Properties

Label 3.34.a.b.1.2
Level $3$
Weight $34$
Character 3.1
Self dual yes
Analytic conductor $20.695$
Analytic rank $1$
Dimension $3$
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] = [3,34,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, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 34 \)
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: \(20.6948486643\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5357605x + 842871622 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2232.09\) 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)
 
\(f(q)\) \(=\) \(q+61269.1 q^{2} -4.30467e7 q^{3} -4.83603e9 q^{4} +2.12383e11 q^{5} -2.63744e12 q^{6} +1.58203e14 q^{7} -8.22597e14 q^{8} +1.85302e15 q^{9} +O(q^{10})\) \(q+61269.1 q^{2} -4.30467e7 q^{3} -4.83603e9 q^{4} +2.12383e11 q^{5} -2.63744e12 q^{6} +1.58203e14 q^{7} -8.22597e14 q^{8} +1.85302e15 q^{9} +1.30125e16 q^{10} -2.62906e17 q^{11} +2.08175e17 q^{12} -3.56670e18 q^{13} +9.69295e18 q^{14} -9.14240e18 q^{15} -8.85864e18 q^{16} +1.73081e19 q^{17} +1.13533e20 q^{18} -5.11529e20 q^{19} -1.02709e21 q^{20} -6.81012e21 q^{21} -1.61080e22 q^{22} -3.17073e22 q^{23} +3.54101e22 q^{24} -7.13087e22 q^{25} -2.18529e23 q^{26} -7.97664e22 q^{27} -7.65074e23 q^{28} +1.64526e23 q^{29} -5.60147e23 q^{30} +1.52635e24 q^{31} +6.52329e24 q^{32} +1.13172e25 q^{33} +1.06045e24 q^{34} +3.35996e25 q^{35} -8.96126e24 q^{36} -4.62071e25 q^{37} -3.13410e25 q^{38} +1.53535e26 q^{39} -1.74706e26 q^{40} -3.97631e26 q^{41} -4.17250e26 q^{42} -3.64043e26 q^{43} +1.27142e27 q^{44} +3.93550e26 q^{45} -1.94268e27 q^{46} -2.74215e27 q^{47} +3.81336e26 q^{48} +1.72972e28 q^{49} -4.36902e27 q^{50} -7.45059e26 q^{51} +1.72487e28 q^{52} +3.12962e28 q^{53} -4.88722e27 q^{54} -5.58368e28 q^{55} -1.30137e29 q^{56} +2.20197e28 q^{57} +1.00804e28 q^{58} -9.28615e28 q^{59} +4.42129e28 q^{60} -8.67336e28 q^{61} +9.35181e28 q^{62} +2.93153e29 q^{63} +4.75772e29 q^{64} -7.57508e29 q^{65} +6.93397e29 q^{66} +4.18932e29 q^{67} -8.37027e28 q^{68} +1.36489e30 q^{69} +2.05862e30 q^{70} -4.44984e30 q^{71} -1.52429e30 q^{72} +1.71800e30 q^{73} -2.83107e30 q^{74} +3.06961e30 q^{75} +2.47377e30 q^{76} -4.15925e31 q^{77} +9.40695e30 q^{78} +2.50450e31 q^{79} -1.88143e30 q^{80} +3.43368e30 q^{81} -2.43625e31 q^{82} +7.18974e31 q^{83} +3.29339e31 q^{84} +3.67596e30 q^{85} -2.23046e31 q^{86} -7.08231e30 q^{87} +2.16266e32 q^{88} -1.21977e32 q^{89} +2.41125e31 q^{90} -5.64263e32 q^{91} +1.53337e32 q^{92} -6.57043e31 q^{93} -1.68009e32 q^{94} -1.08640e32 q^{95} -2.80806e32 q^{96} +5.67244e32 q^{97} +1.05978e33 q^{98} -4.87170e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 136620 q^{2} - 129140163 q^{3} + 14589555600 q^{4} - 260488036134 q^{5} - 5881043023020 q^{6} + 10760698892832 q^{7} + 18\!\cdots\!76 q^{8}+ \cdots + 55\!\cdots\!23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 136620 q^{2} - 129140163 q^{3} + 14589555600 q^{4} - 260488036134 q^{5} - 5881043023020 q^{6} + 10760698892832 q^{7} + 18\!\cdots\!76 q^{8}+ \cdots - 64\!\cdots\!96 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 61269.1 0.661069 0.330535 0.943794i \(-0.392771\pi\)
0.330535 + 0.943794i \(0.392771\pi\)
\(3\) −4.30467e7 −0.577350
\(4\) −4.83603e9 −0.562988
\(5\) 2.12383e11 0.622465 0.311232 0.950334i \(-0.399258\pi\)
0.311232 + 0.950334i \(0.399258\pi\)
\(6\) −2.63744e12 −0.381668
