Properties

Label 2.74.a.b.1.4
Level $2$
Weight $74$
Character 2.1
Self dual yes
Analytic conductor $67.497$
Analytic rank $0$
Dimension $4$
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] = [2,74,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 74, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 74);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 74 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(67.4967947474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2 x^{3} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{14}\cdot 5^{5}\cdot 7^{2}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.89813e14\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.87195e10 q^{2} +4.40707e17 q^{3} +4.72237e21 q^{4} -6.01656e25 q^{5} +3.02852e28 q^{6} +1.36725e30 q^{7} +3.24519e32 q^{8} +1.26637e35 q^{9} +O(q^{10})\) \(q+6.87195e10 q^{2} +4.40707e17 q^{3} +4.72237e21 q^{4} -6.01656e25 q^{5} +3.02852e28 q^{6} +1.36725e30 q^{7} +3.24519e32 q^{8} +1.26637e35 q^{9} -4.13455e36 q^{10} +1.17371e38 q^{11} +2.08118e39 q^{12} -4.64177e40 q^{13} +9.39565e40 q^{14} -2.65154e43 q^{15} +2.23007e43 q^{16} +7.41645e44 q^{17} +8.70246e45 q^{18} +3.86997e46 q^{19} -2.84124e47 q^{20} +6.02555e47 q^{21} +8.06565e48 q^{22} +5.58055e49 q^{23} +1.43018e50 q^{24} +2.56110e51 q^{25} -3.18980e51 q^{26} +2.60247e52 q^{27} +6.45664e51 q^{28} -1.18607e53 q^{29} -1.82212e54 q^{30} +2.47658e52 q^{31} +1.53250e54 q^{32} +5.17260e55 q^{33} +5.09655e55 q^{34} -8.22612e55 q^{35} +5.98028e56 q^{36} +2.03375e57 q^{37} +2.65942e57 q^{38} -2.04566e58 q^{39} -1.95248e58 q^{40} +1.41862e59 q^{41} +4.14073e58 q^{42} +2.36476e59 q^{43} +5.54267e59 q^{44} -7.61921e60 q^{45} +3.83492e60 q^{46} -7.61861e60 q^{47} +9.82809e60 q^{48} -4.73524e61 q^{49} +1.75998e62 q^{50} +3.26848e62 q^{51} -2.19201e62 q^{52} -6.20840e62 q^{53} +1.78841e63 q^{54} -7.06167e63 q^{55} +4.43697e62 q^{56} +1.70552e64 q^{57} -8.15063e63 q^{58} +4.74332e64 q^{59} -1.25215e65 q^{60} +6.52682e64 q^{61} +1.70189e63 q^{62} +1.73145e65 q^{63} +1.05312e65 q^{64} +2.79275e66 q^{65} +3.55459e66 q^{66} -7.25393e66 q^{67} +3.50232e66 q^{68} +2.45939e67 q^{69} -5.65294e66 q^{70} -2.81253e67 q^{71} +4.10962e67 q^{72} +5.22665e67 q^{73} +1.39758e68 q^{74} +1.12870e69 q^{75} +1.82754e68 q^{76} +1.60475e68 q^{77} -1.40577e69 q^{78} +6.42026e68 q^{79} -1.34174e69 q^{80} +2.91046e69 q^{81} +9.74871e69 q^{82} +1.58123e69 q^{83} +2.84549e69 q^{84} -4.46215e70 q^{85} +1.62505e70 q^{86} -5.22711e70 q^{87} +3.80889e70 q^{88} +4.25333e69 q^{89} -5.23588e71 q^{90} -6.34644e70 q^{91} +2.63534e71 q^{92} +1.09144e70 q^{93} -5.23547e71 q^{94} -2.32839e72 q^{95} +6.75381e71 q^{96} -2.34394e72 q^{97} -3.25403e72 q^{98} +1.48635e73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 274877906944 q^{2} + 30\!\cdots\!76 q^{3}+ \cdots + 12\!\cdots\!52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 274877906944 q^{2} + 30\!\cdots\!76 q^{3}+ \cdots + 28\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.87195e10 0.707107
\(3\) 4.40707e17 1.69521 0.847606 0.530626i \(-0.178043\pi\)
0.847606 + 0.530626i \(0.178043\pi\)
\(4\) 4.72237e21 0.500000
\(5\) −6.01656e25 −1.84903 −0.924513 0.381151i \(-0.875528\pi\)
−0.924513 + 0.381151i \(0.875528\pi\)
\(6\) 3.02852e28 1.19870
\(7\) 1.36725e30 0.194880 0.0974402 0.995241i \(-0.468935\pi\)
0.0974402 + 0.995241i \(0.468935\pi\)
\(8\) 3.24519e32 0.353553
\(9\) 1.26637e35 1.87374
\(10\) −4.13455e36 −1.30746
\(11\) 1.17371e38 1.14479 0.572396 0.819978i \(-0.306014\pi\)
0.572396 + 0.819978i \(0.306014\pi\)
\(12\) 2.08118e39 0.847606
\(13\) −4.64177e40 −1.01801 −0.509004 0.860764i \(-0.669986\pi\)
−0.509004 + 0.860764i \(0.669986\pi\)
\(14\) 9.39565e40 0.137801
\(15\) −2.65154e43 −3.13449
\(16\) 2.23007e43 0.250000
\(17\) 7.41645e44 0.909517 0.454758 0.890615i \(-0.349726\pi\)
0.454758 + 0.890615i \(0.349726\pi\)
\(18\) 8.70246e45 1.32494
\(19\) 3.86997e46 0.818844 0.409422 0.912345i \(-0.365730\pi\)
0.409422 + 0.912345i \(0.365730\pi\)
\(20\) −2.84124e47 −0.924513
\(21\) 6.02555e47 0.330364
\(22\) 8.06565e48 0.809490
\(23\) 5.58055e49 1.10563 0.552816 0.833303i \(-0.313553\pi\)
0.552816 + 0.833303i \(0.313553\pi\)
\(24\) 1.43018e50 0.599348
\(25\) 2.56110e51 2.41889
\(26\) −3.18980e51 −0.719840
\(27\) 2.60247e52 1.48118
\(28\) 6.45664e51 0.0974402
\(29\) −1.18607e53 −0.497261 −0.248631 0.968598i \(-0.579981\pi\)
−0.248631 + 0.968598i \(0.579981\pi\)
\(30\) −1.82212e54 −2.21642
\(31\) 2.47658e52 0.00910227 0.00455114 0.999990i \(-0.498551\pi\)
0.00455114 + 0.999990i \(0.498551\pi\)
\(32\) 1.53250e54 0.176777
\(33\) 5.17260e55 1.94066
\(34\) 5.09655e55 0.643125
\(35\) −8.22612e55 −0.360339
\(36\) 5.98028e56 0.936872
\(37\) 2.03375e57 1.17202 0.586010 0.810303i \(-0.300698\pi\)
0.586010 + 0.810303i \(0.300698\pi\)
\(38\) 2.65942e57 0.579010
\(39\) −2.04566e58 −1.72574
\(40\) −1.95248e58 −0.653729
\(41\) 1.41862e59 1.92866 0.964332 0.264696i \(-0.0852717\pi\)
0.964332 + 0.264696i \(0.0852717\pi\)
\(42\) 4.14073e58 0.233602
\(43\) 2.36476e59 0.565182 0.282591 0.959241i \(-0.408806\pi\)
0.282591 + 0.959241i \(0.408806\pi\)
\(44\) 5.54267e59 0.572396
\(45\) −7.61921e60 −3.46460
\(46\) 3.83492e60 0.781800
\(47\) −7.61861e60 −0.708441 −0.354221 0.935162i \(-0.615254\pi\)
−0.354221 + 0.935162i \(0.615254\pi\)
\(48\) 9.82809e60 0.423803
\(49\) −4.73524e61 −0.962022
\(50\) 1.75998e62 1.71042
\(51\) 3.26848e62 1.54182
\(52\) −2.19201e62 −0.509004
\(53\) −6.20840e62 −0.719300 −0.359650 0.933087i \(-0.617104\pi\)
−0.359650 + 0.933087i \(0.617104\pi\)
\(54\) 1.78841e63 1.04735
\(55\) −7.06167e63 −2.11675
