Properties

Label 3.70.a.a.1.4
Level $3$
Weight $70$
Character 3.1
Self dual yes
Analytic conductor $90.454$
Analytic rank $1$
Dimension $6$
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] = [3,70,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 70, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 70);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 70 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(90.4544859877\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{51}\cdot 3^{33}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 17\cdot 23^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.14909e8\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.99182e9 q^{2} -1.66772e16 q^{3} -5.74361e20 q^{4} +2.45458e24 q^{5} -6.65724e25 q^{6} +3.62318e28 q^{7} -4.64911e30 q^{8} +2.78128e32 q^{9} +O(q^{10})\) \(q+3.99182e9 q^{2} -1.66772e16 q^{3} -5.74361e20 q^{4} +2.45458e24 q^{5} -6.65724e25 q^{6} +3.62318e28 q^{7} -4.64911e30 q^{8} +2.78128e32 q^{9} +9.79827e33 q^{10} -8.58823e35 q^{11} +9.57873e36 q^{12} +4.20595e37 q^{13} +1.44631e38 q^{14} -4.09355e40 q^{15} +3.20485e41 q^{16} -3.41417e42 q^{17} +1.11024e42 q^{18} -1.21562e44 q^{19} -1.40982e45 q^{20} -6.04245e44 q^{21} -3.42827e45 q^{22} +1.33316e47 q^{23} +7.75340e46 q^{24} +4.33092e48 q^{25} +1.67894e47 q^{26} -4.63840e48 q^{27} -2.08102e49 q^{28} -1.91784e50 q^{29} -1.63407e50 q^{30} +2.32297e51 q^{31} +4.02367e51 q^{32} +1.43228e52 q^{33} -1.36288e52 q^{34} +8.89341e52 q^{35} -1.59746e53 q^{36} +1.40508e54 q^{37} -4.85256e53 q^{38} -7.01434e53 q^{39} -1.14116e55 q^{40} -5.81808e55 q^{41} -2.41204e54 q^{42} -3.25513e55 q^{43} +4.93275e56 q^{44} +6.82689e56 q^{45} +5.32175e56 q^{46} -3.00434e57 q^{47} -5.34478e57 q^{48} -1.91878e58 q^{49} +1.72883e58 q^{50} +5.69387e58 q^{51} -2.41573e58 q^{52} +1.16179e59 q^{53} -1.85157e58 q^{54} -2.10805e60 q^{55} -1.68446e59 q^{56} +2.02732e60 q^{57} -7.65568e59 q^{58} +2.31881e61 q^{59} +2.35118e61 q^{60} -7.75450e61 q^{61} +9.27289e60 q^{62} +1.00771e61 q^{63} -1.73119e62 q^{64} +1.03239e62 q^{65} +5.71739e61 q^{66} +2.47350e60 q^{67} +1.96097e63 q^{68} -2.22334e63 q^{69} +3.55009e62 q^{70} -2.45908e63 q^{71} -1.29305e63 q^{72} +5.55313e63 q^{73} +5.60884e63 q^{74} -7.22275e64 q^{75} +6.98207e64 q^{76} -3.11167e64 q^{77} -2.80000e63 q^{78} -2.90319e65 q^{79} +7.86656e65 q^{80} +7.73554e64 q^{81} -2.32247e65 q^{82} -8.96198e65 q^{83} +3.47055e65 q^{84} -8.38036e66 q^{85} -1.29939e65 q^{86} +3.19842e66 q^{87} +3.99276e66 q^{88} -1.65103e67 q^{89} +2.72518e66 q^{90} +1.52389e66 q^{91} -7.65717e67 q^{92} -3.87406e67 q^{93} -1.19928e67 q^{94} -2.98385e68 q^{95} -6.71034e67 q^{96} +3.36704e68 q^{97} -7.65942e67 q^{98} -2.38863e68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 869363388 q^{2} - 10\!\cdots\!14 q^{3}+ \cdots + 16\!\cdots\!66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 869363388 q^{2} - 10\!\cdots\!14 q^{3}+ \cdots - 53\!\cdots\!36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.99182e9 0.164300 0.0821498 0.996620i \(-0.473821\pi\)
0.0821498 + 0.996620i \(0.473821\pi\)
\(3\) −1.66772e16 −0.577350
\(4\) −5.74361e20 −0.973006
\(5\) 2.45458e24 1.88587 0.942937 0.332971i \(-0.108051\pi\)
0.942937 + 0.332971i \(0.108051\pi\)
\(6\) −6.65724e25 −0.0948584
\(7\) 3.62318e28 0.253051 0.126526 0.991963i \(-0.459617\pi\)
0.126526 + 0.991963i \(0.459617\pi\)
\(8\) −4.64911e30 −0.324164
\(9\) 2.78128e32 0.333333
\(10\) 9.79827e33 0.309848
\(11\) −8.58823e35 −1.01358 −0.506788 0.862071i \(-0.669167\pi\)
−0.506788 + 0.862071i \(0.669167\pi\)
\(12\) 9.57873e36 0.561765
\(13\) 4.20595e37 0.155890 0.0779450 0.996958i \(-0.475164\pi\)
0.0779450 + 0.996958i \(0.475164\pi\)
\(14\) 1.44631e38 0.0415762
\(15\) −4.09355e40 −1.08881
\(16\) 3.20485e41 0.919746
\(17\) −3.41417e42 −1.21003 −0.605016 0.796213i \(-0.706833\pi\)
−0.605016 + 0.796213i \(0.706833\pi\)
\(18\) 1.11024e42 0.0547665
\(19\) −1.21562e44 −0.928539 −0.464269 0.885694i \(-0.653683\pi\)
−0.464269 + 0.885694i \(0.653683\pi\)
\(20\) −1.40982e45 −1.83497
\(21\) −6.04245e44 −0.146099
\(22\) −3.42827e45 −0.166530
\(23\) 1.33316e47 1.39725 0.698623 0.715490i \(-0.253797\pi\)
0.698623 + 0.715490i \(0.253797\pi\)
\(24\) 7.75340e46 0.187156
\(25\) 4.33092e48 2.55652
\(26\) 1.67894e47 0.0256127
\(27\) −4.63840e48 −0.192450
\(28\) −2.08102e49 −0.246220
\(29\) −1.91784e50 −0.676210 −0.338105 0.941108i \(-0.609786\pi\)
−0.338105 + 0.941108i \(0.609786\pi\)
\(30\) −1.63407e50 −0.178891
\(31\) 2.32297e51 0.820475 0.410237 0.911979i \(-0.365446\pi\)
0.410237 + 0.911979i \(0.365446\pi\)
\(32\) 4.02367e51 0.475278
\(33\) 1.43228e52 0.585188
\(34\) −1.36288e52 −0.198808
\(35\) 8.89341e52 0.477222
\(36\) −1.59746e53 −0.324335
\(37\) 1.40508e54 1.10852 0.554259 0.832344i \(-0.313002\pi\)
0.554259 + 0.832344i \(0.313002\pi\)
\(38\) −4.85256e53 −0.152559
\(39\) −7.01434e53 −0.0900032
\(40\) −1.14116e55 −0.611333
\(41\) −5.81808e55 −1.32965 −0.664824 0.747000i \(-0.731493\pi\)
−0.664824 + 0.747000i \(0.731493\pi\)
\(42\) −2.41204e54 −0.0240040
\(43\) −3.25513e55 −0.143849 −0.0719244 0.997410i \(-0.522914\pi\)
−0.0719244 + 0.997410i \(0.522914\pi\)
\(44\) 4.93275e56 0.986215
\(45\) 6.82689e56 0.628625
\(46\) 5.32175e56 0.229567
\(47\) −3.00434e57 −0.617125 −0.308562 0.951204i \(-0.599848\pi\)
−0.308562 + 0.951204i \(0.599848\pi\)
\(48\) −5.34478e57 −0.531015
\(49\) −1.91878e58 −0.935965
\(50\) 1.72883e58 0.420035
\(51\) 5.69387e58 0.698613
\(52\) −2.41573e58 −0.151682
\(53\) 1.16179e59 0.378104 0.189052 0.981967i \(-0.439458\pi\)
0.189052 + 0.981967i \(0.439458\pi\)
\(54\) −1.85157e58 −0.0316195
\(55\) −2.10805e60 −1.91148