\(7\) 1.58203e14 1.79927 0.899636 0.436641i \(-0.143832\pi\)
0.899636 + 0.436641i \(0.143832\pi\)
\(8\) −8.22597e14 −1.03324
\(9\) 1.85302e15 0.333333
\(10\) 1.30125e16 0.411492
\(11\) −2.62906e17 −1.72513 −0.862563 0.505949i \(-0.831142\pi\)
−0.862563 + 0.505949i \(0.831142\pi\)
\(12\) 2.08175e17 0.325041
\(13\) −3.56670e18 −1.48663 −0.743313 0.668944i \(-0.766747\pi\)
−0.743313 + 0.668944i \(0.766747\pi\)
\(14\) 9.69295e18 1.18944
\(15\) −9.14240e18 −0.359380
\(16\) −8.85864e18 −0.120057
\(17\) 1.73081e19 0.0862667 0.0431334 0.999069i \(-0.486266\pi\)
0.0431334 + 0.999069i \(0.486266\pi\)
\(18\) 1.13533e20 0.220356
\(19\) −5.11529e20 −0.406852 −0.203426 0.979090i \(-0.565208\pi\)
−0.203426 + 0.979090i \(0.565208\pi\)
\(20\) −1.02709e21 −0.350440
\(21\) −6.81012e21 −1.03881
\(22\) −1.61080e22 −1.14043
\(23\) −3.17073e22 −1.07808 −0.539039 0.842281i \(-0.681212\pi\)
−0.539039 + 0.842281i \(0.681212\pi\)
\(24\) 3.54101e22 0.596543
\(25\) −7.13087e22 −0.612537
\(26\) −2.18529e23 −0.982762
\(27\) −7.97664e22 −0.192450
\(28\) −7.65074e23 −1.01297
\(29\) 1.64526e23 0.122087 0.0610433 0.998135i \(-0.480557\pi\)
0.0610433 + 0.998135i \(0.480557\pi\)
\(30\) −5.60147e23 −0.237575
\(31\) 1.52635e24 0.376865 0.188433 0.982086i \(-0.439659\pi\)
0.188433 + 0.982086i \(0.439659\pi\)
\(32\) 6.52329e24 0.953877
\(33\) 1.13172e25 0.996002
\(34\) 1.06045e24 0.0570282
\(35\) 3.35996e25 1.11998
\(36\) −8.96126e24 −0.187663
\(37\) −4.62071e25 −0.615716 −0.307858 0.951432i \(-0.599612\pi\)
−0.307858 + 0.951432i \(0.599612\pi\)
\(38\) −3.13410e25 −0.268957
\(39\) 1.53535e26 0.858304
\(40\) −1.74706e26 −0.643157
\(41\) −3.97631e26 −0.973972 −0.486986 0.873410i \(-0.661904\pi\)
−0.486986 + 0.873410i \(0.661904\pi\)
\(42\) −4.17250e26 −0.686725
\(43\) −3.64043e26 −0.406372 −0.203186 0.979140i \(-0.565130\pi\)
−0.203186 + 0.979140i \(0.565130\pi\)
\(44\) 1.27142e27 0.971225
\(45\) 3.93550e26 0.207488
\(46\) −1.94268e27 −0.712683
\(47\) −2.74215e27 −0.705467 −0.352733 0.935724i \(-0.614748\pi\)
−0.352733 + 0.935724i \(0.614748\pi\)
\(48\) 3.81336e26 0.0693149
\(49\) 1.72972e28 2.23738
\(50\) −4.36902e27 −0.404930
\(51\) −7.45059e26 −0.0498061
\(52\) 1.72487e28 0.836952
\(53\) 3.12962e28 1.10902 0.554511 0.832177i \(-0.312906\pi\)
0.554511 + 0.832177i \(0.312906\pi\)
\(54\) −4.88722e27 −0.127223
\(55\) −5.58368e28 −1.07383
\(56\) −1.30137e29 −1.85908
\(57\) 2.20197e28 0.234896
\(58\) 1.00804e28 0.0807077
\(59\) −9.28615e28 −0.560761 −0.280380 0.959889i \(-0.590461\pi\)
−0.280380 + 0.959889i \(0.590461\pi\)
\(60\) 4.42129e28 0.202327
\(61\) −8.67336e28 −0.302166 −0.151083 0.988521i \(-0.548276\pi\)
−0.151083 + 0.988521i \(0.548276\pi\)
\(62\) 9.35181e28 0.249134
\(63\) 2.93153e29 0.599757
\(64\) 4.75772e29 0.750635
\(65\) −7.57508e29 −0.925372
\(66\) 6.93397e29 0.658426
\(67\) 4.18932e29 0.310391 0.155196 0.987884i \(-0.450399\pi\)
0.155196 + 0.987884i \(0.450399\pi\)
\(68\) −8.37027e28 −0.0485671
\(69\) 1.36489e30 0.622428
\(70\) 2.05862e30 0.740386
\(71\) −4.44984e30 −1.26643 −0.633216 0.773975i \(-0.718266\pi\)
−0.633216 + 0.773975i \(0.718266\pi\)
\(72\) −1.52429e30 −0.344414
\(73\) 1.71800e30 0.309169 0.154584 0.987980i \(-0.450596\pi\)
0.154584 + 0.987980i \(0.450596\pi\)
\(74\) −2.83107e30 −0.407031
\(75\) 3.06961e30 0.353649
\(76\) 2.47377e30 0.229053
\(77\) −4.15925e31 −3.10397
\(78\) 9.40695e30 0.567398
\(79\) 2.50450e31 1.22426 0.612130 0.790757i \(-0.290313\pi\)