\(56\) 4.43697e62 0.0689007
\(57\) 1.70552e64 1.38812
\(58\) −8.15063e63 −0.351617
\(59\) 4.74332e64 1.09644 0.548219 0.836335i \(-0.315306\pi\)
0.548219 + 0.836335i \(0.315306\pi\)
\(60\) −1.25215e65 −1.56725
\(61\) 6.52682e64 0.446851 0.223425 0.974721i \(-0.428276\pi\)
0.223425 + 0.974721i \(0.428276\pi\)
\(62\) 1.70189e63 0.00643628
\(63\) 1.73145e65 0.365156
\(64\) 1.05312e65 0.125000
\(65\) 2.79275e66 1.88232
\(66\) 3.55459e66 1.37226
\(67\) −7.25393e66 −1.61750 −0.808748 0.588156i \(-0.799854\pi\)
−0.808748 + 0.588156i \(0.799854\pi\)
\(68\) 3.50232e66 0.454758
\(69\) 2.45939e67 1.87428
\(70\) −5.65294e66 −0.254798
\(71\) −2.81253e67 −0.755385 −0.377692 0.925931i \(-0.623282\pi\)
−0.377692 + 0.925931i \(0.623282\pi\)
\(72\) 4.10962e67 0.662469
\(73\) 5.22665e67 0.509260 0.254630 0.967039i \(-0.418046\pi\)
0.254630 + 0.967039i \(0.418046\pi\)
\(74\) 1.39758e68 0.828744
\(75\) 1.12870e69 4.10054
\(76\) 1.82754e68 0.409422
\(77\) 1.60475e68 0.223098
\(78\) −1.40577e69 −1.22028
\(79\) 6.42026e68 0.350077 0.175039 0.984562i \(-0.443995\pi\)
0.175039 + 0.984562i \(0.443995\pi\)
\(80\) −1.34174e69 −0.462256
\(81\) 2.91046e69 0.637175
\(82\) 9.74871e69 1.36377
\(83\) 1.58123e69 0.142117 0.0710584 0.997472i \(-0.477362\pi\)
0.0710584 + 0.997472i \(0.477362\pi\)
\(84\) 2.84549e69 0.165182
\(85\) −4.46215e70 −1.68172
\(86\) 1.62505e70 0.399644
\(87\) −5.22711e70 −0.842963
\(88\) 3.80889e70 0.404745
\(89\) 4.25333e69 0.0299222 0.0149611 0.999888i \(-0.495238\pi\)
0.0149611 + 0.999888i \(0.495238\pi\)
\(90\) −5.23588e71 −2.44984
\(91\) −6.34644e70 −0.198390
\(92\) 2.63534e71 0.552816
\(93\) 1.09144e70 0.0154303
\(94\) −5.23547e71 −0.500944
\(95\) −2.32839e72 −1.51406
\(96\) 6.75381e71 0.299674
\(97\) −2.34394e72 −0.712490 −0.356245 0.934393i \(-0.615943\pi\)
−0.356245 + 0.934393i \(0.615943\pi\)
\(98\) −3.25403e72 −0.680252
\(99\) 1.48635e73 2.14505
\(100\) 1.20945e73 1.20945
\(101\) 5.56733e72 0.387183 0.193592 0.981082i \(-0.437986\pi\)
0.193592 + 0.981082i \(0.437986\pi\)
\(102\) 2.24608e73 1.09023
\(103\) 7.78162e72 0.264552 0.132276 0.991213i \(-0.457771\pi\)
0.132276 + 0.991213i \(0.457771\pi\)
\(104\) −1.50634e73 −0.359920
\(105\) −3.62531e73 −0.610851
\(106\) −4.26638e73 −0.508622
\(107\) 1.02093e74 0.863947 0.431974 0.901886i \(-0.357817\pi\)
0.431974 + 0.901886i \(0.357817\pi\)
\(108\) 1.22898e74 0.740591
\(109\) 4.14598e73 0.178468 0.0892340 0.996011i \(-0.471558\pi\)
0.0892340 + 0.996011i \(0.471558\pi\)
\(110\) −4.85274e74 −1.49677
\(111\) 8.96289e74 1.98682
\(112\) 3.04906e73 0.0487201
\(113\) −1.34444e75 −1.55303 −0.776515 0.630099i \(-0.783014\pi\)
−0.776515 + 0.630099i \(0.783014\pi\)
\(114\) 1.17203e75 0.981546
\(115\) −3.35757e75 −2.04434
\(116\) −5.60107e74 −0.248631
\(117\) −5.87822e75 −1.90749
\(118\) 3.25959e75 0.775299
\(119\) 1.01401e75 0.177247
\(120\) −8.60473e75 −1.10821
\(121\) 3.26433e75 0.310547
\(122\) 4.48520e75 0.315971
\(123\) 6.25197e76 3.26949
\(124\) 1.16953e74 0.00455114
\(125\) −9.03875e76 −2.62357
\(126\) 1.18984e76 0.258205
\(127\) 6.87081e76 1.11731 0.558653 0.829401i \(-0.311318\pi\)
0.558653 + 0.829401i \(0.311318\pi\)
\(128\) 7.23701e75 0.0883883
\(129\) 1.04217e77 0.958103
\(130\) 1.91916e77 1.33100
\(131\) −1.63020e77 −0.854747 −0.427373 0.904075i \(-0.640561\pi\)
−0.427373 + 0.904075i \(0.640561\pi\)
\(132\) 2.44269e77 0.970332
\(133\) 5.29121e76 0.159577
\(134\) −4.98486e77 −1.14374
\(135\) −1.56579e78 −2.73874
\(136\) 2.40678e77 0.321563
\(137\) −3.23319e77 −0.330621 −0.165311 0.986242i \(-0.552863\pi\)
−0.165311 + 0.986242i \(0.552863\pi\)
\(138\) 1.69008e78 1.32532
\(139\) 1.33914e78 0.806838 0.403419 0.915015i \(-0.367822\pi\)
0.403419 + 0.915015i \(0.367822\pi\)
\(140\) −3.88467e77 −0.180169
\(141\) −3.35757e78 −1.20096
\(142\) −1.93276e78 −0.534138
\(143\) −5.44807e78 −1.16541
\(144\) 2.82411e78 0.468436
\(145\) 7.13608e78 0.919448
\(146\) 3.59173e78 0.360101
\(147\) −2.08685e79 −1.63083
\(148\) 9.60413e78 0.586010
\(149\) −1.70860e79 −0.815347 −0.407674 0.913128i \(-0.633660\pi\)
−0.407674 + 0.913128i \(0.633660\pi\)
\(150\) 7.75634e79 2.89952
\(151\) 5.81417e78 0.170541 0.0852705 0.996358i \(-0.472825\pi\)
0.0852705 + 0.996358i \(0.472825\pi\)
\(152\) 1.25588e79 0.289505
\(153\) 9.39200e79 1.70420
\(154\) 1.10277e79 0.157754
\(155\) −1.49005e78 −0.0168303
\(156\) −9.66036e79 −0.862870
\(157\) −2.65451e80 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(158\) 4.41197e79 0.247542
\(159\) −2.73608e80 −1.21937
\(160\) −9.22035e79 −0.326865
\(161\) 7.62998e79 0.215466
\(162\) 2.00005e80 0.450551
\(163\) 5.19219e80 0.934335 0.467168 0.884169i \(-0.345274\pi\)
0.467168 + 0.884169i \(0.345274\pi\)
\(164\) 6.69926e80 0.964332
\(165\) −3.11213e81 −3.58834
\(166\) 1.08661e80 0.100492
\(167\) 1.52156e81 1.13015 0.565076 0.825039i \(-0.308847\pi\)
0.565076 + 0.825039i \(0.308847\pi\)
\(168\) 1.95540e80 0.116801
\(169\) 7.55530e79 0.0363402
\(170\) −3.06637e81 −1.18916
\(171\) 4.90083e81 1.53431
\(172\) 1.11673e81 0.282591
\(173\) 1.82325e81 0.373391 0.186695 0.982418i \(-0.440222\pi\)
0.186695 + 0.982418i \(0.440222\pi\)
\(174\) −3.59204e81 −0.596065
\(175\) 3.50166e81 0.471395
\(176\) 2.61745e81 0.286198
\(177\) 2.09042e82 1.85870
\(178\) 2.92287e80 0.0211582
\(179\) −9.69849e80 −0.0572227 −0.0286114 0.999591i \(-0.509109\pi\)
−0.0286114 + 0.999591i \(0.509109\pi\)
\(180\) −3.59807e82 −1.73230
\(181\) −4.27027e81 −0.167953 −0.0839765 0.996468i \(-0.526762\pi\)
−0.0839765 + 0.996468i \(0.526762\pi\)
\(182\) −4.36124e81 −0.140283
\(183\) 2.87641e82 0.757507
\(184\) 1.81099e82 0.390900
\(185\) −1.22362e83 −2.16710