\(56\) −1.68446e59 −0.0820300
\(57\) 2.02732e60 0.536092
\(58\) −7.65568e59 −0.111101
\(59\) 2.31881e61 1.86582 0.932911 0.360106i \(-0.117260\pi\)
0.932911 + 0.360106i \(0.117260\pi\)
\(60\) 2.35118e61 1.05942
\(61\) −7.75450e61 −1.97549 −0.987745 0.156078i \(-0.950115\pi\)
−0.987745 + 0.156078i \(0.950115\pi\)
\(62\) 9.27289e60 0.134804
\(63\) 1.00771e61 0.0843503
\(64\) −1.73119e62 −0.841658
\(65\) 1.03239e62 0.293989
\(66\) 5.71739e61 0.0961462
\(67\) 2.47350e60 0.00247589 0.00123795 0.999999i \(-0.499606\pi\)
0.00123795 + 0.999999i \(0.499606\pi\)
\(68\) 1.96097e63 1.17737
\(69\) −2.22334e63 −0.806700
\(70\) 3.55009e62 0.0784074
\(71\) −2.45908e63 −0.332935 −0.166468 0.986047i \(-0.553236\pi\)
−0.166468 + 0.986047i \(0.553236\pi\)
\(72\) −1.29305e63 −0.108055
\(73\) 5.55313e63 0.288336 0.144168 0.989553i \(-0.453949\pi\)
0.144168 + 0.989553i \(0.453949\pi\)
\(74\) 5.60884e63 0.182129
\(75\) −7.22275e64 −1.47601
\(76\) 6.98207e64 0.903473
\(77\) −3.11167e64 −0.256486
\(78\) −2.80000e63 −0.0147875
\(79\) −2.90319e65 −0.987963 −0.493982 0.869472i \(-0.664459\pi\)
−0.493982 + 0.869472i \(0.664459\pi\)
\(80\) 7.86656e65 1.73452
\(81\) 7.73554e64 0.111111
\(82\) −2.32247e65 −0.218461
\(83\) −8.96198e65 −0.554894 −0.277447 0.960741i \(-0.589488\pi\)
−0.277447 + 0.960741i \(0.589488\pi\)
\(84\) 3.47055e65 0.142155
\(85\) −8.38036e66 −2.28197
\(86\) −1.29939e65 −0.0236343
\(87\) 3.19842e66 0.390410
\(88\) 3.99276e66 0.328565
\(89\) −1.65103e67 −0.920023 −0.460011 0.887913i \(-0.652155\pi\)
−0.460011 + 0.887913i \(0.652155\pi\)
\(90\) 2.72518e66 0.103283
\(91\) 1.52389e66 0.0394481
\(92\) −7.65717e67 −1.35953
\(93\) −3.87406e67 −0.473701
\(94\) −1.19928e67 −0.101393
\(95\) −2.98385e68 −1.75111
\(96\) −6.71034e67 −0.274402
\(97\) 3.36704e68 0.962993 0.481496 0.876448i \(-0.340093\pi\)
0.481496 + 0.876448i \(0.340093\pi\)
\(98\) −7.65942e67 −0.153779
\(99\) −2.38863e68 −0.337859
\(100\) −2.48751e69 −2.48751
\(101\) −1.25892e69 −0.893118 −0.446559 0.894754i \(-0.647351\pi\)
−0.446559 + 0.894754i \(0.647351\pi\)
\(102\) 2.27289e68 0.114782
\(103\) −4.12679e69 −1.48843 −0.744214 0.667941i \(-0.767176\pi\)
−0.744214 + 0.667941i \(0.767176\pi\)
\(104\) −1.95539e68 −0.0505340
\(105\) −1.48317e69 −0.275524
\(106\) 4.63768e68 0.0621224
\(107\) 6.56850e69 0.636394 0.318197 0.948025i \(-0.396923\pi\)
0.318197 + 0.948025i \(0.396923\pi\)
\(108\) 2.66412e69 0.187255
\(109\) −3.66549e70 −1.87464 −0.937321 0.348468i \(-0.886702\pi\)
−0.937321 + 0.348468i \(0.886702\pi\)
\(110\) −8.41498e69 −0.314055
\(111\) −2.34328e70 −0.640003
\(112\) 1.16117e70 0.232743
\(113\) −6.51324e70 −0.960707 −0.480353 0.877075i \(-0.659492\pi\)
−0.480353 + 0.877075i \(0.659492\pi\)
\(114\) 8.09270e69 0.0880797
\(115\) 3.27236e71 2.63503
\(116\) 1.10153e71 0.657956
\(117\) 1.16979e70 0.0519634
\(118\) 9.25627e70 0.306554
\(119\) −1.23702e71 −0.306200
\(120\) 1.90314e71 0.352953
\(121\) 1.96255e70 0.0273354
\(122\) −3.09546e71 −0.324572
\(123\) 9.70292e71 0.767673
\(124\) −1.33422e72 −0.798326
\(125\) 6.47237e72 2.93540
\(126\) 4.02260e70 0.0138587
\(127\) 3.25151e72 0.852819 0.426409 0.904530i \(-0.359778\pi\)
0.426409 + 0.904530i \(0.359778\pi\)
\(128\) −3.06621e72 −0.613562
\(129\) 5.42864e71 0.0830512
\(130\) 4.12110e71 0.0483023
\(131\) 6.67900e72 0.600969 0.300484 0.953787i \(-0.402852\pi\)
0.300484 + 0.953787i \(0.402852\pi\)
\(132\) −8.22643e72 −0.569391
\(133\) −4.40443e72 −0.234968
\(134\) 9.87379e69 0.000406788 0
\(135\) −1.13853e73 −0.362937
\(136\) 1.58728e73 0.392249
\(137\) −7.49685e73 −1.43886 −0.719431 0.694564i \(-0.755598\pi\)
−0.719431 + 0.694564i \(0.755598\pi\)
\(138\) −8.87518e72 −0.132541
\(139\) −9.78034e73 −1.13853 −0.569263 0.822155i \(-0.692772\pi\)
−0.569263 + 0.822155i \(0.692772\pi\)
\(140\) −5.10803e73 −0.464340
\(141\) 5.01039e73 0.356297
\(142\) −9.81622e72 −0.0547011
\(143\) −3.61217e73 −0.158006
\(144\) 8.91359e73 0.306582
\(145\) −4.70750e74 −1.27525
\(146\) 2.21671e73 0.0473735
\(147\) 3.19998e74 0.540380
\(148\) −8.07025e74 −1.07859
\(149\) −4.01723e74 −0.425599 −0.212800 0.977096i \(-0.568258\pi\)
−0.212800 + 0.977096i \(0.568258\pi\)
\(150\) −2.88319e74 −0.242508
\(151\) 2.77648e75 1.85691 0.928453 0.371450i \(-0.121139\pi\)
0.928453 + 0.371450i \(0.121139\pi\)
\(152\) 5.65156e74 0.300999
\(153\) −9.49577e74 −0.403344
\(154\) −1.24213e74 −0.0421406
\(155\) 5.70192e75 1.54731
\(156\) 4.02876e74 0.0875736
\(157\) 1.47342e75 0.256915 0.128458 0.991715i \(-0.458997\pi\)
0.128458 + 0.991715i \(0.458997\pi\)
\(158\) −1.15890e75 −0.162322
\(159\) −1.93754e75 −0.218299
\(160\) 9.87642e75 0.896314
\(161\) 4.83029e75 0.353574
\(162\) 3.08789e74 0.0182555
\(163\) −1.36864e76 −0.654362 −0.327181 0.944962i \(-0.606099\pi\)
−0.327181 + 0.944962i \(0.606099\pi\)
\(164\) 3.34168e76 1.29375
\(165\) 3.51564e76 1.10359
\(166\) −3.57746e75 −0.0911688
\(167\) −7.54924e76 −1.56381 −0.781907 0.623396i \(-0.785753\pi\)
−0.781907 + 0.623396i \(0.785753\pi\)
\(168\) 2.80920e75 0.0473601
\(169\) −7.10243e76 −0.975698
\(170\) −3.34529e76 −0.374927
\(171\) −3.38100e76 −0.309513
\(172\) 1.86962e76 0.139966
\(173\) 1.16564e74 0.000714453 0 0.000357227 1.00000i \(-0.499886\pi\)
0.000357227 1.00000i \(0.499886\pi\)
\(174\) 1.27675e76 0.0641442
\(175\) 1.56917e77 0.646930
\(176\) −2.75240e77 −0.932232
\(177\) −3.86712e77 −1.07723
\(178\) −6.59060e76 −0.151159
\(179\) −2.95990e77 −0.559561 −0.279780 0.960064i \(-0.590262\pi\)
−0.279780 + 0.960064i \(0.590262\pi\)
\(180\) −3.92110e77 −0.611655
\(181\) −4.99123e77 −0.643126 −0.321563 0.946888i \(-0.604208\pi\)
−0.321563 + 0.946888i \(0.604208\pi\)
\(182\) 6.08311e75 0.00648131