0.612130 + 0.790757i \(0.290313\pi\)
\(80\) −1.88143e30 −0.0747313
\(81\) 3.43368e30 0.111111
\(82\) −2.43625e31 −0.643863
\(83\) 7.18974e31 1.55570 0.777848 0.628453i \(-0.216311\pi\)
0.777848 + 0.628453i \(0.216311\pi\)
\(84\) 3.29339e31 0.584837
\(85\) 3.67596e30 0.0536980
\(86\) −2.23046e31 −0.268640
\(87\) −7.08231e30 −0.0704868
\(88\) 2.16266e32 1.78247
\(89\) −1.21977e32 −0.834338 −0.417169 0.908829i \(-0.636978\pi\)
−0.417169 + 0.908829i \(0.636978\pi\)
\(90\) 2.41125e31 0.137164
\(91\) −5.64263e32 −2.67484
\(92\) 1.53337e32 0.606944
\(93\) −6.57043e31 −0.217583
\(94\) −1.68009e32 −0.466362
\(95\) −1.08640e32 −0.253251
\(96\) −2.80806e32 −0.550721
\(97\) 5.67244e32 0.937640 0.468820 0.883294i \(-0.344679\pi\)
0.468820 + 0.883294i \(0.344679\pi\)
\(98\) 1.05978e33 1.47906
\(99\) −4.87170e32 −0.575042
\(100\) 3.44851e32 0.344851
\(101\) −4.34825e32 −0.368987 −0.184494 0.982834i \(-0.559065\pi\)
−0.184494 + 0.982834i \(0.559065\pi\)
\(102\) −4.56491e31 −0.0329253
\(103\) −1.39936e33 −0.859242 −0.429621 0.903009i \(-0.641353\pi\)
−0.429621 + 0.903009i \(0.641353\pi\)
\(104\) 2.93396e33 1.53605
\(105\) −1.44635e33 −0.646623
\(106\) 1.91749e33 0.733139
\(107\) 3.37111e33 1.10393 0.551965 0.833868i \(-0.313878\pi\)
0.551965 + 0.833868i \(0.313878\pi\)
\(108\) 3.85753e32 0.108347
\(109\) −4.20805e33 −1.01518 −0.507591 0.861598i \(-0.669464\pi\)
−0.507591 + 0.861598i \(0.669464\pi\)
\(110\) −3.42107e33 −0.709876
\(111\) 1.98907e33 0.355484
\(112\) −1.40146e33 −0.216015
\(113\) −5.11191e33 −0.680438 −0.340219 0.940346i \(-0.610501\pi\)
−0.340219 + 0.940346i \(0.610501\pi\)
\(114\) 1.34913e33 0.155282
\(115\) −6.73409e33 −0.671065
\(116\) −7.95653e32 −0.0687333
\(117\) −6.60917e33 −0.495542
\(118\) −5.68954e33 −0.370701
\(119\) 2.73820e33 0.155217
\(120\) 7.52051e33 0.371327
\(121\) 4.58943e34 1.97606
\(122\) −5.31409e33 −0.199752
\(123\) 1.71167e34 0.562323
\(124\) −7.38147e33 −0.212170
\(125\) −3.98694e34 −1.00375
\(126\) 1.79612e34 0.396481
\(127\) 9.75563e34 1.89014 0.945070 0.326869i \(-0.105993\pi\)
0.945070 + 0.326869i \(0.105993\pi\)
\(128\) −2.68845e34 −0.457655
\(129\) 1.56709e34 0.234619
\(130\) −4.64118e34 −0.611735
\(131\) −8.73187e34 −1.01422 −0.507109 0.861882i \(-0.669286\pi\)
−0.507109 + 0.861882i \(0.669286\pi\)
\(132\) −5.47305e34 −0.560737
\(133\) −8.09254e34 −0.732037
\(134\) 2.56676e34 0.205190
\(135\) −1.69410e34 −0.119793
\(136\) −1.42376e34 −0.0891345
\(137\) 1.13700e34 0.0630770 0.0315385 0.999503i \(-0.489959\pi\)
0.0315385 + 0.999503i \(0.489959\pi\)
\(138\) 8.36259e34 0.411468
\(139\) 7.98052e33 0.0348568 0.0174284 0.999848i \(-0.494452\pi\)
0.0174284 + 0.999848i \(0.494452\pi\)
\(140\) −1.62489e35 −0.630537
\(141\) 1.18041e35 0.407301
\(142\) −2.72638e35 −0.837199
\(143\) 9.37707e35 2.56462
\(144\) −1.64152e34 −0.0400190
\(145\) 3.49426e34 0.0759947
\(146\) 1.05260e35 0.204382
\(147\) −7.44586e35 −1.29175
\(148\) 2.23459e35 0.346641
\(149\) −1.73859e35 −0.241336 −0.120668 0.992693i \(-0.538504\pi\)
−0.120668 + 0.992693i \(0.538504\pi\)
\(150\) 1.88072e35 0.233786
\(151\) 8.75792e35 0.975622 0.487811 0.872949i \(-0.337796\pi\)
0.487811 + 0.872949i \(0.337796\pi\)
\(152\) 4.20783e35 0.420377
\(153\) 3.20723e34 0.0287556
\(154\) −2.54833e36 −2.05194
\(155\) 3.24171e35 0.234585
\(156\) −7.42499e35 −0.483215
\(157\) 5.38159e35 0.315186 0.157593 0.987504i \(-0.449627\pi\)