\(186\) 7.50035e80 0.0109109
\(187\) 8.70474e82 1.04121
\(188\) −3.59779e82 −0.354221
\(189\) 3.55822e82 0.288654
\(190\) −1.60006e83 −1.07061
\(191\) −2.67648e83 −1.47859 −0.739293 0.673384i \(-0.764840\pi\)
−0.739293 + 0.673384i \(0.764840\pi\)
\(192\) 4.64119e82 0.211902
\(193\) −1.63168e83 −0.616300 −0.308150 0.951338i \(-0.599710\pi\)
−0.308150 + 0.951338i \(0.599710\pi\)
\(194\) −1.61075e83 −0.503806
\(195\) 1.23078e84 3.19094
\(196\) −2.23615e83 −0.481011
\(197\) 4.78478e83 0.854761 0.427380 0.904072i \(-0.359436\pi\)
0.427380 + 0.904072i \(0.359436\pi\)
\(198\) 1.02141e84 1.51678
\(199\) 7.72352e83 0.954285 0.477143 0.878826i \(-0.341673\pi\)
0.477143 + 0.878826i \(0.341673\pi\)
\(200\) 8.31126e83 0.855208
\(201\) −3.19686e84 −2.74200
\(202\) 3.82584e83 0.273780
\(203\) −1.62165e83 −0.0969065
\(204\) 1.54350e84 0.770912
\(205\) −8.53523e84 −3.56615
\(206\) 5.34749e83 0.187067
\(207\) 7.06706e84 2.07167
\(208\) −1.03515e84 −0.254502
\(209\) 4.54221e84 0.937406
\(210\) −2.49129e84 −0.431937
\(211\) −2.41050e84 −0.351397 −0.175698 0.984444i \(-0.556218\pi\)
−0.175698 + 0.984444i \(0.556218\pi\)
\(212\) −2.93183e84 −0.359650
\(213\) −1.23950e85 −1.28054
\(214\) 7.01576e84 0.610903
\(215\) −1.42277e85 −1.04504
\(216\) 8.44551e84 0.523677
\(217\) 3.38609e82 0.00177386
\(218\) 2.84910e84 0.126196
\(219\) 2.30342e85 0.863303
\(220\) −3.33478e85 −1.05837
\(221\) −3.44255e85 −0.925895
\(222\) 6.15925e85 1.40490
\(223\) 2.03817e84 0.0394560 0.0197280 0.999805i \(-0.493720\pi\)
0.0197280 + 0.999805i \(0.493720\pi\)
\(224\) 2.09530e84 0.0344503
\(225\) 3.24332e86 4.53239
\(226\) −9.23895e85 −1.09816
\(227\) −1.34214e86 −1.35786 −0.678929 0.734204i \(-0.737556\pi\)
−0.678929 + 0.734204i \(0.737556\pi\)
\(228\) 8.05411e85 0.694058
\(229\) −2.03245e85 −0.149288 −0.0746438 0.997210i \(-0.523782\pi\)
−0.0746438 + 0.997210i \(0.523782\pi\)
\(230\) −2.30730e86 −1.44557
\(231\) 7.07223e85 0.378198
\(232\) −3.84903e85 −0.175808
\(233\) −2.65619e86 −1.03698 −0.518488 0.855085i \(-0.673505\pi\)
−0.518488 + 0.855085i \(0.673505\pi\)
\(234\) −4.03948e86 −1.34880
\(235\) 4.58378e86 1.30993
\(236\) 2.23997e86 0.548219
\(237\) 2.82945e86 0.593455
\(238\) 6.96824e85 0.125333
\(239\) −1.21129e87 −1.86950 −0.934752 0.355301i \(-0.884378\pi\)
−0.934752 + 0.355301i \(0.884378\pi\)
\(240\) −5.91313e86 −0.783623
\(241\) −5.87394e86 −0.668817 −0.334409 0.942428i \(-0.608537\pi\)
−0.334409 + 0.942428i \(0.608537\pi\)
\(242\) 2.24323e86 0.219590
\(243\) −4.76226e86 −0.401036
\(244\) 3.08220e86 0.223425
\(245\) 2.84898e87 1.77880
\(246\) 4.29632e87 2.31188
\(247\) −1.79635e87 −0.833590
\(248\) 8.03695e84 0.00321814
\(249\) 6.96860e86 0.240918
\(250\) −6.21138e87 −1.85515
\(251\) 6.83827e87 1.76545 0.882726 0.469889i \(-0.155706\pi\)
0.882726 + 0.469889i \(0.155706\pi\)
\(252\) 8.17652e86 0.182578
\(253\) 6.54992e87 1.26572
\(254\) 4.72159e87 0.790055
\(255\) −1.96650e88 −2.85087
\(256\) 4.97323e86 0.0625000
\(257\) −1.37152e88 −1.49501 −0.747504 0.664257i \(-0.768748\pi\)
−0.747504 + 0.664257i \(0.768748\pi\)
\(258\) 7.16171e87 0.677481
\(259\) 2.78064e87 0.228404
\(260\) 1.31884e88 0.941161
\(261\) −1.50201e88 −0.931740
\(262\) −1.12026e88 −0.604397
\(263\) −1.37975e88 −0.647763 −0.323881 0.946098i \(-0.604988\pi\)
−0.323881 + 0.946098i \(0.604988\pi\)
\(264\) 1.67861e88 0.686128
\(265\) 3.73532e88 1.33000
\(266\) 3.63609e87 0.112838
\(267\) 1.87447e87 0.0507245
\(268\) −3.42557e88 −0.808748
\(269\) −5.70510e88 −1.17572 −0.587862 0.808962i \(-0.700030\pi\)
−0.587862 + 0.808962i \(0.700030\pi\)
\(270\) −1.07600e89 −1.93658
\(271\) 5.21634e88 0.820326 0.410163 0.912012i \(-0.365472\pi\)
0.410163 + 0.912012i \(0.365472\pi\)
\(272\) 1.65392e88 0.227379
\(273\) −2.79692e88 −0.336313
\(274\) −2.22183e88 −0.233785
\(275\) 3.00598e89 2.76913
\(276\) 1.16141e89 0.937141
\(277\) 1.77935e89 1.25820 0.629100 0.777324i \(-0.283424\pi\)
0.629100 + 0.777324i \(0.283424\pi\)
\(278\) 9.20251e88 0.570521
\(279\) 3.13627e87 0.0170553
\(280\) −2.66953e88 −0.127399
\(281\) −3.49527e89 −1.46453 −0.732267 0.681017i \(-0.761538\pi\)
−0.732267 + 0.681017i \(0.761538\pi\)
\(282\) −2.30731e89 −0.849206
\(283\) 5.88769e89 1.90432 0.952160 0.305600i \(-0.0988571\pi\)
0.952160 + 0.305600i \(0.0988571\pi\)
\(284\) −1.32818e89 −0.377692
\(285\) −1.02614e90 −2.56666
\(286\) −3.74389e89 −0.824067
\(287\) 1.93961e89 0.375859
\(288\) 1.94071e89 0.331234
\(289\) −1.14885e89 −0.172780
\(290\) 4.90387e89 0.650148
\(291\) −1.03299e90 −1.20782
\(292\) 2.46822e89 0.254630
\(293\) 2.14538e90 1.95361 0.976803 0.214141i \(-0.0686953\pi\)
0.976803 + 0.214141i \(0.0686953\pi\)
\(294\) −1.43407e90 −1.15317
\(295\) −2.85385e90 −2.02734
\(296\) 6.59991e89 0.414372
\(297\) 3.05454e90 1.69565
\(298\) −1.17414e90 −0.576538
\(299\) −2.59036e90 −1.12554
\(300\) 5.33012e90 2.05027
\(301\) 3.23321e89 0.110143
\(302\) 3.99547e89 0.120591
\(303\) 2.45356e90 0.656358
\(304\) 8.63033e89 0.204711
\(305\) −3.92690e90 −0.826239
\(306\) 6.45414e90 1.20505
\(307\) −5.86814e90 −0.972635 −0.486317 0.873782i \(-0.661660\pi\)
−0.486317 + 0.873782i \(0.661660\pi\)
\(308\) 7.57820e89 0.111549
\(309\) 3.42941e90 0.448472
\(310\) −1.02395e89 −0.0119008
\(311\) 7.44177e90 0.768993 0.384496 0.923127i \(-0.374375\pi\)
0.384496 + 0.923127i \(0.374375\pi\)
\(312\) −6.63855e90 −0.610141
\(313\) 6.11264e89 0.0499874 0.0249937 0.999688i \(-0.492043\pi\)
0.0249937 + 0.999688i \(0.492043\pi\)
\(314\) −1.82417e91 −1.32780
\(315\) −1.04173e91 −0.675183
\(316\) 3.03188e90 0.175039
\(317\) 4.50179e90 0.231591 0.115796 0.993273i \(-0.463058\pi\)
0.115796 + 0.993273i \(0.463058\pi\)