\(183\) 1.29323e78 1.14055
\(184\) −6.19801e77 −0.452937
\(185\) 3.44890e78 2.09052
\(186\) −1.54646e77 −0.0778289
\(187\) 2.93217e78 1.22646
\(188\) 1.72558e78 0.600466
\(189\) −1.68058e77 −0.0486997
\(190\) −1.19110e78 −0.287706
\(191\) −7.53132e78 −1.51782 −0.758909 0.651197i \(-0.774267\pi\)
−0.758909 + 0.651197i \(0.774267\pi\)
\(192\) 2.88714e78 0.485931
\(193\) 1.51305e78 0.212876 0.106438 0.994319i \(-0.466055\pi\)
0.106438 + 0.994319i \(0.466055\pi\)
\(194\) 1.34406e78 0.158219
\(195\) −1.72173e78 −0.169735
\(196\) 1.10207e79 0.910699
\(197\) 1.05867e79 0.733967 0.366983 0.930227i \(-0.380391\pi\)
0.366983 + 0.930227i \(0.380391\pi\)
\(198\) −9.53499e77 −0.0555100
\(199\) 2.47010e79 1.20860 0.604300 0.796757i \(-0.293453\pi\)
0.604300 + 0.796757i \(0.293453\pi\)
\(200\) −2.01349e79 −0.828732
\(201\) −4.12511e76 −0.00142946
\(202\) −5.02537e78 −0.146739
\(203\) −6.94869e78 −0.171116
\(204\) −3.27034e79 −0.679754
\(205\) −1.42810e80 −2.50755
\(206\) −1.64734e79 −0.244548
\(207\) 3.70790e79 0.465749
\(208\) 1.34794e79 0.143379
\(209\) 1.04401e80 0.941144
\(210\) −5.92055e78 −0.0452686
\(211\) −1.98758e80 −1.28997 −0.644984 0.764196i \(-0.723136\pi\)
−0.644984 + 0.764196i \(0.723136\pi\)
\(212\) −6.67289e79 −0.367898
\(213\) 4.10105e79 0.192220
\(214\) 2.62203e79 0.104559
\(215\) −7.98999e79 −0.271281
\(216\) 2.15644e79 0.0623854
\(217\) 8.41655e79 0.207622
\(218\) −1.46320e80 −0.308003
\(219\) −9.26106e79 −0.166471
\(220\) 1.21078e81 1.85988
\(221\) −1.43598e80 −0.188632
\(222\) −9.35397e79 −0.105152
\(223\) 7.81695e80 0.752522 0.376261 0.926514i \(-0.377210\pi\)
0.376261 + 0.926514i \(0.377210\pi\)
\(224\) 1.45785e80 0.120270
\(225\) 1.20455e81 0.852174
\(226\) −2.59997e80 −0.157844
\(227\) 7.53245e80 0.392685 0.196343 0.980535i \(-0.437094\pi\)
0.196343 + 0.980535i \(0.437094\pi\)
\(228\) −1.16441e81 −0.521621
\(229\) −1.05540e81 −0.406532 −0.203266 0.979124i \(-0.565156\pi\)
−0.203266 + 0.979124i \(0.565156\pi\)
\(230\) 1.30627e81 0.432934
\(231\) 5.18940e80 0.148082
\(232\) 8.91624e80 0.219203
\(233\) 7.14672e81 1.51470 0.757351 0.653008i \(-0.226493\pi\)
0.757351 + 0.653008i \(0.226493\pi\)
\(234\) 4.66961e79 0.00853756
\(235\) −7.37441e81 −1.16382
\(236\) −1.33183e82 −1.81546
\(237\) 4.84170e81 0.570401
\(238\) −4.93795e80 −0.0503085
\(239\) −5.20728e81 −0.459075 −0.229537 0.973300i \(-0.573721\pi\)
−0.229537 + 0.973300i \(0.573721\pi\)
\(240\) −1.31192e82 −1.00143
\(241\) −8.58844e81 −0.567971 −0.283986 0.958829i \(-0.591657\pi\)
−0.283986 + 0.958829i \(0.591657\pi\)
\(242\) 7.83416e79 0.00449120
\(243\) −1.29007e81 −0.0641500
\(244\) 4.45389e82 1.92216
\(245\) −4.70980e82 −1.76511
\(246\) 3.87323e81 0.126128
\(247\) −5.11285e81 −0.144750
\(248\) −1.07997e82 −0.265968
\(249\) 1.49461e82 0.320368
\(250\) 2.58366e82 0.482286
\(251\) 1.36885e82 0.222646 0.111323 0.993784i \(-0.464491\pi\)
0.111323 + 0.993784i \(0.464491\pi\)
\(252\) −5.78790e81 −0.0820734
\(253\) −1.14495e83 −1.41621
\(254\) 1.29795e82 0.140118
\(255\) 1.39761e83 1.31750
\(256\) 8.99516e82 0.740850
\(257\) 1.35870e83 0.978209 0.489104 0.872225i \(-0.337324\pi\)
0.489104 + 0.872225i \(0.337324\pi\)
\(258\) 2.16702e81 0.0136453
\(259\) 5.09088e82 0.280511
\(260\) −5.92962e82 −0.286053
\(261\) −5.33406e82 −0.225403
\(262\) 2.66614e82 0.0987389
\(263\) −2.74224e82 −0.0890495 −0.0445247 0.999008i \(-0.514177\pi\)
−0.0445247 + 0.999008i \(0.514177\pi\)
\(264\) −6.65880e82 −0.189697
\(265\) 2.85172e83 0.713057
\(266\) −1.75817e82 −0.0386051
\(267\) 2.75345e83 0.531175
\(268\) −1.42069e81 −0.00240906
\(269\) −3.09103e83 −0.460945 −0.230473 0.973079i \(-0.574027\pi\)
−0.230473 + 0.973079i \(0.574027\pi\)
\(270\) −4.54483e82 −0.0596303
\(271\) 1.04318e84 1.20481 0.602404 0.798191i \(-0.294210\pi\)
0.602404 + 0.798191i \(0.294210\pi\)
\(272\) −1.09419e84 −1.11292
\(273\) −2.54142e82 −0.0227754
\(274\) −2.99261e83 −0.236405
\(275\) −3.71949e84 −2.59123
\(276\) 1.27700e84 0.784924
\(277\) −2.70579e84 −1.46806 −0.734028 0.679119i \(-0.762362\pi\)
−0.734028 + 0.679119i \(0.762362\pi\)
\(278\) −3.90414e83 −0.187059
\(279\) 6.46084e83 0.273492
\(280\) −4.13464e83 −0.154698
\(281\) −1.62527e84 −0.537721 −0.268861 0.963179i \(-0.586647\pi\)
−0.268861 + 0.963179i \(0.586647\pi\)
\(282\) 2.00006e83 0.0585395
\(283\) 5.71736e84 1.48103 0.740514 0.672041i \(-0.234582\pi\)
0.740514 + 0.672041i \(0.234582\pi\)
\(284\) 1.41240e84 0.323948
\(285\) 4.97622e84 1.01100
\(286\) −1.44191e83 −0.0259604
\(287\) −2.10800e84 −0.336469
\(288\) 1.11910e84 0.158426
\(289\) 3.69540e84 0.464180
\(290\) −1.87915e84 −0.209523
\(291\) −5.61527e84 −0.555984
\(292\) −3.18950e84 −0.280553
\(293\) −1.89352e85 −1.48025 −0.740127 0.672467i \(-0.765235\pi\)
−0.740127 + 0.672467i \(0.765235\pi\)
\(294\) 1.27738e84 0.0887842
\(295\) 5.69171e85 3.51871
\(296\) −6.53238e84 −0.359342
\(297\) 3.98356e84 0.195063
\(298\) −1.60361e84 −0.0699258
\(299\) 5.60722e84 0.217817
\(300\) 4.14847e85 1.43616
\(301\) −1.17939e84 −0.0364011
\(302\) 1.10832e85 0.305089
\(303\) 2.09952e85 0.515642
\(304\) −3.89589e85 −0.854019
\(305\) −1.90341e86 −3.72552
\(306\) −3.79054e84 −0.0662693
\(307\) −6.25855e84 −0.0977686 −0.0488843 0.998804i \(-0.515567\pi\)
−0.0488843 + 0.998804i \(0.515567\pi\)
\(308\) 1.78723e85 0.249563
\(309\) 6.88232e85 0.859344
\(310\) 2.27611e85 0.254223
\(311\) −7.80584e85 −0.780165 −0.390082 0.920780i \(-0.627554\pi\)
−0.390082 + 0.920780i \(0.627554\pi\)
\(312\) 3.26104e84 0.0291758
\(313\) −8.60592e85 −0.689474 −0.344737 0.938699i \(-0.612032\pi\)
−0.344737 + 0.938699i \(0.612032\pi\)
\(314\) 5.88165e84 0.0422110
\(315\) 2.47351e85 0.159074