0.157593 + 0.987504i \(0.449627\pi\)
\(158\) 1.53448e36 0.809321
\(159\) −1.34720e36 −0.640294
\(160\) 1.38544e36 0.593755
\(161\) −5.01619e36 −1.93975
\(162\) 2.10379e35 0.0734521
\(163\) −4.22882e36 −1.33390 −0.666951 0.745102i \(-0.732401\pi\)
−0.666951 + 0.745102i \(0.732401\pi\)
\(164\) 1.92295e36 0.548334
\(165\) 2.40359e36 0.619977
\(166\) 4.40509e36 1.02842
\(167\) 5.06821e35 0.107160 0.0535800 0.998564i \(-0.482937\pi\)
0.0535800 + 0.998564i \(0.482937\pi\)
\(168\) 5.60198e36 1.07334
\(169\) 6.96524e36 1.21006
\(170\) 2.25223e35 0.0354981
\(171\) −9.47874e35 −0.135617
\(172\) 1.76052e36 0.228782
\(173\) −5.77607e36 −0.682138 −0.341069 0.940038i \(-0.610789\pi\)
−0.341069 + 0.940038i \(0.610789\pi\)
\(174\) −4.33927e35 −0.0465966
\(175\) −1.12812e37 −1.10212
\(176\) 2.32899e36 0.207114
\(177\) 3.99738e36 0.323755
\(178\) −7.47341e36 −0.551555
\(179\) −7.40660e36 −0.498361 −0.249180 0.968457i \(-0.580161\pi\)
−0.249180 + 0.968457i \(0.580161\pi\)
\(180\) −1.90322e36 −0.116813
\(181\) 1.00139e37 0.560928 0.280464 0.959865i \(-0.409512\pi\)
0.280464 + 0.959865i \(0.409512\pi\)
\(182\) −3.45719e37 −1.76826
\(183\) 3.73360e36 0.174455
\(184\) 2.60823e37 1.11392
\(185\) −9.81361e36 −0.383262
\(186\) −4.02565e36 −0.143837
\(187\) −4.55041e36 −0.148821
\(188\) 1.32611e37 0.397169
\(189\) −1.26193e37 −0.346270
\(190\) −6.65629e36 −0.167416
\(191\) 7.94277e37 1.83198 0.915991 0.401198i \(-0.131406\pi\)
0.915991 + 0.401198i \(0.131406\pi\)
\(192\) −2.04804e37 −0.433380
\(193\) 2.66529e37 0.517666 0.258833 0.965922i \(-0.416662\pi\)
0.258833 + 0.965922i \(0.416662\pi\)
\(194\) 3.47545e37 0.619845
\(195\) 3.26082e37 0.534264
\(196\) −8.36496e37 −1.25962
\(197\) 8.31876e36 0.115177 0.0575885 0.998340i \(-0.481659\pi\)
0.0575885 + 0.998340i \(0.481659\pi\)
\(198\) −2.98485e37 −0.380143
\(199\) −7.30042e36 −0.0855603 −0.0427801 0.999085i \(-0.513621\pi\)
−0.0427801 + 0.999085i \(0.513621\pi\)
\(200\) 5.86584e37 0.632900
\(201\) −1.80337e37 −0.179204
\(202\) −2.66413e37 −0.243926
\(203\) 2.60285e37 0.219667
\(204\) 3.60313e36 0.0280402
\(205\) −8.44501e37 −0.606263
\(206\) −8.57377e37 −0.568019
\(207\) −5.87543e37 −0.359359
\(208\) 3.15962e37 0.178480
\(209\) 1.34484e38 0.701871
\(210\) −8.86168e37 −0.427462
\(211\) −2.24819e38 −1.00270 −0.501352 0.865243i \(-0.667164\pi\)
−0.501352 + 0.865243i \(0.667164\pi\)
\(212\) −1.51349e38 −0.624365
\(213\) 1.91551e38 0.731175
\(214\) 2.06545e38 0.729773
\(215\) −7.73166e37 −0.252952
\(216\) 6.56156e37 0.198848
\(217\) 2.41473e38 0.678083
\(218\) −2.57824e38 −0.671105
\(219\) −7.39542e37 −0.178499
\(220\) 2.70028e38 0.604554
\(221\) −6.17330e37 −0.128246
\(222\) 1.21868e38 0.234999
\(223\) 9.65044e38 1.72789 0.863947 0.503583i \(-0.167985\pi\)
0.863947 + 0.503583i \(0.167985\pi\)
\(224\) 1.03200e39 1.71628
\(225\) −1.32137e38 −0.204179
\(226\) −3.13202e38 −0.449817
\(227\) −1.45369e39 −1.94109 −0.970545 0.240921i \(-0.922551\pi\)
−0.970545 + 0.240921i \(0.922551\pi\)
\(228\) −1.06488e38 −0.132244
\(229\) −8.85760e38 −1.02337 −0.511683 0.859174i \(-0.670978\pi\)
−0.511683 + 0.859174i \(0.670978\pi\)
\(230\) −4.12592e38 −0.443620
\(231\) 1.79042e39 1.79208
\(232\) −1.35339e38 −0.126145
\(233\) −1.96369e39 −1.70491 −0.852453 0.522804i \(-0.824886\pi\)
−0.852453 + 0.522804i \(0.824886\pi\)
\(234\) −4.04938e38 −0.327587
\(235\) −5.82386e38 −0.439128
\(236\) 4.49081e38 0.315701
