\pi\)
\(822\) −1.44950e13 −1.10738
\(823\) −4.72106e12 −0.358707 −0.179354 0.983785i \(-0.557401\pi\)
−0.179354 + 0.983785i \(0.557401\pi\)
\(824\) 4.88423e12 0.369082
\(825\) −2.83770e12 −0.213267
\(826\) 1.43798e12 0.107484
\(827\) 6.76103e12 0.502618 0.251309 0.967907i \(-0.419139\pi\)
0.251309 + 0.967907i \(0.419139\pi\)
\(828\) 2.32815e13 1.72137
\(829\) 2.09255e13 1.53880 0.769398 0.638770i \(-0.220556\pi\)
0.769398 + 0.638770i \(0.220556\pi\)
\(830\) 7.53357e12 0.550997
\(831\) −1.60553e13 −1.16792
\(832\) 6.17293e12 0.446618
\(833\) −1.02995e12 −0.0741160
\(834\) −1.99958e13 −1.43117
\(835\) −4.10620e12 −0.292315
\(836\) −1.95289e12 −0.138277
\(837\) 2.35479e12 0.165839
\(838\) −3.65388e13 −2.55950
\(839\) −8.97849e12 −0.625568 −0.312784 0.949824i \(-0.601262\pi\)
−0.312784 + 0.949824i \(0.601262\pi\)
\(840\) −1.17752e13 −0.816039
\(841\) 1.96049e12 0.135140
\(842\) 1.61043e13 1.10417
\(843\) −7.56358e12 −0.515826
\(844\) −1.18826e13 −0.806065
\(845\) 1.84652e12 0.124595
\(846\) −2.54866e13 −1.71059
\(847\) 5.60283e12 0.374052
\(848\) −7.01319e10 −0.00465730
\(849\) 1.54365e13 1.01968
\(850\) 2.08122e13 1.36752
\(851\) −2.10174e13 −1.37372
\(852\) −6.99971e12 −0.455095
\(853\) 3.05360e13 1.97489 0.987443 0.157974i \(-0.0504962\pi\)
0.987443 + 0.157974i \(0.0504962\pi\)
\(854\) −1.05383e13 −0.677971
\(855\) −1.40285e13 −0.897768
\(856\) 2.25213e13 1.43371
\(857\) −7.68700e12 −0.486792 −0.243396 0.969927i \(-0.578261\pi\)
−0.243396 + 0.969927i \(0.578261\pi\)
\(858\) −9.38961e11 −0.0591500
\(859\) 2.39704e13 1.50212 0.751062 0.660232i \(-0.229542\pi\)
0.751062 + 0.660232i \(0.229542\pi\)
\(860\) −5.77557e13 −3.60041
\(861\) 3.50614e12 0.217428
\(862\) 4.40327e13 2.71639
\(863\) −1.85204e12 −0.113658 −0.0568292 0.998384i \(-0.518099\pi\)
−0.0568292 + 0.998384i \(0.518099\pi\)
\(864\) 6.80848e12 0.415660
\(865\) −3.17077e13 −1.92572
\(866\) 2.31163e13 1.39665
\(867\) 1.56996e13 0.943630
\(868\) −3.98981e12 −0.238569
\(869\) 2.35361e12 0.140006
\(870\) −6.11293e13 −3.61754
\(871\) −5.80152e12 −0.341555
\(872\) −1.75032e13 −1.02516
\(873\) −7.91337e12 −0.461102
\(874\) 3.67033e13 2.12767
\(875\) −6.61893e12 −0.381726
\(876\) −2.00119e13 −1.14821
\(877\) 1.17900e13 0.673001 0.336501 0.941683i \(-0.390757\pi\)
0.336501 + 0.941683i \(0.390757\pi\)
\(878\) −3.18499e13 −1.80877
\(879\) 2.39408e12 0.135266
\(880\) 1.17880e11 0.00662624
\(881\) −2.04680e13 −1.14468 −0.572339 0.820017i \(-0.693964\pi\)
−0.572339 + 0.820017i \(0.693964\pi\)
\(882\) −2.78083e12 −0.154727
\(883\) −2.67886e13 −1.48295 −0.741475 0.670981i \(-0.765873\pi\)
−0.741475 + 0.670981i \(0.765873\pi\)
\(884\) 4.27389e12 0.235390
\(885\) −6.68499e12 −0.366316
\(886\) 9.66507e12 0.526930
\(887\) 1.60633e13 0.871320 0.435660 0.900111i \(-0.356515\pi\)
0.435660 + 0.900111i \(0.356515\pi\)
\(888\) −2.15105e13 −1.16089
\(889\) −1.19651e13 −0.642478
\(890\) 3.38929e12 0.181073
\(891\) −2.33899e12 −0.124331
\(892\) 3.84049e13 2.03116
\(893\) −2.49363e13 −1.31220
\(894\) −7.55737e13 −3.95687
\(895\) −2.08070e13 −1.08394
\(896\) −1.20119e13 −0.622624
\(897\) 1.09521e13 0.564850
\(898\) 2.07252e12 0.106354
\(899\) −8.05111e12 −0.411090
\(900\) 3.48741e13 1.77179
\(901\) 1.18867e12 0.0600898
\(902\) −1.46303e12 −0.0735910
\(903\) −1.32491e13 −0.663121
\(904\) −2.76495e13 −1.37699
\(905\) −5.67633e13 −2.81287
\(906\) −6.30343e13 −3.10814
\(907\) 6.32202e12 0.310186 0.155093 0.987900i \(-0.450432\pi\)
0.155093 + 0.987900i \(0.450432\pi\)
\(908\) 3.06655e13 1.49714
\(909\) −1.64744e13 −0.800338
\(910\) −5.70257e12 −0.275667
\(911\) −2.44911e13 −1.17808 −0.589041 0.808103i \(-0.700494\pi\)