\(318\) −1.88022e91 −0.862223
\(319\) −1.39210e91 −0.569260
\(320\) −6.33617e90 −0.231128
\(321\) 4.49930e91 1.46457
\(322\) 5.24328e90 0.152358
\(323\) 2.87015e91 0.744753
\(324\) 1.37443e91 0.318587
\(325\) −1.18881e92 −2.46245
\(326\) 3.56804e91 0.660675
\(327\) 1.82716e91 0.302541
\(328\) 4.60370e91 0.681886
\(329\) −1.04165e91 −0.138061
\(330\) −2.13864e92 −2.53734
\(331\) 4.58554e91 0.487157 0.243578 0.969881i \(-0.421679\pi\)
0.243578 + 0.969881i \(0.421679\pi\)
\(332\) 7.46716e90 0.0710584
\(333\) 2.57549e92 2.19607
\(334\) 1.04561e92 0.799138
\(335\) 4.36437e92 2.99079
\(336\) 1.34374e91 0.0825910
\(337\) −5.44105e91 −0.300049 −0.150024 0.988682i \(-0.547935\pi\)
−0.150024 + 0.988682i \(0.547935\pi\)
\(338\) 5.19196e90 0.0256964
\(339\) −5.92506e92 −2.63272
\(340\) −2.10719e92 −0.840860
\(341\) 2.90677e90 0.0104202
\(342\) 3.36783e92 1.08492
\(343\) −1.32041e92 −0.382360
\(344\) 7.67408e91 0.199822
\(345\) −1.47970e93 −3.46559
\(346\) 1.25293e92 0.264027
\(347\) 4.82676e92 0.915439 0.457720 0.889097i \(-0.348666\pi\)
0.457720 + 0.889097i \(0.348666\pi\)
\(348\) −2.46843e92 −0.421482
\(349\) 2.26931e92 0.348951 0.174476 0.984661i \(-0.444177\pi\)
0.174476 + 0.984661i \(0.444177\pi\)
\(350\) 2.40632e92 0.333327
\(351\) −1.20801e93 −1.50786
\(352\) 1.79870e92 0.202372
\(353\) 3.79349e92 0.384824 0.192412 0.981314i \(-0.438369\pi\)
0.192412 + 0.981314i \(0.438369\pi\)
\(354\) 1.43652e93 1.31430
\(355\) 1.69218e93 1.39673
\(356\) 2.00858e91 0.0149611
\(357\) 4.46882e92 0.300471
\(358\) −6.66475e91 −0.0404626
\(359\) 4.26527e92 0.233883 0.116942 0.993139i \(-0.462691\pi\)
0.116942 + 0.993139i \(0.462691\pi\)
\(360\) −2.47258e93 −1.22492
\(361\) −7.35970e92 −0.329494
\(362\) −2.93451e92 −0.118761
\(363\) 1.43861e93 0.526444
\(364\) −2.99702e92 −0.0991949
\(365\) −3.14465e93 −0.941634
\(366\) 1.97666e93 0.535638
\(367\) 1.28566e93 0.315365 0.157683 0.987490i \(-0.449598\pi\)
0.157683 + 0.987490i \(0.449598\pi\)
\(368\) 1.24450e93 0.276408
\(369\) 1.79651e94 3.61382
\(370\) −8.40865e93 −1.53237
\(371\) −8.48841e92 −0.140178
\(372\) 5.15420e91 0.00771514
\(373\) −5.43996e93 −0.738284 −0.369142 0.929373i \(-0.620348\pi\)
−0.369142 + 0.929373i \(0.620348\pi\)
\(374\) 5.98185e93 0.736244
\(375\) −3.98344e94 −4.44751
\(376\) −2.47238e93 −0.250472
\(377\) 5.50548e93 0.506216
\(378\) 2.44519e93 0.204109
\(379\) 5.07066e93 0.384356 0.192178 0.981360i \(-0.438445\pi\)
0.192178 + 0.981360i \(0.438445\pi\)
\(380\) −1.09955e94 −0.757032
\(381\) 3.02802e94 1.89407
\(382\) −1.83927e94 −1.04552
\(383\) 1.73605e94 0.897032 0.448516 0.893775i \(-0.351953\pi\)
0.448516 + 0.893775i \(0.351953\pi\)
\(384\) 3.18940e93 0.149837
\(385\) −9.65504e93 −0.412513
\(386\) −1.12128e94 −0.435790
\(387\) 2.99467e94 1.05901
\(388\) −1.10690e94 −0.356245
\(389\) −5.19087e94 −1.52083 −0.760414 0.649439i \(-0.775004\pi\)
−0.760414 + 0.649439i \(0.775004\pi\)
\(390\) 8.45788e94 2.25633
\(391\) 4.13879e94 1.00559
\(392\) −1.53667e94 −0.340126
\(393\) −7.18439e94 −1.44898
\(394\) 3.28808e94 0.604407
\(395\) −3.86279e94 −0.647302
\(396\) 7.01909e94 1.07252
\(397\) 9.79236e94 1.36469 0.682345 0.731030i \(-0.260960\pi\)
0.682345 + 0.731030i \(0.260960\pi\)
\(398\) 5.30757e94 0.674782
\(399\) 2.33187e94 0.270517
\(400\) 5.71145e94 0.604724
\(401\) −1.67677e95 −1.62071 −0.810354 0.585941i \(-0.800725\pi\)
−0.810354 + 0.585941i \(0.800725\pi\)
\(402\) −2.19686e95 −1.93889
\(403\) −1.14957e93 −0.00926619
\(404\) 2.62910e94 0.193592
\(405\) −1.75110e95 −1.17815
\(406\) −1.11439e94 −0.0685232
\(407\) 2.38703e95 1.34172
\(408\) 1.06068e95 0.545117
\(409\) −1.70027e95 −0.799127 −0.399564 0.916705i \(-0.630838\pi\)
−0.399564 + 0.916705i \(0.630838\pi\)
\(410\) −5.86537e95 −2.52165
\(411\) −1.42489e95 −0.560473
\(412\) 3.67476e94 0.132276
\(413\) 6.48529e94 0.213674
\(414\) 4.85645e95 1.46489
\(415\) −9.51357e94 −0.262777
\(416\) −7.11349e94 −0.179960
\(417\) 5.90169e95 1.36776
\(418\) 3.12138e95 0.662846
\(419\) 4.66273e95 0.907463 0.453731 0.891138i \(-0.350093\pi\)
0.453731 + 0.891138i \(0.350093\pi\)
\(420\) −1.71200e95 −0.305426
\(421\) 6.58342e95 1.07685 0.538425 0.842674i \(-0.319020\pi\)
0.538425 + 0.842674i \(0.319020\pi\)
\(422\) −1.65648e95 −0.248475
\(423\) −9.64801e95 −1.32744
\(424\) −2.01474e95 −0.254311
\(425\) 1.89943e96 2.20002
\(426\) −8.51780e95 −0.905477
\(427\) 8.92377e94 0.0870825
\(428\) 4.82120e95 0.431974
\(429\) −2.40100e96 −1.97561
\(430\) −9.77720e95 −0.738951
\(431\) −1.71148e96 −1.18837 −0.594184 0.804329i \(-0.702525\pi\)
−0.594184 + 0.804329i \(0.702525\pi\)
\(432\) 5.80371e95 0.370296
\(433\) 2.15999e96 1.26661 0.633306 0.773901i \(-0.281697\pi\)
0.633306 + 0.773901i \(0.281697\pi\)
\(434\) 2.32690e93 0.00125431
\(435\) 3.14492e96 1.55866
\(436\) 1.95789e95 0.0892340
\(437\) 2.15966e96 0.905341
\(438\) 1.58290e96 0.610448
\(439\) −3.71212e96 −1.31725 −0.658624 0.752472i \(-0.728861\pi\)
−0.658624 + 0.752472i \(0.728861\pi\)
\(440\) −2.29164e96 −0.748383
\(441\) −5.99658e96 −1.80258
\(442\) −2.36570e96 −0.654707
\(443\) −2.95072e96 −0.751956 −0.375978 0.926629i \(-0.622693\pi\)
−0.375978 + 0.926629i \(0.622693\pi\)
\(444\) 4.23261e96 0.993412
\(445\) −2.55904e95 −0.0553270
\(446\) 1.40062e95 0.0278996
\(447\) −7.52994e96 −1.38219
\(448\) 1.43988e95 0.0243601
\(449\) −7.64002e96 −1.19153 −0.595763 0.803161i \(-0.703150\pi\)
−0.595763 + 0.803161i \(0.703150\pi\)
\(450\) 2.22879e97 3.20488
\(451\) 1.66505e97 2.20792
\(452\) −6.34896e96 −0.776515
\(453\) 2.56234e96 0.289103
\(454\) −9.22313e96 −0.960151
\(455\) 3.81837e96 0.366828
\(456\) 5.53474e96 0.490773