\(237\) −1.07810e39 −0.706827
\(238\) 1.67767e38 0.102609
\(239\) 1.13364e39 0.647008 0.323504 0.946227i \(-0.395139\pi\)
0.323504 + 0.946227i \(0.395139\pi\)
\(240\) 8.09892e37 0.0431461
\(241\) −8.77821e38 −0.436641 −0.218321 0.975877i \(-0.570058\pi\)
−0.218321 + 0.975877i \(0.570058\pi\)
\(242\) 2.81191e39 1.30631
\(243\) −1.47809e38 −0.0641500
\(244\) 4.19446e38 0.170115
\(245\) 3.67363e39 1.39269
\(246\) 1.04873e39 0.371734
\(247\) 1.82447e39 0.604836
\(248\) −1.25557e39 −0.389393
\(249\) −3.09495e39 −0.898181
\(250\) −2.44276e39 −0.663547
\(251\) −4.39897e39 −1.11875 −0.559377 0.828913i \(-0.688960\pi\)
−0.559377 + 0.828913i \(0.688960\pi\)
\(252\) −1.41770e39 −0.337656
\(253\) 8.33603e39 1.85982
\(254\) 5.97719e39 1.24951
\(255\) −1.58238e38 −0.0310026
\(256\) −5.73404e39 −1.05318
\(257\) −6.53419e38 −0.112537 −0.0562685 0.998416i \(-0.517920\pi\)
−0.0562685 + 0.998416i \(0.517920\pi\)
\(258\) 9.60141e38 0.155099
\(259\) −7.31010e39 −1.10784
\(260\) 3.66333e39 0.520973
\(261\) 3.04870e38 0.0406955
\(262\) −5.34994e39 −0.670467
\(263\) 1.97088e39 0.231948 0.115974 0.993252i \(-0.463001\pi\)
0.115974 + 0.993252i \(0.463001\pi\)
\(264\) −9.30953e39 −1.02911
\(265\) 6.64678e39 0.690327
\(266\) −4.95823e39 −0.483927
\(267\) 5.25070e39 0.481705
\(268\) −2.02597e39 −0.174746
\(269\) −1.40997e40 −1.14366 −0.571830 0.820372i \(-0.693766\pi\)
−0.571830 + 0.820372i \(0.693766\pi\)
\(270\) −1.03796e39 −0.0791917
\(271\) 2.95789e39 0.212319 0.106160 0.994349i \(-0.466145\pi\)
0.106160 + 0.994349i \(0.466145\pi\)
\(272\) −1.53327e38 −0.0103569
\(273\) 2.42897e40 1.54432
\(274\) 6.96630e38 0.0416983
\(275\) 1.87475e40 1.05670
\(276\) −6.60067e39 −0.350419
\(277\) 1.84352e40 0.922000 0.461000 0.887400i \(-0.347491\pi\)
0.461000 + 0.887400i \(0.347491\pi\)
\(278\) 4.88960e38 0.0230427
\(279\) 2.82836e39 0.125622
\(280\) −2.76390e40 −1.15721
\(281\) −3.15556e40 −1.24572 −0.622862 0.782332i \(-0.714030\pi\)
−0.622862 + 0.782332i \(0.714030\pi\)
\(282\) 7.23224e39 0.269254
\(283\) 2.91693e40 1.02435 0.512176 0.858880i \(-0.328840\pi\)
0.512176 + 0.858880i \(0.328840\pi\)
\(284\) 2.15196e40 0.712986
\(285\) 4.67660e39 0.146214
\(286\) 5.74525e40 1.69539
\(287\) −6.29064e40 −1.75244
\(288\) 1.20878e40 0.317959
\(289\) −3.99549e40 −0.992558
\(290\) 2.14090e39 0.0502377
\(291\) −2.44180e40 −0.541347
\(292\) −8.30829e39 −0.174058
\(293\) −7.83490e40 −1.55138 −0.775689 0.631115i \(-0.782598\pi\)
−0.775689 + 0.631115i \(0.782598\pi\)
\(294\) −4.56202e40 −0.853937
\(295\) −1.97222e40 −0.349054
\(296\) 3.80098e40 0.636184
\(297\) 2.09711e40 0.332001
\(298\) −1.06522e40 −0.159540
\(299\) 1.13091e41 1.60270
\(300\) −1.48447e40 −0.199100
\(301\) −5.75927e40 −0.731173
\(302\) 5.36590e40 0.644953
\(303\) 1.87178e40 0.213035
\(304\) 4.53146e39 0.0488454
\(305\) −1.84207e40 −0.188087
\(306\) 1.96504e39 0.0190094
\(307\) 1.14909e41 1.05335 0.526673 0.850068i \(-0.323439\pi\)
0.526673 + 0.850068i \(0.323439\pi\)
\(308\) 2.01142e41 1.74750
\(309\) 6.02379e40 0.496084
\(310\) 1.98617e40 0.155077
\(311\) −1.32546e41 −0.981342 −0.490671 0.871345i \(-0.663248\pi\)
−0.490671 + 0.871345i \(0.663248\pi\)
\(312\) −1.26297e41 −0.886836
\(313\) −1.30815e41 −0.871319 −0.435660 0.900111i \(-0.643485\pi\)
−0.435660 + 0.900111i \(0.643485\pi\)
\(314\) 3.29726e40 0.208360
\(315\) 6.22608e40 0.373328
\(316\) −1.21118e41 −0.689244