\(316\) 1.66748e86 0.961294
\(317\) −2.04537e86 −1.05737 −0.528684 0.848819i \(-0.677314\pi\)
−0.528684 + 0.848819i \(0.677314\pi\)
\(318\) −7.73434e84 −0.0358664
\(319\) 1.64709e86 0.685390
\(320\) −4.24935e86 −1.58726
\(321\) −1.09544e86 −0.367422
\(322\) 1.92817e85 0.0580921
\(323\) 4.15035e86 1.12356
\(324\) −4.44299e85 −0.108112
\(325\) 1.82156e86 0.398536
\(326\) −5.46338e85 −0.107511
\(327\) 6.11301e86 1.08232
\(328\) 2.70489e86 0.431024
\(329\) −1.08853e86 −0.156164
\(330\) 1.40338e86 0.181320
\(331\) 2.28247e86 0.265668 0.132834 0.991138i \(-0.457592\pi\)
0.132834 + 0.991138i \(0.457592\pi\)
\(332\) 5.14741e86 0.539915
\(333\) 3.90794e86 0.369506
\(334\) −3.01352e86 −0.256934
\(335\) 6.07143e84 0.00466922
\(336\) −1.93651e86 −0.134374
\(337\) −1.58037e87 −0.989754 −0.494877 0.868963i \(-0.664787\pi\)
−0.494877 + 0.868963i \(0.664787\pi\)
\(338\) −2.83516e86 −0.160307
\(339\) 1.08623e87 0.554664
\(340\) 4.81336e87 2.22037
\(341\) −1.99502e87 −0.831613
\(342\) −1.34963e86 −0.0508528
\(343\) −1.43798e87 −0.489898
\(344\) 1.51334e86 0.0466306
\(345\) −5.45737e87 −1.52134
\(346\) 4.65303e83 0.000117384 0
\(347\) 1.43275e85 0.00327193 0.00163597 0.999999i \(-0.499479\pi\)
0.00163597 + 0.999999i \(0.499479\pi\)
\(348\) −1.83705e87 −0.379871
\(349\) −7.98599e87 −1.49572 −0.747862 0.663854i \(-0.768920\pi\)
−0.747862 + 0.663854i \(0.768920\pi\)
\(350\) 6.26385e86 0.106290
\(351\) −1.95089e86 −0.0300011
\(352\) −3.45562e87 −0.481730
\(353\) 4.54672e87 0.574740 0.287370 0.957820i \(-0.407219\pi\)
0.287370 + 0.957820i \(0.407219\pi\)
\(354\) −1.54369e87 −0.176989
\(355\) −6.03602e87 −0.627874
\(356\) 9.48285e87 0.895187
\(357\) 2.06299e87 0.176785
\(358\) −1.18154e87 −0.0919356
\(359\) 1.80684e88 1.27691 0.638456 0.769659i \(-0.279574\pi\)
0.638456 + 0.769659i \(0.279574\pi\)
\(360\) −3.17390e87 −0.203778
\(361\) −2.36210e87 −0.137816
\(362\) −1.99241e87 −0.105665
\(363\) −3.27298e86 −0.0157821
\(364\) −8.75265e86 −0.0383833
\(365\) 1.36306e88 0.543766
\(366\) 5.16236e87 0.187392
\(367\) −7.20654e87 −0.238094 −0.119047 0.992889i \(-0.537984\pi\)
−0.119047 + 0.992889i \(0.537984\pi\)
\(368\) 4.27258e88 1.28511
\(369\) −1.61817e88 −0.443216
\(370\) 1.37674e88 0.343472
\(371\) 4.20939e87 0.0956796
\(372\) 2.22511e88 0.460914
\(373\) 3.41896e88 0.645565 0.322782 0.946473i \(-0.395382\pi\)
0.322782 + 0.946473i \(0.395382\pi\)
\(374\) 1.17047e88 0.201507
\(375\) −1.07941e89 −1.69476
\(376\) 1.39675e88 0.200050
\(377\) −8.06634e87 −0.105414
\(378\) −6.70857e86 −0.00800134
\(379\) −8.74868e88 −0.952555 −0.476277 0.879295i \(-0.658014\pi\)
−0.476277 + 0.879295i \(0.658014\pi\)
\(380\) 1.71381e89 1.70384
\(381\) −5.42260e88 −0.492375
\(382\) −3.00637e88 −0.249377
\(383\) −5.77543e88 −0.437751 −0.218875 0.975753i \(-0.570239\pi\)
−0.218875 + 0.975753i \(0.570239\pi\)
\(384\) 5.11358e88 0.354240
\(385\) −7.63787e88 −0.483701
\(386\) 6.03984e87 0.0349754
\(387\) −9.05344e87 −0.0479496
\(388\) −1.93390e89 −0.936998
\(389\) 9.43356e88 0.418229 0.209115 0.977891i \(-0.432942\pi\)
0.209115 + 0.977891i \(0.432942\pi\)
\(390\) −6.87284e87 −0.0278873
\(391\) −4.55164e89 −1.69071
\(392\) 8.92060e88 0.303406
\(393\) −1.11387e89 −0.346969
\(394\) 4.22603e88 0.120590
\(395\) −7.12612e89 −1.86317
\(396\) 1.37194e89 0.328738
\(397\) 6.06080e89 1.33124 0.665621 0.746290i \(-0.268167\pi\)
0.665621 + 0.746290i \(0.268167\pi\)
\(398\) 9.86019e88 0.198573
\(399\) 7.34535e88 0.135659
\(400\) 1.38799e90 2.35135
\(401\) −6.19345e88 −0.0962614 −0.0481307 0.998841i \(-0.515326\pi\)
−0.0481307 + 0.998841i \(0.515326\pi\)
\(402\) −1.64667e86 −0.000234859 0
\(403\) 9.77030e88 0.127904
\(404\) 7.23072e89 0.869009
\(405\) 1.89875e89 0.209542
\(406\) −2.77379e88 −0.0281142
\(407\) −1.20672e90 −1.12357
\(408\) −2.64714e89 −0.226465
\(409\) −1.19150e90 −0.936788 −0.468394 0.883520i \(-0.655167\pi\)
−0.468394 + 0.883520i \(0.655167\pi\)
\(410\) −5.70071e89 −0.411989
\(411\) 1.25026e90 0.830728
\(412\) 2.37027e90 1.44825
\(413\) 8.40147e89 0.472148
\(414\) 1.48013e89 0.0765223
\(415\) −2.19979e90 −1.04646
\(416\) 1.69233e89 0.0740911
\(417\) 1.63108e90 0.657329
\(418\) 4.16749e89 0.154630
\(419\) 1.29706e90 0.443175 0.221588 0.975140i \(-0.428876\pi\)
0.221588 + 0.975140i \(0.428876\pi\)
\(420\) 8.51875e89 0.268087
\(421\) −5.23705e88 −0.0151829 −0.00759144 0.999971i \(-0.502416\pi\)
−0.00759144 + 0.999971i \(0.502416\pi\)
\(422\) −7.93405e89 −0.211941
\(423\) −8.35592e89 −0.205708
\(424\) −5.40130e89 −0.122568
\(425\) −1.47865e91 −3.09348
\(426\) 1.63707e89 0.0315817
\(427\) −2.80960e90 −0.499900
\(428\) −3.77269e90 −0.619214
\(429\) 6.02408e89 0.0912250
\(430\) −3.18946e89 −0.0445713
\(431\) −5.57328e90 −0.718861 −0.359430 0.933172i \(-0.617029\pi\)
−0.359430 + 0.933172i \(0.617029\pi\)
\(432\) −1.48653e90 −0.177005
\(433\) 1.32658e91 1.45848 0.729242 0.684256i \(-0.239873\pi\)
0.729242 + 0.684256i \(0.239873\pi\)
\(434\) 3.35974e89 0.0341122
\(435\) 7.85078e90 0.736264
\(436\) 2.10532e91 1.82404
\(437\) −1.62062e91 −1.29740
\(438\) −3.69685e89 −0.0273511
\(439\) −6.30768e90 −0.431364 −0.215682 0.976464i \(-0.569197\pi\)
−0.215682 + 0.976464i \(0.569197\pi\)
\(440\) 9.80056e90 0.619632
\(441\) −5.33666e90 −0.311988
\(442\) −5.73219e89 −0.0309922
\(443\) 2.35910e91 1.17982 0.589912 0.807467i \(-0.299162\pi\)
0.589912 + 0.807467i \(0.299162\pi\)
\(444\) 1.34589e91 0.622726
\(445\) −4.05258e91 −1.73505
\(446\) 3.12039e90 0.123639
\(447\) 6.69961e90 0.245720
\(448\) −6.27242e90 −0.212982
\(449\) 2.35122e91 0.739257 0.369628 0.929180i \(-0.379485\pi\)
0.369628 + 0.929180i \(0.379485\pi\)
\(450\) 4.80835e90 0.140012
\(451\) 4.99670e91 1.34770