\(457\) 5.24571e96 0.429401 0.214700 0.976680i \(-0.431122\pi\)
0.214700 + 0.976680i \(0.431122\pi\)
\(458\) −1.39669e96 −0.105562
\(459\) 1.93011e97 1.34716
\(460\) −1.58557e97 −1.02217
\(461\) 1.20990e97 0.720550 0.360275 0.932846i \(-0.382683\pi\)
0.360275 + 0.932846i \(0.382683\pi\)
\(462\) 4.86000e96 0.267426
\(463\) −2.83398e97 −1.44109 −0.720544 0.693409i \(-0.756108\pi\)
−0.720544 + 0.693409i \(0.756108\pi\)
\(464\) −2.64503e96 −0.124315
\(465\) −6.56674e95 −0.0285310
\(466\) −1.82532e97 −0.733252
\(467\) 2.09309e97 0.777537 0.388768 0.921336i \(-0.372901\pi\)
0.388768 + 0.921336i \(0.372901\pi\)
\(468\) −2.77591e97 −0.953744
\(469\) −9.91791e96 −0.315218
\(470\) 3.14995e97 0.926258
\(471\) −1.16986e98 −3.18325
\(472\) 1.53930e97 0.387650
\(473\) 2.77553e97 0.647015
\(474\) 1.94439e97 0.419636
\(475\) 9.91140e97 1.98070
\(476\) 4.78854e96 0.0886235
\(477\) −7.86215e97 −1.34779
\(478\) −8.32395e97 −1.32194
\(479\) 9.84743e97 1.44903 0.724514 0.689260i \(-0.242064\pi\)
0.724514 + 0.689260i \(0.242064\pi\)
\(480\) −4.06347e97 −0.554105
\(481\) −9.44021e97 −1.19313
\(482\) −4.03654e97 −0.472925
\(483\) 3.36259e97 0.365261
\(484\) 1.54154e97 0.155274
\(485\) 1.41025e98 1.31741
\(486\) −3.27260e97 −0.283576
\(487\) 1.79002e98 1.43897 0.719485 0.694508i \(-0.244378\pi\)
0.719485 + 0.694508i \(0.244378\pi\)
\(488\) 2.11807e97 0.157986
\(489\) 2.28823e98 1.58390
\(490\) 1.95781e98 1.25780
\(491\) −1.44665e98 −0.862761 −0.431380 0.902170i \(-0.641973\pi\)
−0.431380 + 0.902170i \(0.641973\pi\)
\(492\) 2.95241e98 1.63475
\(493\) −8.79646e97 −0.452267
\(494\) −1.23444e98 −0.589437
\(495\) −8.94272e98 −3.96625
\(496\) 5.52295e95 0.00227557
\(497\) −3.84543e97 −0.147210
\(498\) 4.78879e97 0.170355
\(499\) 2.56109e97 0.0846749 0.0423374 0.999103i \(-0.486520\pi\)
0.0423374 + 0.999103i \(0.486520\pi\)
\(500\) −4.26843e98 −1.31179
\(501\) 6.70560e98 1.91585
\(502\) 4.69923e98 1.24836
\(503\) −5.69558e98 −1.40704 −0.703520 0.710675i \(-0.748389\pi\)
−0.703520 + 0.710675i \(0.748389\pi\)
\(504\) 5.61886e97 0.129102
\(505\) −3.34961e98 −0.715912
\(506\) 4.50107e98 0.894998
\(507\) 3.32967e97 0.0616043
\(508\) 3.24465e98 0.558653
\(509\) −1.31519e97 −0.0210761 −0.0105381 0.999944i \(-0.503354\pi\)
−0.0105381 + 0.999944i \(0.503354\pi\)
\(510\) −1.35137e99 −2.01587
\(511\) 7.14612e97 0.0992448
\(512\) 3.41758e97 0.0441942
\(513\) 1.00715e99 1.21286
\(514\) −9.42503e98 −1.05713
\(515\) −4.68185e98 −0.489163
\(516\) 4.92149e98 0.479051
\(517\) −8.94201e98 −0.811018
\(518\) 1.91084e98 0.161506
\(519\) 8.03520e98 0.632977
\(520\) 9.06298e98 0.665501
\(521\) 9.33703e98 0.639193 0.319596 0.947554i \(-0.396453\pi\)
0.319596 + 0.947554i \(0.396453\pi\)
\(522\) −1.03217e99 −0.658840
\(523\) 1.72766e99 1.02837 0.514183 0.857681i \(-0.328095\pi\)
0.514183 + 0.857681i \(0.328095\pi\)
\(524\) −7.69838e98 −0.427373
\(525\) 1.54321e99 0.799115
\(526\) −9.48158e98 −0.458038
\(527\) 1.83674e97 0.00827867
\(528\) 1.15353e99 0.485166
\(529\) 5.66646e98 0.222423
\(530\) 2.56689e99 0.940455
\(531\) 6.00682e99 2.05445
\(532\) 2.49870e98 0.0797884
\(533\) −6.58492e99 −1.96339
\(534\) 1.28813e98 0.0358677
\(535\) −6.14247e99 −1.59746
\(536\) −2.35403e99 −0.571871
\(537\) −4.27419e98 −0.0970047
\(538\) −3.92052e99 −0.831362
\(539\) −5.55778e99 −1.10131
\(540\) −7.39424e99 −1.36937
\(541\) 5.02383e99 0.869631 0.434816 0.900520i \(-0.356814\pi\)
0.434816 + 0.900520i \(0.356814\pi\)
\(542\) 3.58464e99 0.580058
\(543\) −1.88194e99 −0.284716
\(544\) 1.13657e99 0.160781
\(545\) −2.49446e99 −0.329992
\(546\) −1.92203e99 −0.237809
\(547\) 7.38676e99 0.854902 0.427451 0.904038i \(-0.359412\pi\)
0.427451 + 0.904038i \(0.359412\pi\)
\(548\) −1.52683e99 −0.165311
\(549\) 8.26540e99 0.837285
\(550\) 2.06570e100 1.95807
\(551\) −4.59007e99 −0.407180
\(552\) 7.98116e99 0.662659
\(553\) 8.77808e98 0.0682233
\(554\) 1.22276e100 0.889682
\(555\) −5.39258e100 −3.67369
\(556\) 6.32392e99 0.403419
\(557\) 2.60182e100 1.55440 0.777200 0.629254i \(-0.216639\pi\)
0.777200 + 0.629254i \(0.216639\pi\)
\(558\) 2.15523e98 0.0120599
\(559\) −1.09767e100 −0.575359
\(560\) −1.83449e99 −0.0900847
\(561\) 3.83624e100 1.76507
\(562\) −2.40193e100 −1.03558
\(563\) −3.27071e100 −1.32156 −0.660778 0.750582i \(-0.729773\pi\)
−0.660778 + 0.750582i \(0.729773\pi\)
\(564\) −1.58557e100 −0.600479
\(565\) 8.08893e100 2.87159
\(566\) 4.04599e100 1.34656
\(567\) 3.97932e99 0.124173
\(568\) −9.12720e99 −0.267069
\(569\) 1.42662e100 0.391480 0.195740 0.980656i \(-0.437289\pi\)
0.195740 + 0.980656i \(0.437289\pi\)
\(570\) −7.05157e100 −1.81490
\(571\) −4.71485e100 −1.13828 −0.569141 0.822240i \(-0.692724\pi\)
−0.569141 + 0.822240i \(0.692724\pi\)
\(572\) −2.57278e100 −0.582703
\(573\) −1.17954e101 −2.50652
\(574\) 1.33289e100 0.265772
\(575\) 1.42924e101 2.67441
\(576\) 1.33365e100 0.234218
\(577\) −5.71235e100 −0.941668 −0.470834 0.882222i \(-0.656047\pi\)
−0.470834 + 0.882222i \(0.656047\pi\)
\(578\) −7.89484e99 −0.122174
\(579\) −7.19091e100 −1.04476
\(580\) 3.36992e100 0.459724
\(581\) 2.16193e99 0.0276958
\(582\) −7.09867e100 −0.854059
\(583\) −7.28683e100 −0.823449
\(584\) 1.69615e100 0.180050
\(585\) 3.53666e101 3.52699
\(586\) 1.47430e101 1.38141
\(587\) −1.41978e101 −1.25006 −0.625030 0.780601i \(-0.714913\pi\)
−0.625030 + 0.780601i \(0.714913\pi\)
\(588\) −9.85488e100 −0.815415
\(589\) 9.58428e98 0.00745335
\(590\) −1.96115e101 −1.43355
\(591\) 2.10869e101 1.44900
\(592\) 4.53542e100 0.293005
\(593\) −1.84307e101 −1.11956 −0.559778 0.828642i \(-0.689114\pi\)
−0.559778 + 0.828642i \(0.689114\pi\)
\(594\) 2.09906e101 1.19900