\(317\) 1.44144e41 0.778607 0.389303 0.921110i \(-0.372716\pi\)
0.389303 + 0.921110i \(0.372716\pi\)
\(318\) −8.25417e40 −0.423278
\(319\) −4.32549e40 −0.210615
\(320\) 1.01046e41 0.467244
\(321\) −1.45115e41 −0.637354
\(322\) −3.07337e41 −1.28231
\(323\) −8.85362e39 −0.0350978
\(324\) −1.66054e40 −0.0625542
\(325\) 2.54337e41 0.910614
\(326\) −2.59096e41 −0.881801
\(327\) 1.81143e41 0.586115
\(328\) 3.27090e41 1.00635
\(329\) −4.33816e41 −1.26933
\(330\) 1.47266e41 0.409847
\(331\) 3.16472e41 0.837864 0.418932 0.908018i \(-0.362405\pi\)
0.418932 + 0.908018i \(0.362405\pi\)
\(332\) −3.47698e41 −0.875837
\(333\) −8.56227e40 −0.205239
\(334\) 3.10525e40 0.0708402
\(335\) 8.89741e40 0.193208
\(336\) 6.03284e40 0.124716
\(337\) −3.75601e41 −0.739321 −0.369660 0.929167i \(-0.620526\pi\)
−0.369660 + 0.929167i \(0.620526\pi\)
\(338\) 4.26754e41 0.799931
\(339\) 2.20051e41 0.392851
\(340\) −1.77770e40 −0.0302313
\(341\) −4.01286e41 −0.650140
\(342\) −5.80754e40 −0.0896523
\(343\) 1.51340e42 2.22638
\(344\) 2.99461e41 0.419881
\(345\) 2.89881e41 0.387440
\(346\) −3.53895e41 −0.450940
\(347\) −2.69449e41 −0.327372 −0.163686 0.986513i \(-0.552338\pi\)
−0.163686 + 0.986513i \(0.552338\pi\)
\(348\) 3.42503e40 0.0396832
\(349\) 2.33704e41 0.258253 0.129127 0.991628i \(-0.458783\pi\)
0.129127 + 0.991628i \(0.458783\pi\)
\(350\) −6.91192e41 −0.728578
\(351\) 2.84503e41 0.286101
\(352\) −1.71501e42 −1.64556
\(353\) 5.73724e41 0.525316 0.262658 0.964889i \(-0.415401\pi\)
0.262658 + 0.964889i \(0.415401\pi\)
\(354\) 2.44916e41 0.214025
\(355\) −9.45071e41 −0.788309
\(356\) 5.89883e41 0.469722
\(357\) −1.17870e41 −0.0896147
\(358\) −4.53796e41 −0.329451
\(359\) −1.52912e42 −1.06019 −0.530095 0.847938i \(-0.677844\pi\)
−0.530095 + 0.847938i \(0.677844\pi\)
\(360\) −3.23733e41 −0.214386
\(361\) −1.31911e42 −0.834472
\(362\) 6.13543e41 0.370812
\(363\) −1.97560e42 −1.14088
\(364\) 2.72879e42 1.50590
\(365\) 3.64874e41 0.192447
\(366\) 2.28754e41 0.115327
\(367\) 2.43178e42 1.17202 0.586010 0.810304i \(-0.300698\pi\)
0.586010 + 0.810304i \(0.300698\pi\)
\(368\) 2.80884e41 0.129431
\(369\) −7.36818e41 −0.324657
\(370\) −6.01272e41 −0.253362
\(371\) 4.95115e42 1.99543
\(372\) 3.17748e41 0.122497
\(373\) −3.22348e42 −1.18886 −0.594428 0.804149i \(-0.702621\pi\)
−0.594428 + 0.804149i \(0.702621\pi\)
\(374\) −2.78800e41 −0.0983810
\(375\) 1.71625e42 0.579514
\(376\) 2.25568e42 0.728918
\(377\) −5.86816e41 −0.181497
\(378\) −7.73172e41 −0.228908
\(379\) −5.43594e41 −0.154073 −0.0770366 0.997028i \(-0.524546\pi\)
−0.0770366 + 0.997028i \(0.524546\pi\)
\(380\) 5.25387e41 0.142577
\(381\) −4.19948e42 −1.09127
\(382\) 4.86646e42 1.21107
\(383\) 5.18860e42 1.23672 0.618359 0.785896i \(-0.287798\pi\)
0.618359 + 0.785896i \(0.287798\pi\)
\(384\) 1.15729e42 0.264227
\(385\) −8.83354e42 −1.93211
\(386\) 1.63300e42 0.342213
\(387\) −6.74580e41 −0.135457
\(388\) −2.74321e42 −0.527880
\(389\) −9.09777e42 −1.67790 −0.838951 0.544207i \(-0.816831\pi\)
−0.838951 + 0.544207i \(0.816831\pi\)
\(390\) 1.99788e42 0.353185
\(391\) −5.48794e41 −0.0930022
\(392\) −1.42286e43 −2.31176
\(393\) 3.75878e42 0.585558
\(394\) 5.09683e41 0.0761400
\(395\) 5.31913e42 0.762059
\(396\) 2.35597e42 0.323742
\(397\) 9.82606e42 1.29520 0.647599 0.761981i \(-0.275773\pi\)
0.647599 + 0.761981i \(0.275773\pi\)
\(398\) −4.47291e41 −0.0565612
\(399\) 3.48357e42 0.422642
\(400\) 6.31699e41 0.0735394