\(452\) 3.74095e91 0.934773
\(453\) −4.63039e91 −1.07209
\(454\) 3.00682e90 0.0645180
\(455\) 3.74052e90 0.0743942
\(456\) −9.42522e90 −0.173782
\(457\) 4.87186e91 0.832887 0.416444 0.909162i \(-0.363276\pi\)
0.416444 + 0.909162i \(0.363276\pi\)
\(458\) −4.21298e90 −0.0667930
\(459\) 1.58363e91 0.232871
\(460\) −1.87952e92 −2.56390
\(461\) −6.07739e91 −0.769193 −0.384596 0.923085i \(-0.625659\pi\)
−0.384596 + 0.923085i \(0.625659\pi\)
\(462\) 2.07152e90 0.0243299
\(463\) 8.29614e91 0.904341 0.452170 0.891932i \(-0.350650\pi\)
0.452170 + 0.891932i \(0.350650\pi\)
\(464\) −6.14638e91 −0.621941
\(465\) −9.50920e91 −0.893341
\(466\) 2.85284e91 0.248865
\(467\) 1.73212e92 1.40329 0.701644 0.712528i \(-0.252450\pi\)
0.701644 + 0.712528i \(0.252450\pi\)
\(468\) −6.71884e90 −0.0505606
\(469\) 8.96196e88 0.000626527 0
\(470\) −2.94373e91 −0.191215
\(471\) −2.45725e91 −0.148330
\(472\) −1.07804e92 −0.604833
\(473\) 2.79558e91 0.145802
\(474\) 1.93272e91 0.0937166
\(475\) −5.26477e92 −2.37383
\(476\) 7.10494e91 0.297934
\(477\) 3.23128e91 0.126035
\(478\) −2.07866e91 −0.0754258
\(479\) 3.66321e92 1.23676 0.618382 0.785878i \(-0.287789\pi\)
0.618382 + 0.785878i \(0.287789\pi\)
\(480\) −1.64711e92 −0.517487
\(481\) 5.90971e91 0.172807
\(482\) −3.42835e91 −0.0933174
\(483\) −8.05557e91 −0.204136
\(484\) −1.12721e91 −0.0265975
\(485\) 8.26468e92 1.81608
\(486\) −5.14973e90 −0.0105398
\(487\) 5.74085e92 1.09453 0.547264 0.836960i \(-0.315669\pi\)
0.547264 + 0.836960i \(0.315669\pi\)
\(488\) 3.60515e92 0.640383
\(489\) 2.28251e92 0.377796
\(490\) −1.88007e92 −0.290007
\(491\) −1.00239e92 −0.144121 −0.0720604 0.997400i \(-0.522957\pi\)
−0.0720604 + 0.997400i \(0.522957\pi\)
\(492\) −5.57298e92 −0.746950
\(493\) 6.54783e92 0.818236
\(494\) −2.04096e91 −0.0237824
\(495\) −5.86310e92 −0.637159
\(496\) 7.44476e92 0.754628
\(497\) −8.90970e91 −0.0842496
\(498\) 5.96620e91 0.0526363
\(499\) 5.99629e92 0.493644 0.246822 0.969061i \(-0.420614\pi\)
0.246822 + 0.969061i \(0.420614\pi\)
\(500\) −3.71748e93 −2.85617
\(501\) 1.25900e93 0.902868
\(502\) 5.46423e91 0.0365806
\(503\) 2.62893e93 1.64317 0.821587 0.570083i \(-0.193089\pi\)
0.821587 + 0.570083i \(0.193089\pi\)
\(504\) −4.68495e91 −0.0273433
\(505\) −3.09011e93 −1.68431
\(506\) −4.57044e92 −0.232683
\(507\) 1.18448e93 0.563320
\(508\) −1.86754e93 −0.829797
\(509\) −2.27493e93 −0.944504 −0.472252 0.881464i \(-0.656559\pi\)
−0.472252 + 0.881464i \(0.656559\pi\)
\(510\) 5.57901e92 0.216464
\(511\) 2.01200e92 0.0729638
\(512\) 2.16904e93 0.735283
\(513\) 5.63855e92 0.178697
\(514\) 5.42370e92 0.160719
\(515\) −1.01295e94 −2.80699
\(516\) −3.11800e92 −0.0808093
\(517\) 2.58020e93 0.625503
\(518\) 2.03219e92 0.0460879
\(519\) −1.94396e90 −0.000412490 0
\(520\) −4.79967e92 −0.0953007
\(521\) 8.07097e93 1.49977 0.749884 0.661569i \(-0.230109\pi\)
0.749884 + 0.661569i \(0.230109\pi\)
\(522\) −2.12926e92 −0.0370337
\(523\) 5.06536e93 0.824712 0.412356 0.911023i \(-0.364706\pi\)
0.412356 + 0.911023i \(0.364706\pi\)
\(524\) −3.83616e93 −0.584746
\(525\) −2.61693e93 −0.373505
\(526\) −1.09465e92 −0.0146308
\(527\) −7.93101e93 −0.992801
\(528\) 4.59022e93 0.538224
\(529\) 8.66947e93 0.952296
\(530\) 1.13836e93 0.117155
\(531\) 6.44926e93 0.621941
\(532\) 2.52973e93 0.228625
\(533\) −2.44706e93 −0.207279
\(534\) 1.09913e93 0.0872719
\(535\) 1.61229e94 1.20016
\(536\) −1.14996e91 −0.000802596 0
\(537\) 4.93628e93 0.323063
\(538\) −1.23388e93 −0.0757332
\(539\) 1.64789e94 0.948671
\(540\) 6.53929e93 0.353139
\(541\) −1.95910e94 −0.992546 −0.496273 0.868167i \(-0.665298\pi\)
−0.496273 + 0.868167i \(0.665298\pi\)
\(542\) 4.16418e93 0.197950
\(543\) 8.32396e93 0.371309
\(544\) −1.37375e94 −0.575102
\(545\) −8.99726e94 −3.53534
\(546\) −1.01449e92 −0.00374199
\(547\) −3.66228e92 −0.0126820 −0.00634101 0.999980i \(-0.502018\pi\)
−0.00634101 + 0.999980i \(0.502018\pi\)
\(548\) 4.30590e94 1.40002
\(549\) −2.15675e94 −0.658497
\(550\) −1.48476e94 −0.425738
\(551\) 2.33137e94 0.627887
\(552\) 1.03365e94 0.261503
\(553\) −1.05188e94 −0.250005
\(554\) −1.08011e94 −0.241201
\(555\) −5.75178e94 −1.20697
\(556\) 5.61745e94 1.10779
\(557\) 4.28418e94 0.794078 0.397039 0.917802i \(-0.370038\pi\)
0.397039 + 0.917802i \(0.370038\pi\)
\(558\) 2.57905e93 0.0449345
\(559\) −1.36909e93 −0.0224246
\(560\) 2.85020e94 0.438923
\(561\) −4.89003e94 −0.708097
\(562\) −6.48778e93 −0.0883473
\(563\) −3.80080e94 −0.486783 −0.243391 0.969928i \(-0.578260\pi\)
−0.243391 + 0.969928i \(0.578260\pi\)
\(564\) −2.87777e94 −0.346679
\(565\) −1.59873e95 −1.81177
\(566\) 2.28227e94 0.243332
\(567\) 2.80273e93 0.0281168
\(568\) 1.14325e94 0.107926
\(569\) −1.11959e94 −0.0994682 −0.0497341 0.998762i \(-0.515837\pi\)
−0.0497341 + 0.998762i \(0.515837\pi\)
\(570\) 1.98642e94 0.166107
\(571\) 3.88842e94 0.306074 0.153037 0.988220i \(-0.451095\pi\)
0.153037 + 0.988220i \(0.451095\pi\)
\(572\) 2.07469e94 0.153741
\(573\) 1.25601e95 0.876313
\(574\) −8.41475e93 −0.0552817
\(575\) 5.77382e95 3.57209
\(576\) −4.81493e94 −0.280553
\(577\) −1.13415e95 −0.622451 −0.311226 0.950336i \(-0.600739\pi\)
−0.311226 + 0.950336i \(0.600739\pi\)
\(578\) 1.47514e94 0.0762645
\(579\) −2.52334e94 −0.122904
\(580\) 2.70380e95 1.24082
\(581\) −3.24709e94 −0.140416
\(582\) −2.24152e94 −0.0913480
\(583\) −9.97775e94 −0.383237
\(584\) −2.58171e94 −0.0934682
\(585\) 2.87136e94 0.0979964
\(586\) −7.55859e94 −0.243205
\(587\) −3.85716e95 −1.17018 −0.585090 0.810968i \(-0.698941\pi\)
−0.585090 + 0.810968i \(0.698941\pi\)
\(588\) −1.83794e95 −0.525793
\(589\) −2.82386e95 −0.761842
\(590\) 2.27203e95 0.578122
\(591\) −1.76557e95 −0.423756