−0.589041 + 0.808103i \(0.700494\pi\)
\(912\) 9.01202e11 0.0431366
\(913\) −4.47552e11 −0.0213169
\(914\) −4.12654e13 −1.95582
\(915\) 4.89913e13 2.31060
\(916\) −3.86281e13 −1.81290
\(917\) 1.02953e13 0.480813
\(918\) 7.79004e12 0.362032
\(919\) 1.06492e13 0.492491 0.246245 0.969208i \(-0.420803\pi\)
0.246245 + 0.969208i \(0.420803\pi\)
\(920\) −5.73118e13 −2.63754
\(921\) 2.32273e13 1.06373
\(922\) −1.08718e13 −0.495466
\(923\) −1.31766e12 −0.0597581
\(924\) 1.79965e12 0.0812204
\(925\) −3.14828e13 −1.41395
\(926\) 6.02636e13 2.69343
\(927\) −5.36229e12 −0.238502
\(928\) −2.32784e13 −1.03036
\(929\) 3.06163e13 1.34860 0.674299 0.738458i \(-0.264446\pi\)
0.674299 + 0.738458i \(0.264446\pi\)
\(930\) 2.98865e13 1.31009
\(931\) −2.72079e12 −0.118692
\(932\) −6.46020e13 −2.80462
\(933\) −2.39978e13 −1.03682
\(934\) 4.24559e13 1.82548
\(935\) −1.99796e12 −0.0854937
\(936\) 4.48544e12 0.191013
\(937\) 2.74612e13 1.16384 0.581918 0.813247i \(-0.302302\pi\)
0.581918 + 0.813247i \(0.302302\pi\)
\(938\) 1.79167e13 0.755693
\(939\) −1.31057e13 −0.550129
\(940\) 1.00173e14 4.18480
\(941\) 7.57137e12 0.314790 0.157395 0.987536i \(-0.449690\pi\)
0.157395 + 0.987536i \(0.449690\pi\)
\(942\) 4.56072e13 1.88714
\(943\) 1.70650e13 0.702754
\(944\) 1.71849e11 0.00704325
\(945\) −6.45078e12 −0.263129
\(946\) 5.52856e12 0.224441
\(947\) −1.16957e12 −0.0472552 −0.0236276 0.999721i \(-0.507522\pi\)
−0.0236276 + 0.999721i \(0.507522\pi\)
\(948\) −7.22829e13 −2.90669
\(949\) −3.76716e12 −0.150770
\(950\) 5.49792e13 2.18999
\(951\) 2.33293e13 0.924890
\(952\) −5.13051e12 −0.202439
\(953\) −3.67280e13 −1.44238 −0.721190 0.692737i \(-0.756404\pi\)
−0.721190 + 0.692737i \(0.756404\pi\)
\(954\) 3.20939e12 0.125446
\(955\) 5.89539e13 2.29349
\(956\) −5.79097e13 −2.24229
\(957\) 3.63155e12 0.139955
\(958\) 6.87444e13 2.63689
\(959\) 5.22979e12 0.199664
\(960\) 8.86247e13 3.36771
\(961\) −2.25034e13 −0.851124
\(962\) −1.04173e13 −0.392162
\(963\) −2.47256e13 −0.926465
\(964\) 3.50540e13 1.30735
\(965\) 7.71476e13 2.86384
\(966\) −3.38233e13 −1.24974
\(967\) 3.20474e13 1.17862 0.589310 0.807907i \(-0.299400\pi\)
0.589310 + 0.807907i \(0.299400\pi\)
\(968\) 2.79096e13 1.02168
\(969\) −1.52746e13 −0.556560
\(970\) 5.01157e13 1.81761
\(971\) −3.55791e13 −1.28442 −0.642211 0.766528i \(-0.721983\pi\)
−0.642211 + 0.766528i \(0.721983\pi\)
\(972\) 5.22667e13 1.87814
\(973\) 7.21448e12 0.258046
\(974\) 3.38550e13 1.20533
\(975\) 1.64056e13 0.581395
\(976\) −1.25941e12 −0.0444264
\(977\) 1.87873e13 0.659688 0.329844 0.944036i \(-0.393004\pi\)
0.329844 + 0.944036i \(0.393004\pi\)
\(978\) −3.86776e13 −1.35187
\(979\) −2.01350e11 −0.00700534
\(980\) 1.09298e13 0.378525
\(981\) 1.92164e13 0.662462
\(982\) 5.16917e13 1.77386
\(983\) −2.46921e13 −0.843466 −0.421733 0.906720i \(-0.638578\pi\)
−0.421733 + 0.906720i \(0.638578\pi\)
\(984\) 1.74653e13 0.593880
\(985\) −8.37279e13 −2.83405
\(986\) −2.66344e13 −0.897422
\(987\) 2.29796e13 0.770754
\(988\) 1.12903e13 0.376962
\(989\) −6.44858e13 −2.14329
\(990\) −5.39444e12 −0.178479
\(991\) 4.81567e13 1.58608 0.793040 0.609170i \(-0.208497\pi\)
0.793040 + 0.609170i \(0.208497\pi\)
\(992\) 1.13810e13 0.373144
\(993\) −1.57898e13 −0.515353
\(994\) 4.06931e12 0.132215
\(995\) −5.92924e13 −1.91776
\(996\) 1.37450e13 0.442565
\(997\) −2.46353e13 −0.789641 −0.394820 0.918758i \(-0.629193\pi\)
−0.394820 + 0.918758i \(0.629193\pi\)
\(998\) −1.78244e13 −0.568757
\(999\) −1.17841e13 −0.374326
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 91.10.a.b.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.b.1.2 13 1.1 even 1 trivial