\(595\) −6.10086e100 −0.327734
\(596\) −8.06866e100 −0.407674
\(597\) 3.40381e101 1.61772
\(598\) −1.78008e101 −0.795879
\(599\) −3.83951e100 −0.161509 −0.0807546 0.996734i \(-0.525733\pi\)
−0.0807546 + 0.996734i \(0.525733\pi\)
\(600\) 3.66283e101 1.44976
\(601\) 2.44570e101 0.910929 0.455465 0.890254i \(-0.349473\pi\)
0.455465 + 0.890254i \(0.349473\pi\)
\(602\) 2.22184e100 0.0778828
\(603\) −9.18619e101 −3.03077
\(604\) 2.74566e100 0.0852705
\(605\) −1.96400e101 −0.574210
\(606\) 1.68607e101 0.464115
\(607\) 7.40100e101 1.91824 0.959120 0.283000i \(-0.0913297\pi\)
0.959120 + 0.283000i \(0.0913297\pi\)
\(608\) 5.93072e100 0.144753
\(609\) −7.14674e100 −0.164277
\(610\) −2.69854e101 −0.584239
\(611\) 3.53638e101 0.721199
\(612\) 4.43525e101 0.852101
\(613\) 1.17887e101 0.213382 0.106691 0.994292i \(-0.465974\pi\)
0.106691 + 0.994292i \(0.465974\pi\)
\(614\) −4.03256e101 −0.687757
\(615\) −3.76154e102 −6.04538
\(616\) 5.20770e100 0.0788769
\(617\) −8.71142e101 −1.24360 −0.621799 0.783177i \(-0.713598\pi\)
−0.621799 + 0.783177i \(0.713598\pi\)
\(618\) 2.35667e101 0.317117
\(619\) 7.46816e101 0.947339 0.473669 0.880703i \(-0.342929\pi\)
0.473669 + 0.880703i \(0.342929\pi\)
\(620\) −7.03654e99 −0.00841517
\(621\) 1.45232e102 1.63764
\(622\) 5.11395e101 0.543760
\(623\) 5.81535e99 0.00583126
\(624\) −4.56197e101 −0.431435
\(625\) 2.72654e102 2.43216
\(626\) 4.20057e100 0.0353464
\(627\) 2.00178e102 1.58910
\(628\) −1.25356e102 −0.938895
\(629\) 1.50832e102 1.06597
\(630\) −7.15874e101 −0.477427
\(631\) 1.75887e102 1.10704 0.553518 0.832837i \(-0.313285\pi\)
0.553518 + 0.832837i \(0.313285\pi\)
\(632\) 2.08349e101 0.123771
\(633\) −1.06232e102 −0.595692
\(634\) 3.09360e101 0.163760
\(635\) −4.13386e102 −2.06593
\(636\) −1.29208e102 −0.609683
\(637\) 2.19799e102 0.979346
\(638\) −9.56645e101 −0.402528
\(639\) −3.56172e102 −1.41540
\(640\) −4.35419e101 −0.163432
\(641\) 5.11680e102 1.81418 0.907091 0.420935i \(-0.138298\pi\)
0.907091 + 0.420935i \(0.138298\pi\)
\(642\) 3.09190e102 1.03561
\(643\) −2.35088e101 −0.0743925 −0.0371963 0.999308i \(-0.511843\pi\)
−0.0371963 + 0.999308i \(0.511843\pi\)
\(644\) 3.60316e101 0.107733
\(645\) −6.27025e102 −1.77156
\(646\) 1.97235e102 0.526620
\(647\) −5.13307e102 −1.29530 −0.647651 0.761937i \(-0.724249\pi\)
−0.647651 + 0.761937i \(0.724249\pi\)
\(648\) 9.44499e101 0.225275
\(649\) 5.56727e102 1.25519
\(650\) −8.16941e102 −1.74122
\(651\) 1.49227e100 0.00300706
\(652\) 2.45194e102 0.467168
\(653\) −1.01178e103 −1.82286 −0.911430 0.411454i \(-0.865021\pi\)
−0.911430 + 0.411454i \(0.865021\pi\)
\(654\) 1.25562e102 0.213929
\(655\) 9.80816e102 1.58045
\(656\) 3.16364e102 0.482166
\(657\) 6.61890e102 0.954223
\(658\) −7.15818e101 −0.0976241
\(659\) 1.89029e102 0.243900 0.121950 0.992536i \(-0.461085\pi\)
0.121950 + 0.992536i \(0.461085\pi\)
\(660\) −1.46966e103 −1.79417
\(661\) 9.13646e100 0.0105542 0.00527709 0.999986i \(-0.498320\pi\)
0.00527709 + 0.999986i \(0.498320\pi\)
\(662\) 3.15116e102 0.344472
\(663\) −1.51715e103 −1.56959
\(664\) 5.13139e101 0.0502459
\(665\) −3.18348e102 −0.295062
\(666\) 1.76987e103 1.55285
\(667\) −6.61894e102 −0.549788
\(668\) 7.18535e102 0.565076
\(669\) 8.98236e101 0.0668863
\(670\) 2.99917e103 2.11481
\(671\) 7.66057e102 0.511551
\(672\) 9.23413e101 0.0584006
\(673\) −2.13125e103 −1.27669 −0.638344 0.769751i \(-0.720380\pi\)
−0.638344 + 0.769751i \(0.720380\pi\)
\(674\) −3.73906e102 −0.212166
\(675\) 6.66520e103 3.58283
\(676\) 3.56789e101 0.0181701
\(677\) −5.31475e102 −0.256446 −0.128223 0.991745i \(-0.540927\pi\)
−0.128223 + 0.991745i \(0.540927\pi\)
\(678\) −4.07167e103 −1.86161
\(679\) −3.20475e102 −0.138850
\(680\) −1.44805e103 −0.594578
\(681\) −5.91491e103 −2.30186
\(682\) 1.99752e101 0.00736820
\(683\) −5.72510e102 −0.200183 −0.100092 0.994978i \(-0.531914\pi\)
−0.100092 + 0.994978i \(0.531914\pi\)
\(684\) 2.31435e103 0.767153
\(685\) 1.94527e103 0.611327
\(686\) −9.07376e102 −0.270369
\(687\) −8.95713e102 −0.253074
\(688\) 5.27359e102 0.141295
\(689\) 2.88179e103 0.732254
\(690\) −1.01684e104 −2.45055
\(691\) 4.29996e103 0.982914 0.491457 0.870902i \(-0.336464\pi\)
0.491457 + 0.870902i \(0.336464\pi\)
\(692\) 8.61006e102 0.186695
\(693\) 2.03221e103 0.418028
\(694\) 3.31692e103 0.647313
\(695\) −8.05702e103 −1.49186
\(696\) −1.69629e103 −0.298032
\(697\) 1.05212e104 1.75415
\(698\) 1.55946e103 0.246746
\(699\) −1.17060e104 −1.75789
\(700\) 1.65361e103 0.235698
\(701\) −1.07964e104 −1.46074 −0.730369 0.683053i \(-0.760652\pi\)
−0.730369 + 0.683053i \(0.760652\pi\)
\(702\) −8.30136e103 −1.06622
\(703\) 7.87057e103 0.959703
\(704\) 1.23606e103 0.143099
\(705\) 2.02010e104 2.22060
\(706\) 2.60687e103 0.272112
\(707\) 7.61191e102 0.0754545
\(708\) 9.87171e103 0.929348
\(709\) −4.86603e103 −0.435098 −0.217549 0.976049i \(-0.569806\pi\)
−0.217549 + 0.976049i \(0.569806\pi\)
\(710\) 1.16286e104 0.987634
\(711\) 8.13046e103 0.655956
\(712\) 1.38028e102 0.0105791
\(713\) 1.38207e102 0.0100638
\(714\) 3.07095e103 0.212465
\(715\) 3.27786e104 2.15487
\(716\) −4.57998e102 −0.0286114
\(717\) −5.33826e104 −3.16921
\(718\) 2.93107e103 0.165381
\(719\) −2.92832e103 −0.157041 −0.0785207 0.996912i \(-0.525020\pi\)
−0.0785207 + 0.996912i \(0.525020\pi\)
\(720\) −1.69914e104 −0.866150
\(721\) 1.06394e103 0.0515560
\(722\) −5.05755e103 −0.232987
\(723\) −2.58869e104 −1.13379
\(724\) −2.01658e103 −0.0839765
\(725\) −3.03766e104 −1.20282
\(726\) 9.88607e103 0.372252
\(727\) 2.53306e104 0.907069 0.453534 0.891239i \(-0.350163\pi\)
0.453534 + 0.891239i \(0.350163\pi\)
\(728\) −2.05954e103 −0.0701414
\(729\) −4.06580e104 −1.31702