\(592\) 4.50308e95 1.01955
\(593\) 3.26456e95 0.697329 0.348664 0.937248i \(-0.386635\pi\)
0.348664 + 0.937248i \(0.386635\pi\)
\(594\) 1.59017e94 0.0320487
\(595\) −3.03636e95 −0.577455
\(596\) 2.30734e95 0.414110
\(597\) −4.11943e95 −0.697786
\(598\) 2.23830e94 0.0357872
\(599\) −1.28868e96 −1.94501 −0.972504 0.232887i \(-0.925183\pi\)
−0.972504 + 0.232887i \(0.925183\pi\)
\(600\) 3.35793e95 0.478469
\(601\) 4.73264e95 0.636698 0.318349 0.947974i \(-0.396872\pi\)
0.318349 + 0.947974i \(0.396872\pi\)
\(602\) −4.70793e93 −0.00598069
\(603\) 6.87952e92 0.000825298 0
\(604\) −1.59470e96 −1.80678
\(605\) 4.81725e94 0.0515512
\(606\) 8.38090e94 0.0847198
\(607\) −1.65228e94 −0.0157788 −0.00788939 0.999969i \(-0.502511\pi\)
−0.00788939 + 0.999969i \(0.502511\pi\)
\(608\) −4.89126e95 −0.441314
\(609\) 1.15885e95 0.0987936
\(610\) −7.59807e95 −0.612102
\(611\) −1.26361e95 −0.0962036
\(612\) 5.45400e95 0.392456
\(613\) 2.85303e96 1.94053 0.970265 0.242045i \(-0.0778181\pi\)
0.970265 + 0.242045i \(0.0778181\pi\)
\(614\) −2.49830e94 −0.0160633
\(615\) 2.38166e96 1.44773
\(616\) 1.44665e95 0.0831436
\(617\) 6.46108e95 0.351129 0.175564 0.984468i \(-0.443825\pi\)
0.175564 + 0.984468i \(0.443825\pi\)
\(618\) 2.74730e95 0.141190
\(619\) 6.71691e95 0.326469 0.163234 0.986587i \(-0.447807\pi\)
0.163234 + 0.986587i \(0.447807\pi\)
\(620\) −3.27496e96 −1.50554
\(621\) −6.18374e95 −0.268900
\(622\) −3.11596e95 −0.128181
\(623\) −5.98197e95 −0.232813
\(624\) −2.24799e95 −0.0827800
\(625\) 8.55012e96 2.97928
\(626\) −3.43533e95 −0.113280
\(627\) −1.74111e96 −0.543370
\(628\) −8.46277e95 −0.249980
\(629\) −4.79719e96 −1.34134
\(630\) 9.87382e94 0.0261358
\(631\) −7.92321e95 −0.198558 −0.0992792 0.995060i \(-0.531654\pi\)
−0.0992792 + 0.995060i \(0.531654\pi\)
\(632\) 1.34972e96 0.320262
\(633\) 3.31472e96 0.744763
\(634\) −8.16474e95 −0.173725
\(635\) 7.98110e96 1.60831
\(636\) 1.11285e96 0.212406
\(637\) −8.07028e95 −0.145908
\(638\) 6.57487e95 0.112609
\(639\) −6.83940e95 −0.110978
\(640\) −7.52628e96 −1.15710
\(641\) −4.54296e96 −0.661816 −0.330908 0.943663i \(-0.607355\pi\)
−0.330908 + 0.943663i \(0.607355\pi\)
\(642\) −4.37280e95 −0.0603673
\(643\) −5.85972e96 −0.766652 −0.383326 0.923613i \(-0.625221\pi\)
−0.383326 + 0.923613i \(0.625221\pi\)
\(644\) −2.77433e96 −0.344030
\(645\) 1.33250e96 0.156624
\(646\) 1.65674e96 0.184601
\(647\) −1.30267e97 −1.37606 −0.688031 0.725681i \(-0.741525\pi\)
−0.688031 + 0.725681i \(0.741525\pi\)
\(648\) −3.59633e95 −0.0360182
\(649\) −1.99145e97 −1.89115
\(650\) 7.27135e95 0.0654794
\(651\) −1.40364e96 −0.119871
\(652\) 7.86095e96 0.636698
\(653\) 2.42489e97 1.86289 0.931445 0.363883i \(-0.118549\pi\)
0.931445 + 0.363883i \(0.118549\pi\)
\(654\) 2.44020e96 0.177825
\(655\) 1.63942e97 1.13335
\(656\) −1.86460e97 −1.22294
\(657\) 1.54448e96 0.0961121
\(658\) −4.34521e95 −0.0256577
\(659\) 4.44616e96 0.249137 0.124568 0.992211i \(-0.460245\pi\)
0.124568 + 0.992211i \(0.460245\pi\)
\(660\) −2.01925e97 −1.07380
\(661\) 2.46337e97 1.24331 0.621655 0.783291i \(-0.286460\pi\)
0.621655 + 0.783291i \(0.286460\pi\)
\(662\) 9.11123e95 0.0436492
\(663\) 2.39481e96 0.108907
\(664\) 4.16652e96 0.179877
\(665\) −1.08110e97 −0.443119
\(666\) 1.55998e96 0.0607097
\(667\) −2.55679e97 −0.944831
\(668\) 4.33599e97 1.52160
\(669\) −1.30365e97 −0.434469
\(670\) 2.42361e94 0.000767152 0
\(671\) 6.65975e97 2.00231
\(672\) −2.43128e96 −0.0694377
\(673\) −3.58072e97 −0.971519 −0.485759 0.874093i \(-0.661457\pi\)
−0.485759 + 0.874093i \(0.661457\pi\)
\(674\) −6.30856e96 −0.162616
\(675\) −2.00885e97 −0.492003
\(676\) 4.07936e97 0.949360
\(677\) 2.65612e97 0.587407 0.293704 0.955897i \(-0.405112\pi\)
0.293704 + 0.955897i \(0.405112\pi\)
\(678\) 4.33602e96 0.0911311
\(679\) 1.21994e97 0.243686
\(680\) 3.89612e97 0.739733
\(681\) −1.25620e97 −0.226717
\(682\) −7.96377e96 −0.136634
\(683\) 2.28473e97 0.372667 0.186333 0.982487i \(-0.440340\pi\)
0.186333 + 0.982487i \(0.440340\pi\)
\(684\) 1.94191e97 0.301158
\(685\) −1.84017e98 −2.71351
\(686\) −5.74016e96 −0.0804900
\(687\) 1.76011e97 0.234711
\(688\) −1.04322e97 −0.132304
\(689\) 4.88645e96 0.0589427
\(690\) −2.17849e97 −0.249955
\(691\) 6.64879e97 0.725688 0.362844 0.931850i \(-0.381806\pi\)
0.362844 + 0.931850i \(0.381806\pi\)
\(692\) −6.69498e94 −0.000695167 0
\(693\) −8.65445e96 −0.0854954
\(694\) 5.71930e94 0.000537577 0
\(695\) −2.40067e98 −2.14712
\(696\) −1.48698e97 −0.126557
\(697\) 1.98639e98 1.60892
\(698\) −3.18787e97 −0.245747
\(699\) −1.19187e98 −0.874514
\(700\) −9.01271e97 −0.629467
\(701\) −1.82266e98 −1.21181 −0.605903 0.795538i \(-0.707188\pi\)
−0.605903 + 0.795538i \(0.707188\pi\)
\(702\) −7.78760e95 −0.00492916
\(703\) −1.70805e98 −1.02930
\(704\) 1.48679e98 0.853084
\(705\) 1.22984e98 0.671932
\(706\) 1.81497e97 0.0944295
\(707\) −4.56128e97 −0.226005
\(708\) 2.22112e98 1.04815
\(709\) 2.78531e98 1.25193 0.625963 0.779853i \(-0.284706\pi\)
0.625963 + 0.779853i \(0.284706\pi\)
\(710\) −2.40947e97 −0.103159
\(711\) −8.07460e97 −0.329321
\(712\) 7.67579e97 0.298238
\(713\) 3.09690e98 1.14640
\(714\) 8.23511e96 0.0290457
\(715\) −8.86637e97 −0.297980
\(716\) 1.70005e98 0.544456
\(717\) 8.68428e97 0.265047
\(718\) 7.21258e97 0.209796
\(719\) −1.07858e98 −0.299023 −0.149511 0.988760i \(-0.547770\pi\)
−0.149511 + 0.988760i \(0.547770\pi\)
\(720\) 2.18791e98 0.578175
\(721\) −1.49521e98 −0.376648
\(722\) −9.42907e96 −0.0226431
\(723\) 1.43231e98 0.327918
\(724\) 2.86677e98 0.625766
\(725\) −8.30600e98 −1.72875
\(726\) −1.30652e96 −0.00259300
\(727\) −1.83421e98 −0.347145 −0.173572 0.984821i \(-0.555531\pi\)
−0.173572 + 0.984821i \(0.555531\pi\)