\(401\) −1.98710e42 −0.221992 −0.110996 0.993821i \(-0.535404\pi\)
−0.110996 + 0.993821i \(0.535404\pi\)
\(402\) −1.10491e42 −0.118467
\(403\) −5.44403e42 −0.560257
\(404\) 2.10282e42 0.207735
\(405\) 7.29256e41 0.0691628
\(406\) 1.59475e42 0.145215
\(407\) 1.21481e43 1.06219
\(408\) 6.12883e41 0.0514618
\(409\) 3.59927e42 0.290255 0.145127 0.989413i \(-0.453641\pi\)
0.145127 + 0.989413i \(0.453641\pi\)
\(410\) −5.17418e42 −0.400782
\(411\) −4.89441e41 −0.0364175
\(412\) 6.76735e42 0.483743
\(413\) −1.46910e43 −1.00896
\(414\) −3.59982e42 −0.237561
\(415\) 1.52698e43 0.968366
\(416\) −2.32667e43 −1.41806
\(417\) −3.43535e41 −0.0201246
\(418\) 8.23972e42 0.463985
\(419\) 3.46936e43 1.87810 0.939049 0.343783i \(-0.111708\pi\)
0.939049 + 0.343783i \(0.111708\pi\)
\(420\) 6.99461e42 0.364041
\(421\) 2.99185e42 0.149722 0.0748609 0.997194i \(-0.476149\pi\)
0.0748609 + 0.997194i \(0.476149\pi\)
\(422\) −1.37745e43 −0.662856
\(423\) −5.08126e42 −0.235156
\(424\) −2.57442e43 −1.14589
\(425\) −1.23422e42 −0.0528416
\(426\) 1.17362e43 0.483357
\(427\) −1.37215e43 −0.543678
\(428\) −1.63028e43 −0.621499
\(429\) −4.03652e43 −1.48068
\(430\) −4.73712e42 −0.167219
\(431\) 4.10600e43 1.39490 0.697452 0.716631i \(-0.254317\pi\)
0.697452 + 0.716631i \(0.254317\pi\)
\(432\) 7.06622e41 0.0231050
\(433\) −5.85447e43 −1.84263 −0.921313 0.388822i \(-0.872882\pi\)
−0.921313 + 0.388822i \(0.872882\pi\)
\(434\) 1.47948e43 0.448260
\(435\) −1.50416e42 −0.0438755
\(436\) 2.03502e43 0.571535
\(437\) 1.62192e43 0.438617
\(438\) −4.53111e42 −0.118000
\(439\) −1.83589e42 −0.0460450 −0.0230225 0.999735i \(-0.507329\pi\)
−0.0230225 + 0.999735i \(0.507329\pi\)
\(440\) 4.59312e43 1.10953
\(441\) 3.20520e43 0.745793
\(442\) −3.78233e42 −0.0847797
\(443\) −5.41463e43 −1.16925 −0.584625 0.811303i \(-0.698758\pi\)
−0.584625 + 0.811303i \(0.698758\pi\)
\(444\) −9.61918e42 −0.200133
\(445\) −2.59058e43 −0.519346
\(446\) 5.91274e43 1.14226
\(447\) 7.48404e42 0.139336
\(448\) 7.52685e43 1.35060
\(449\) 6.67635e43 1.15471 0.577357 0.816492i \(-0.304084\pi\)
0.577357 + 0.816492i \(0.304084\pi\)
\(450\) −8.09589e42 −0.134977
\(451\) 1.04539e44 1.68022
\(452\) 2.47213e43 0.383079
\(453\) −3.77000e43 −0.563276
\(454\) −8.90664e43 −1.28319
\(455\) −1.19840e44 −1.66500
\(456\) −1.81133e43 −0.242705
\(457\) −6.52277e43 −0.842975 −0.421487 0.906834i \(-0.638492\pi\)
−0.421487 + 0.906834i \(0.638492\pi\)
\(458\) −5.42698e43 −0.676515
\(459\) −1.38061e42 −0.0166020
\(460\) 3.25663e43 0.377802
\(461\) −8.70813e43 −0.974675 −0.487337 0.873214i \(-0.662032\pi\)
−0.487337 + 0.873214i \(0.662032\pi\)
\(462\) 1.09697e44 1.18469
\(463\) 7.08765e43 0.738611 0.369306 0.929308i \(-0.379596\pi\)
0.369306 + 0.929308i \(0.379596\pi\)
\(464\) −1.45748e42 −0.0146574
\(465\) −1.39545e43 −0.135438
\(466\) −1.20314e44 −1.12706
\(467\) −1.42327e44 −1.28694 −0.643469 0.765473i \(-0.722505\pi\)
−0.643469 + 0.765473i \(0.722505\pi\)
\(468\) 3.19621e43 0.278984
\(469\) 6.62763e43 0.558478
\(470\) −3.56823e43 −0.290294
\(471\) −2.31660e43 −0.181973
\(472\) 7.63876e43 0.579402
\(473\) 9.57091e43 0.701043
\(474\) −6.60545e43 −0.467261
\(475\) 3.64765e43 0.249212
\(476\) −1.32420e43 −0.0873854
\(477\) 5.79925e43 0.369674
\(478\) 6.94571e43 0.427717
\(479\) 3.66112e43 0.217810 0.108905 0.994052i \(-0.465266\pi\)
0.108905 + 0.994052i \(0.465266\pi\)
\(480\) −5.96385e43 −0.342805
\(481\) 1.64807e44 0.915339