\(730\) −2.16098e104 −0.665836
\(731\) 1.75381e104 0.514042
\(732\) 1.35835e104 0.378754
\(733\) −9.16740e103 −0.243193 −0.121596 0.992580i \(-0.538801\pi\)
−0.121596 + 0.992580i \(0.538801\pi\)
\(734\) 8.83497e103 0.222997
\(735\) 1.25557e105 3.01545
\(736\) 8.55216e103 0.195450
\(737\) −8.51398e104 −1.85169
\(738\) 1.23455e105 2.55536
\(739\) −5.93173e104 −1.16858 −0.584292 0.811544i \(-0.698628\pi\)
−0.584292 + 0.811544i \(0.698628\pi\)
\(740\) −5.77838e104 −1.08355
\(741\) −7.91665e104 −1.41311
\(742\) −5.83319e103 −0.0991205
\(743\) 1.03004e105 1.66633 0.833167 0.553021i \(-0.186525\pi\)
0.833167 + 0.553021i \(0.186525\pi\)
\(744\) 3.54194e102 0.00545543
\(745\) 1.02799e105 1.50760
\(746\) −3.73831e104 −0.522046
\(747\) 2.00243e104 0.266291
\(748\) 4.11070e104 0.520603
\(749\) 1.39586e104 0.168366
\(750\) −2.73740e105 −3.14487
\(751\) −1.02419e105 −1.12078 −0.560392 0.828228i \(-0.689349\pi\)
−0.560392 + 0.828228i \(0.689349\pi\)
\(752\) −1.69901e104 −0.177110
\(753\) 3.01368e105 2.99281
\(754\) 3.78334e104 0.357949
\(755\) −3.49813e104 −0.315335
\(756\) 1.68032e104 0.144327
\(757\) −1.34912e104 −0.110421 −0.0552103 0.998475i \(-0.517583\pi\)
−0.0552103 + 0.998475i \(0.517583\pi\)
\(758\) 3.48453e104 0.271781
\(759\) 2.88660e105 2.14566
\(760\) −7.55606e104 −0.535303
\(761\) 1.84315e104 0.124458 0.0622288 0.998062i \(-0.480179\pi\)
0.0622288 + 0.998062i \(0.480179\pi\)
\(762\) 2.08084e105 1.33931
\(763\) 5.66858e103 0.0347799
\(764\) −1.26393e105 −0.739293
\(765\) −5.65075e105 −3.15111
\(766\) 1.19301e105 0.634298
\(767\) −2.20174e105 −1.11618
\(768\) 2.19174e104 0.105951
\(769\) 1.37771e104 0.0635107 0.0317554 0.999496i \(-0.489890\pi\)
0.0317554 + 0.999496i \(0.489890\pi\)
\(770\) −6.63489e104 −0.291691
\(771\) −6.04439e105 −2.53436
\(772\) −7.70537e104 −0.308150
\(773\) 1.82843e105 0.697472 0.348736 0.937221i \(-0.386611\pi\)
0.348736 + 0.937221i \(0.386611\pi\)
\(774\) 2.05792e105 0.748830
\(775\) 6.34277e103 0.0220174
\(776\) −7.60653e104 −0.251903
\(777\) 1.22545e105 0.387193
\(778\) −3.56714e105 −1.07539
\(779\) 5.49004e105 1.57928
\(780\) 5.81221e105 1.59547
\(781\) −3.30109e105 −0.864758
\(782\) 2.84415e105 0.711060
\(783\) −3.08672e105 −0.736535
\(784\) −1.05599e105 −0.240505
\(785\) 1.59710e106 3.47208
\(786\) −4.93707e105 −1.02458
\(787\) −3.90064e105 −0.772784 −0.386392 0.922335i \(-0.626279\pi\)
−0.386392 + 0.922335i \(0.626279\pi\)
\(788\) 2.25955e105 0.427380
\(789\) −6.08066e105 −1.09810
\(790\) −2.65449e105 −0.457712
\(791\) −1.83819e105 −0.302655
\(792\) 4.82348e105 0.758389
\(793\) −3.02960e105 −0.454898
\(794\) 6.72926e105 0.964981
\(795\) 1.64618e106 2.25464
\(796\) 3.64733e105 0.477143
\(797\) −8.67794e105 −1.08440 −0.542198 0.840251i \(-0.682408\pi\)
−0.542198 + 0.840251i \(0.682408\pi\)
\(798\) 1.60245e105 0.191284
\(799\) −5.65031e105 −0.644339
\(800\) 3.92488e105 0.427604
\(801\) 5.38631e104 0.0560666
\(802\) −1.15227e106 −1.14601
\(803\) 6.13455e105 0.582996
\(804\) −1.50967e106 −1.37100
\(805\) −4.59062e105 −0.398402
\(806\) −7.89978e103 −0.00655218
\(807\) −2.51428e106 −1.99310
\(808\) 1.80670e105 0.136890
\(809\) 1.18602e106 0.858955 0.429478 0.903077i \(-0.358698\pi\)
0.429478 + 0.903077i \(0.358698\pi\)
\(810\) −1.20334e106 −0.833079
\(811\) 1.04206e106 0.689653 0.344826 0.938666i \(-0.387938\pi\)
0.344826 + 0.938666i \(0.387938\pi\)
\(812\) −7.65805e104 −0.0484532
\(813\) 2.29888e106 1.39063
\(814\) 1.64035e106 0.948739
\(815\) −3.12391e106 −1.72761
\(816\) 7.28896e105 0.385456
\(817\) 9.15155e105 0.462796
\(818\) −1.16841e106 −0.565068
\(819\) −8.03697e105 −0.371732
\(820\) −4.03065e106 −1.78307
\(821\) −3.06697e106 −1.29773 −0.648864 0.760904i \(-0.724756\pi\)
−0.648864 + 0.760904i \(0.724756\pi\)
\(822\) −9.79177e105 −0.396315
\(823\) −6.90406e105 −0.267307 −0.133654 0.991028i \(-0.542671\pi\)
−0.133654 + 0.991028i \(0.542671\pi\)
\(824\) 2.52528e105 0.0935333
\(825\) 1.32476e107 4.69426
\(826\) 4.45666e105 0.151091
\(827\) 4.71629e106 1.52985 0.764926 0.644119i \(-0.222776\pi\)
0.764926 + 0.644119i \(0.222776\pi\)
\(828\) 3.33733e106 1.03584
\(829\) −6.66879e106 −1.98065 −0.990323 0.138781i \(-0.955682\pi\)
−0.990323 + 0.138781i \(0.955682\pi\)
\(830\) −6.53768e105 −0.185812
\(831\) 7.84172e106 2.13292
\(832\) −4.88835e105 −0.127251
\(833\) −3.51187e106 −0.874975
\(834\) 4.05561e106 0.967154
\(835\) −9.15453e106 −2.08968
\(836\) 2.14500e106 0.468703
\(837\) 6.44522e104 0.0134821
\(838\) 3.20421e106 0.641673
\(839\) −3.53926e106 −0.678579 −0.339290 0.940682i \(-0.610187\pi\)
−0.339290 + 0.940682i \(0.610187\pi\)
\(840\) −1.17648e106 −0.215968
\(841\) −4.28247e106 −0.752731
\(842\) 4.52409e106 0.761447
\(843\) −1.54039e107 −2.48270
\(844\) −1.13833e106 −0.175698
\(845\) −4.54569e105 −0.0671939
\(846\) −6.63006e106 −0.938641
\(847\) 4.46314e105 0.0605196
\(848\) −1.38452e106 −0.179825
\(849\) 2.59474e107 3.22823
\(850\) 1.30528e107 1.55565
\(851\) 1.13495e107 1.29582
\(852\) −5.85339e106 −0.640269
\(853\) −9.52082e105 −0.0997780 −0.0498890 0.998755i \(-0.515887\pi\)
−0.0498890 + 0.998755i \(0.515887\pi\)
\(854\) 6.13237e105 0.0615766
\(855\) −2.94861e107 −2.83697
\(856\) 3.31310e106 0.305452
\(857\) 6.55956e106 0.579529 0.289764 0.957098i \(-0.406423\pi\)
0.289764 + 0.957098i \(0.406423\pi\)
\(858\) −1.64996e107 −1.39697
\(859\) −1.41476e107 −1.14798 −0.573989 0.818863i \(-0.694605\pi\)
−0.573989 + 0.818863i \(0.694605\pi\)
\(860\) −6.71884e106 −0.522518
\(861\) 8.54799e106 0.637161
\(862\) −1.17612e107 −0.840304
\(863\) 4.99653e105 0.0342195 0.0171098 0.999854i \(-0.494554\pi\)
0.0171098 + 0.999854i \(0.494554\pi\)