\(728\) −7.08474e96 −0.0127877
\(729\) 2.15147e97 0.0370370
\(730\) 5.44111e97 0.0893405
\(731\) 1.11136e98 0.174062
\(732\) −7.42783e98 −1.10976
\(733\) 8.73500e98 1.24502 0.622509 0.782612i \(-0.286113\pi\)
0.622509 + 0.782612i \(0.286113\pi\)
\(734\) −2.87672e97 −0.0391187
\(735\) 7.85462e98 1.01909
\(736\) 5.36420e98 0.664080
\(737\) −2.12430e96 −0.00250951
\(738\) −6.45946e97 −0.0728202
\(739\) 6.62422e98 0.712692 0.356346 0.934354i \(-0.384023\pi\)
0.356346 + 0.934354i \(0.384023\pi\)
\(740\) −1.98091e99 −2.03409
\(741\) 8.52680e97 0.0835714
\(742\) 1.68032e97 0.0157201
\(743\) −1.53661e99 −1.37230 −0.686150 0.727460i \(-0.740701\pi\)
−0.686150 + 0.727460i \(0.740701\pi\)
\(744\) 1.80109e98 0.153557
\(745\) −9.86064e98 −0.802627
\(746\) 1.36479e98 0.106066
\(747\) −2.49258e98 −0.184965
\(748\) −1.68412e99 −1.19335
\(749\) 2.37989e98 0.161040
\(750\) −4.30881e98 −0.278448
\(751\) −1.21982e99 −0.752864 −0.376432 0.926444i \(-0.622849\pi\)
−0.376432 + 0.926444i \(0.622849\pi\)
\(752\) −9.62845e98 −0.567598
\(753\) −2.28286e98 −0.128544
\(754\) −3.21994e97 −0.0173195
\(755\) 6.81510e99 3.50189
\(756\) 9.65258e97 0.0473851
\(757\) −1.23583e99 −0.579629 −0.289815 0.957083i \(-0.593594\pi\)
−0.289815 + 0.957083i \(0.593594\pi\)
\(758\) −3.49232e98 −0.156504
\(759\) 1.90946e99 0.817652
\(760\) 1.38722e99 0.567646
\(761\) −5.18239e98 −0.202656 −0.101328 0.994853i \(-0.532309\pi\)
−0.101328 + 0.994853i \(0.532309\pi\)
\(762\) −2.16461e98 −0.0808970
\(763\) −1.32808e99 −0.474380
\(764\) 4.32569e99 1.47685
\(765\) −2.33082e99 −0.760657
\(766\) −2.30545e98 −0.0719223
\(767\) 9.75279e98 0.290863
\(768\) −1.50014e99 −0.427730
\(769\) 5.71448e98 0.155782 0.0778911 0.996962i \(-0.475181\pi\)
0.0778911 + 0.996962i \(0.475181\pi\)
\(770\) −3.04890e98 −0.0794719
\(771\) −2.26593e99 −0.564769
\(772\) −8.69038e98 −0.207130
\(773\) 1.16534e99 0.265619 0.132809 0.991142i \(-0.457600\pi\)
0.132809 + 0.991142i \(0.457600\pi\)
\(774\) −3.61397e97 −0.00787810
\(775\) 1.00606e100 2.09756
\(776\) −1.56537e99 −0.312168
\(777\) −8.49015e98 −0.161953
\(778\) 3.76571e98 0.0687149
\(779\) 7.07260e99 1.23463
\(780\) 9.88894e98 0.165153
\(781\) 2.11192e99 0.337455
\(782\) −1.81694e99 −0.277784
\(783\) 8.89570e98 0.130137
\(784\) −6.14938e99 −0.860850
\(785\) 3.61664e99 0.484509
\(786\) −4.44637e98 −0.0570069
\(787\) −1.38928e100 −1.70475 −0.852375 0.522931i \(-0.824838\pi\)
−0.852375 + 0.522931i \(0.824838\pi\)
\(788\) −6.08060e99 −0.714154
\(789\) 4.57328e98 0.0514127
\(790\) −2.84462e99 −0.306119
\(791\) −2.35987e99 −0.243108
\(792\) 1.11050e99 0.109522
\(793\) −3.26151e99 −0.307959
\(794\) 2.41936e99 0.218723
\(795\) −4.75587e99 −0.411684
\(796\) −1.41873e100 −1.17598
\(797\) −8.96893e99 −0.711916 −0.355958 0.934502i \(-0.615845\pi\)
−0.355958 + 0.934502i \(0.615845\pi\)
\(798\) 2.93213e98 0.0222887
\(799\) 1.02573e100 0.746741
\(800\) 1.74262e100 1.21506
\(801\) −4.59197e99 −0.306674
\(802\) −2.47232e98 −0.0158157
\(803\) −4.76916e99 −0.292251
\(804\) 2.36930e97 0.00139087
\(805\) 1.18564e100 0.666797
\(806\) 3.90013e98 0.0210146
\(807\) 5.15497e99 0.266127
\(808\) 5.85283e99 0.289517
\(809\) −1.26179e100 −0.598087 −0.299044 0.954239i \(-0.596668\pi\)
−0.299044 + 0.954239i \(0.596668\pi\)
\(810\) 7.57949e98 0.0344276
\(811\) −2.52772e99 −0.110030 −0.0550148 0.998486i \(-0.517521\pi\)
−0.0550148 + 0.998486i \(0.517521\pi\)
\(812\) 3.99106e99 0.166496
\(813\) −1.73973e100 −0.695596
\(814\) −4.81701e99 −0.184602
\(815\) −3.35945e100 −1.23404
\(816\) 1.82480e100 0.642546
\(817\) 3.95701e99 0.133569
\(818\) −4.75627e99 −0.153914
\(819\) 4.23838e98 0.0131494
\(820\) 8.20243e100 2.43986
\(821\) 2.15046e100 0.613328 0.306664 0.951818i \(-0.400787\pi\)
0.306664 + 0.951818i \(0.400787\pi\)
\(822\) 4.99083e99 0.136488
\(823\) 9.20346e99 0.241355 0.120678 0.992692i \(-0.461493\pi\)
0.120678 + 0.992692i \(0.461493\pi\)
\(824\) 1.91859e100 0.482495
\(825\) 6.20306e100 1.49605
\(826\) 3.35372e99 0.0775738
\(827\) −7.13320e100 −1.58250 −0.791251 0.611492i \(-0.790570\pi\)
−0.791251 + 0.611492i \(0.790570\pi\)
\(828\) −2.12968e100 −0.453176
\(829\) 4.90975e100 1.00214 0.501070 0.865407i \(-0.332940\pi\)
0.501070 + 0.865407i \(0.332940\pi\)
\(830\) −8.78118e99 −0.171933
\(831\) 4.51250e100 0.847583
\(832\) −7.28130e99 −0.131206
\(833\) 6.55103e100 1.13255
\(834\) 6.51100e99 0.107999
\(835\) −1.85302e101 −2.94915
\(836\) −5.99637e100 −0.915739
\(837\) −1.07749e100 −0.157900
\(838\) 5.17763e99 0.0728135
\(839\) −4.61585e100 −0.622965 −0.311483 0.950252i \(-0.600826\pi\)
−0.311483 + 0.950252i \(0.600826\pi\)
\(840\) 6.89541e99 0.0893151
\(841\) −4.36570e100 −0.542740
\(842\) −2.09054e98 −0.00249454
\(843\) 2.71049e100 0.310453
\(844\) 1.14159e101 1.25515
\(845\) −1.74335e101 −1.84004
\(846\) −3.33554e99 −0.0337978
\(847\) 7.11069e98 0.00691726
\(848\) 3.72337e100 0.347760
\(849\) −9.53494e100 −0.855072
\(850\) −5.90250e100 −0.508257
\(851\) 1.87320e101 1.54887
\(852\) −2.35549e100 −0.187031
\(853\) 1.85595e101 1.41522 0.707611 0.706603i \(-0.249773\pi\)
0.707611 + 0.706603i \(0.249773\pi\)
\(854\) −1.12154e100 −0.0821333
\(855\) −8.29894e100 −0.583702
\(856\) −3.05376e100 −0.206296
\(857\) −1.43016e101 −0.927996 −0.463998 0.885836i \(-0.653586\pi\)
−0.463998 + 0.885836i \(0.653586\pi\)
\(858\) 2.40471e99 0.0149882
\(859\) −8.89938e100 −0.532839 −0.266419 0.963857i \(-0.585841\pi\)
−0.266419 + 0.963857i \(0.585841\pi\)
\(860\) 4.58914e100 0.263958
\(861\) 3.51555e100 0.194260
\(862\) −2.22475e100 −0.118109
\(863\) 2.29685e99 0.0117155 0.00585774 0.999983i \(-0.498135\pi\)
0.00585774 + 0.999983i \(0.498135\pi\)
\(864\) −1.86634e100 −0.0914673