\(864\) 3.98828e106 0.261839
\(865\) −1.09697e107 −0.690409
\(866\) 1.48434e107 0.895631
\(867\) −5.06306e106 −0.292898
\(868\) 1.59904e104 0.000886928 0
\(869\) 7.53550e106 0.400766
\(870\) 2.16117e107 1.10214
\(871\) 3.36711e107 1.64662
\(872\) 1.34545e106 0.0630980
\(873\) −2.96831e107 −1.33502
\(874\) 1.48410e107 0.640173
\(875\) −1.23582e107 −0.511283
\(876\) 1.08776e107 0.431652
\(877\) 2.87704e107 1.09512 0.547558 0.836767i \(-0.315557\pi\)
0.547558 + 0.836767i \(0.315557\pi\)
\(878\) −2.55095e107 −0.931435
\(879\) 9.45486e107 3.31178
\(880\) −1.57480e107 −0.529187
\(881\) 5.29762e107 1.70789 0.853946 0.520362i \(-0.174203\pi\)
0.853946 + 0.520362i \(0.174203\pi\)
\(882\) −4.12082e107 −1.27462
\(883\) 5.20945e107 1.54606 0.773030 0.634369i \(-0.218740\pi\)
0.773030 + 0.634369i \(0.218740\pi\)
\(884\) −1.62570e107 −0.462948
\(885\) −1.25771e108 −3.43678
\(886\) −2.02772e107 −0.531713
\(887\) −6.50483e107 −1.63691 −0.818453 0.574574i \(-0.805168\pi\)
−0.818453 + 0.574574i \(0.805168\pi\)
\(888\) 2.90863e107 0.702448
\(889\) 9.39410e106 0.217741
\(890\) −1.75856e106 −0.0391221
\(891\) 3.41603e107 0.729432
\(892\) 9.62499e105 0.0197280
\(893\) −2.94838e107 −0.580103
\(894\) −5.17453e107 −0.977354
\(895\) 5.83515e106 0.105806
\(896\) 9.89477e105 0.0172252
\(897\) −1.14159e108 −1.90803
\(898\) −5.25018e107 −0.842535
\(899\) −2.93740e105 −0.00452621
\(900\) 1.53161e108 2.26620
\(901\) −4.60443e107 −0.654216
\(902\) 1.14421e108 1.56123
\(903\) 1.42490e107 0.186716
\(904\) −4.36297e107 −0.549079
\(905\) 2.56924e107 0.310549
\(906\) 1.76083e107 0.204427
\(907\) −5.23703e107 −0.584008 −0.292004 0.956417i \(-0.594322\pi\)
−0.292004 + 0.956417i \(0.594322\pi\)
\(908\) −6.33809e107 −0.678929
\(909\) 7.05032e107 0.725483
\(910\) 2.62397e107 0.259387
\(911\) 6.95039e107 0.660067 0.330034 0.943969i \(-0.392940\pi\)
0.330034 + 0.943969i \(0.392940\pi\)
\(912\) 3.80344e107 0.347029
\(913\) 1.85590e107 0.162694
\(914\) 3.60482e107 0.303632
\(915\) −1.73061e108 −1.40065
\(916\) −9.59795e106 −0.0746438
\(917\) −2.22888e107 −0.166573
\(918\) 1.32636e108 0.952586
\(919\) 1.60470e108 1.10759 0.553794 0.832654i \(-0.313179\pi\)
0.553794 + 0.832654i \(0.313179\pi\)
\(920\) −1.08959e108 −0.722784
\(921\) −2.58613e108 −1.64882
\(922\) 8.31434e107 0.509506
\(923\) 1.30551e108 0.768988
\(924\) 3.33976e107 0.189099
\(925\) 5.20865e108 2.83500
\(926\) −1.94750e108 −1.01900
\(927\) 9.85444e107 0.495703
\(928\) −1.81765e107 −0.0879042
\(929\) −1.44798e108 −0.673268 −0.336634 0.941635i \(-0.609289\pi\)
−0.336634 + 0.941635i \(0.609289\pi\)
\(930\) −4.51263e106 −0.0201745
\(931\) −1.83252e108 −0.787746
\(932\) −1.25435e108 −0.518488
\(933\) 3.27964e108 1.30361
\(934\) 1.43836e108 0.549802
\(935\) −5.23725e108 −1.92522
\(936\) −1.90759e108 −0.674399
\(937\) −4.06681e108 −1.38280 −0.691400 0.722472i \(-0.743006\pi\)
−0.691400 + 0.722472i \(0.743006\pi\)
\(938\) −6.81554e107 −0.222893
\(939\) 2.69388e107 0.0847392
\(940\) 2.16463e108 0.654963
\(941\) −1.02707e108 −0.298935 −0.149468 0.988767i \(-0.547756\pi\)
−0.149468 + 0.988767i \(0.547756\pi\)
\(942\) −8.03922e108 −2.25090
\(943\) 7.91670e108 2.13239
\(944\) 1.05780e108 0.274110
\(945\) −2.14082e108 −0.533728
\(946\) 1.90733e108 0.457509
\(947\) 4.82553e107 0.111371 0.0556853 0.998448i \(-0.482266\pi\)
0.0556853 + 0.998448i \(0.482266\pi\)
\(948\) 1.33617e108 0.296728
\(949\) −2.42609e108 −0.518430
\(950\) 6.81106e108 1.40057
\(951\) 1.98397e108 0.392596
\(952\) 3.29066e107 0.0626663
\(953\) −7.60641e108 −1.39408 −0.697041 0.717031i \(-0.745501\pi\)
−0.697041 + 0.717031i \(0.745501\pi\)
\(954\) −5.40283e108 −0.953028
\(955\) 1.61032e109 2.73394
\(956\) −5.72018e108 −0.934752
\(957\) −6.13509e108 −0.965017
\(958\) 6.76710e108 1.02462
\(959\) −4.42057e107 −0.0644317
\(960\) −2.79240e108 −0.391811
\(961\) −7.40232e108 −0.999917
\(962\) −6.48727e108 −0.843668
\(963\) 1.29288e109 1.61882
\(964\) −2.77389e108 −0.334409
\(965\) 9.81707e108 1.13955
\(966\) 2.31075e108 0.258278
\(967\) −2.33112e108 −0.250899 −0.125449 0.992100i \(-0.540037\pi\)
−0.125449 + 0.992100i \(0.540037\pi\)
\(968\) 1.05933e108 0.109795
\(969\) 1.26489e109 1.26251
\(970\) 9.69114e108 0.931551
\(971\) 3.57624e108 0.331074 0.165537 0.986204i \(-0.447064\pi\)
0.165537 + 0.986204i \(0.447064\pi\)
\(972\) −2.24891e108 −0.200518
\(973\) 1.83094e108 0.157237
\(974\) 1.23010e109 1.01751
\(975\) −5.23915e109 −4.17438
\(976\) 1.45553e108 0.111713
\(977\) −3.42553e108 −0.253266 −0.126633 0.991950i \(-0.540417\pi\)
−0.126633 + 0.991950i \(0.540417\pi\)
\(978\) 1.57246e109 1.11998
\(979\) 4.99216e107 0.0342547
\(980\) 1.34539e109 0.889401
\(981\) 5.25037e108 0.334404
\(982\) −9.94134e108 −0.610064
\(983\) −1.73690e108 −0.102700 −0.0513502 0.998681i \(-0.516352\pi\)
−0.0513502 + 0.998681i \(0.516352\pi\)
\(984\) 2.02888e109 1.15594
\(985\) −2.87879e109 −1.58047
\(986\) −6.04488e108 −0.319801
\(987\) −4.59063e108 −0.234043
\(988\) −8.48303e108 −0.416795
\(989\) 1.31966e109 0.624883
\(990\) −6.14539e109 −2.80456
\(991\) −1.78550e109 −0.785363 −0.392682 0.919675i \(-0.628453\pi\)
−0.392682 + 0.919675i \(0.628453\pi\)
\(992\) 3.79534e106 0.00160907
\(993\) 2.02088e109 0.825834
\(994\) −2.64256e108 −0.104093
\(995\) −4.64690e109 −1.76450
\(996\) 3.29083e108 0.120459
\(997\) 6.27538e108 0.221446 0.110723 0.993851i \(-0.464683\pi\)
0.110723 + 0.993851i \(0.464683\pi\)
\(998\) 1.75997e108 0.0598742
\(999\) 5.29279e109 1.73598
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 2.74.a.b.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.74.a.b.1.4 4 1.1 even 1 trivial