\(865\) 2.86116e98 0.00134737
\(866\) 5.29548e100 0.239628
\(867\) −6.16289e100 −0.267994
\(868\) −4.83414e100 −0.202017
\(869\) 2.49333e101 1.00138
\(870\) 3.13389e100 0.120968
\(871\) 1.04034e98 0.000385967 0
\(872\) 1.70413e101 0.607691
\(873\) 9.36469e100 0.320998
\(874\) −6.46925e100 −0.213162
\(875\) 2.34506e101 0.742807
\(876\) 5.31919e100 0.161977
\(877\) 1.19553e101 0.350004 0.175002 0.984568i \(-0.444007\pi\)
0.175002 + 0.984568i \(0.444007\pi\)
\(878\) −2.51792e100 −0.0708729
\(879\) 3.15785e101 0.854625
\(880\) −6.75599e101 −1.75807
\(881\) 1.41431e101 0.353896 0.176948 0.984220i \(-0.443378\pi\)
0.176948 + 0.984220i \(0.443378\pi\)
\(882\) −2.13030e100 −0.0512596
\(883\) 8.01247e101 1.85405 0.927027 0.374995i \(-0.122356\pi\)
0.927027 + 0.374995i \(0.122356\pi\)
\(884\) 8.24772e100 0.183540
\(885\) −9.49217e101 −2.03153
\(886\) 9.41710e100 0.193845
\(887\) 5.39746e101 1.06862 0.534311 0.845288i \(-0.320571\pi\)
0.534311 + 0.845288i \(0.320571\pi\)
\(888\) 1.08942e101 0.207466
\(889\) 1.17808e101 0.215807
\(890\) −1.61772e101 −0.285068
\(891\) −6.64346e100 −0.112620
\(892\) −4.48975e101 −0.732209
\(893\) 3.65215e101 0.573024
\(894\) 2.67437e100 0.0403717
\(895\) −7.26532e101 −1.05526
\(896\) −1.11095e101 −0.155262
\(897\) −9.35126e100 −0.125757
\(898\) 9.38567e100 0.121460
\(899\) −4.45508e101 −0.554813
\(900\) −6.91847e101 −0.829170
\(901\) −3.96656e101 −0.457518
\(902\) 1.99460e101 0.221426
\(903\) 1.96690e100 0.0210162
\(904\) 3.02807e101 0.311427
\(905\) −1.22514e102 −1.21286
\(906\) −1.84837e101 −0.176143
\(907\) −9.24045e101 −0.847699 −0.423850 0.905733i \(-0.639322\pi\)
−0.423850 + 0.905733i \(0.639322\pi\)
\(908\) −4.32635e101 −0.382085
\(909\) −3.50140e101 −0.297706
\(910\) 1.49315e100 0.0122229
\(911\) 1.06011e102 0.835543 0.417772 0.908552i \(-0.362811\pi\)
0.417772 + 0.908552i \(0.362811\pi\)
\(912\) 6.49724e101 0.493068
\(913\) 7.69675e101 0.562427
\(914\) 1.94476e101 0.136843
\(915\) 3.17435e102 2.15093
\(916\) 6.06183e101 0.395557
\(917\) 2.41992e101 0.152076
\(918\) 6.32156e100 0.0382606
\(919\) −2.97590e102 −1.73474 −0.867368 0.497668i \(-0.834190\pi\)
−0.867368 + 0.497668i \(0.834190\pi\)
\(920\) −1.52135e102 −0.854182
\(921\) 1.04375e101 0.0564467
\(922\) −2.42599e101 −0.126378
\(923\) −1.03428e101 −0.0519013
\(924\) −2.98059e101 −0.144085
\(925\) 6.08530e102 2.83395
\(926\) 3.31167e101 0.148583
\(927\) −1.14778e102 −0.496143
\(928\) −7.71675e101 −0.321388
\(929\) −3.26208e102 −1.30904 −0.654520 0.756045i \(-0.727129\pi\)
−0.654520 + 0.756045i \(0.727129\pi\)
\(930\) −3.79591e101 −0.146776
\(931\) 2.33251e102 0.869080
\(932\) −4.10480e102 −1.47381
\(933\) 1.30179e102 0.450428
\(934\) 6.91434e101 0.230560
\(935\) 7.19725e102 2.31295
\(936\) −5.43850e100 −0.0168447
\(937\) 4.99948e102 1.49248 0.746241 0.665675i \(-0.231856\pi\)
0.746241 + 0.665675i \(0.231856\pi\)
\(938\) 3.57746e98 0.000102938 0
\(939\) 1.43523e102 0.398068
\(940\) 4.23557e102 1.13240
\(941\) −6.74589e102 −1.73859 −0.869295 0.494293i \(-0.835427\pi\)
−0.869295 + 0.494293i \(0.835427\pi\)
\(942\) −9.80893e100 −0.0243706
\(943\) −7.75645e102 −1.85784
\(944\) 7.43142e102 1.71608
\(945\) −4.12512e101 −0.0918415
\(946\) 1.11595e101 0.0239552
\(947\) −6.40994e102 −1.32672 −0.663361 0.748300i \(-0.730870\pi\)
−0.663361 + 0.748300i \(0.730870\pi\)
\(948\) −2.78089e102 −0.555003
\(949\) 2.33562e101 0.0449488
\(950\) −2.10160e102 −0.390019
\(951\) 3.41109e102 0.610472
\(952\) 5.75102e101 0.0992590
\(953\) −3.74284e102 −0.623012 −0.311506 0.950244i \(-0.600833\pi\)
−0.311506 + 0.950244i \(0.600833\pi\)
\(954\) 1.28987e101 0.0207075
\(955\) −1.84862e103 −2.86241
\(956\) 2.99086e102 0.446682
\(957\) −2.74687e102 −0.395710
\(958\) 1.46229e102 0.203200
\(959\) −2.71625e102 −0.364106
\(960\) 7.08672e102 0.916405
\(961\) −2.61980e102 −0.326821
\(962\) 2.35905e101 0.0283921
\(963\) 1.82689e102 0.212131
\(964\) 4.93287e102 0.552639
\(965\) 3.71391e102 0.401457
\(966\) −3.21564e101 −0.0335395
\(967\) 1.14430e103 1.15166 0.575832 0.817568i \(-0.304678\pi\)
0.575832 + 0.817568i \(0.304678\pi\)
\(968\) −9.12411e100 −0.00886116
\(969\) −6.92161e102 −0.648689
\(970\) 3.29911e102 0.298382
\(971\) −1.09503e103 −0.955786 −0.477893 0.878418i \(-0.658599\pi\)
−0.477893 + 0.878418i \(0.658599\pi\)
\(972\) 7.40966e101 0.0624183
\(973\) −3.54360e102 −0.288105
\(974\) 2.29165e102 0.179831
\(975\) −3.03785e102 −0.230095
\(976\) −2.48520e103 −1.81695
\(977\) 6.27156e102 0.442602 0.221301 0.975206i \(-0.428970\pi\)
0.221301 + 0.975206i \(0.428970\pi\)
\(978\) 9.11138e101 0.0620717
\(979\) 1.41794e103 0.932513
\(980\) 2.70513e103 1.71746
\(981\) −1.01948e103 −0.624880
\(982\) −4.00137e101 −0.0236790
\(983\) 9.29318e102 0.530968 0.265484 0.964115i \(-0.414468\pi\)
0.265484 + 0.964115i \(0.414468\pi\)
\(984\) −4.51099e102 −0.248852
\(985\) 2.59860e103 1.38417
\(986\) 2.61378e102 0.134436
\(987\) 1.81536e102 0.0901614
\(988\) 2.93662e102 0.140843
\(989\) −4.33962e102 −0.200992
\(990\) −2.34044e102 −0.104685
\(991\) 8.94969e102 0.386605 0.193302 0.981139i \(-0.438080\pi\)
0.193302 + 0.981139i \(0.438080\pi\)
\(992\) 9.34685e102 0.389953
\(993\) −3.80652e102 −0.153384
\(994\) −3.55660e101 −0.0138422
\(995\) 6.06306e103 2.27927
\(996\) −8.58443e102 −0.311720
\(997\) −3.91155e103 −1.37204 −0.686019 0.727583i \(-0.740643\pi\)
−0.686019 + 0.727583i \(0.740643\pi\)
\(998\) 2.39361e102 0.0811054
\(999\) −6.51734e102 −0.213334
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 3.70.a.a.1.4 6
3.2 odd 2 9.70.a.d.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.70.a.a.1.4 6 1.1 even 1 trivial
9.70.a.d.1.3 6 3.2 odd 2