\(482\) −5.37833e43 −0.288650
\(483\) 2.15930e44 1.11992
\(484\) −2.21946e44 −1.11250
\(485\) 1.20473e44 0.583648
\(486\) −9.05612e42 −0.0424076
\(487\) −2.96641e44 −1.34277 −0.671387 0.741107i \(-0.734301\pi\)
−0.671387 + 0.741107i \(0.734301\pi\)
\(488\) 7.13468e43 0.312210
\(489\) 1.82037e44 0.770129
\(490\) 2.25080e44 0.920664
\(491\) 9.62405e43 0.380639 0.190319 0.981722i \(-0.439048\pi\)
0.190319 + 0.981722i \(0.439048\pi\)
\(492\) −8.27769e43 −0.316581
\(493\) 2.84764e42 0.0105320
\(494\) 1.11784e44 0.399838
\(495\) −1.03467e44 −0.357944
\(496\) −1.35214e43 −0.0452453
\(497\) −7.03978e44 −2.27866
\(498\) −1.89625e44 −0.593760
\(499\) 2.79044e43 0.0845305 0.0422652 0.999106i \(-0.486543\pi\)
0.0422652 + 0.999106i \(0.486543\pi\)
\(500\) 1.92810e44 0.565098
\(501\) −2.18170e43 −0.0618689
\(502\) −2.69521e44 −0.739573
\(503\) 5.95546e44 1.58141 0.790704 0.612199i \(-0.209715\pi\)
0.790704 + 0.612199i \(0.209715\pi\)
\(504\) −2.41147e44 −0.619695
\(505\) −9.23494e43 −0.229682
\(506\) 5.10742e44 1.22947
\(507\) −2.99831e44 −0.698626
\(508\) −4.71785e44 −1.06413
\(509\) −9.54525e43 −0.208422 −0.104211 0.994555i \(-0.533232\pi\)
−0.104211 + 0.994555i \(0.533232\pi\)
\(510\) −9.69510e42 −0.0204948
\(511\) 2.71792e44 0.556279
\(512\) −1.20383e44 −0.238568
\(513\) 4.08029e43 0.0782986
\(514\) −4.00344e43 −0.0743947
\(515\) −2.97201e44 −0.534848
\(516\) −7.57848e43 −0.132088
\(517\) 7.20927e44 1.21702
\(518\) −4.47884e44 −0.732359
\(519\) 2.48641e44 0.393832
\(520\) 6.23123e44 0.956134
\(521\) 4.01987e44 0.597571 0.298786 0.954320i \(-0.403418\pi\)
0.298786 + 0.954320i \(0.403418\pi\)
\(522\) 1.86791e43 0.0269026
\(523\) −1.14424e45 −1.59677 −0.798383 0.602150i \(-0.794311\pi\)
−0.798383 + 0.602150i \(0.794311\pi\)
\(524\) 4.22276e44 0.570992
\(525\) 4.85621e44 0.636310
\(526\) 1.20754e44 0.153334
\(527\) 2.64183e43 0.0325109
\(528\) −1.00255e44 −0.119577
\(529\) 1.40347e44 0.162250
\(530\) 4.07243e44 0.456354
\(531\) −1.72074e44 −0.186920
\(532\) 3.91358e44 0.412128
\(533\) 1.41823e45 1.44793
\(534\) 3.21706e44 0.318440
\(535\) 7.15968e44 0.687157
\(536\) −3.44612e44 −0.320710
\(537\) 3.18830e44 0.287729
\(538\) −8.63876e44 −0.756039
\(539\) −4.54753e45 −3.85976
\(540\) 8.19274e43 0.0674422
\(541\) −4.23426e44 −0.338082 −0.169041 0.985609i \(-0.554067\pi\)
−0.169041 + 0.985609i \(0.554067\pi\)
\(542\) 1.81228e44 0.140358
\(543\) −4.31066e44 −0.323852
\(544\) 1.12906e44 0.0822878
\(545\) −8.93719e44 −0.631915
\(546\) 1.48821e45 1.02090
\(547\) −2.88922e45 −1.92305 −0.961523 0.274724i \(-0.911414\pi\)
−0.961523 + 0.274724i \(0.911414\pi\)
\(548\) −5.49857e43 −0.0355116
\(549\) −1.60719e44 −0.100722
\(550\) 1.14864e45 0.698555
\(551\) −8.41600e43 −0.0496712
\(552\) −1.12276e45 −0.643119
\(553\) 3.96219e45 2.20278
\(554\) 1.12951e45 0.609506
\(555\) 4.22444e44 0.221276
\(556\) −3.85940e43 −0.0196239
\(557\) −3.21957e44 −0.158923 −0.0794615 0.996838i \(-0.525320\pi\)
−0.0794615 + 0.996838i \(0.525320\pi\)
\(558\) 1.73291e44 0.0830446
\(559\) 1.29843e45 0.604123
\(560\) −2.97647e44 −0.134462
\(561\) 1.95880e44 0.0859218
\(562\) −1.93338e45 −0.823510
\(563\) 1.66489e45 0.688647 0.344324 0.938851i \(-0.388108\pi\)
0.344324 + 0.938851i \(0.388108\pi\)
\(564\) −5.70847e44 −0.229306
\(565\) −1.08568e45 −0.423549
\(566\) 1.78718e45 0.677168
\(567\) 5.43219e44 0.199919
\(568\) 3.66043e45 1.30853
\(569\) 1.27473e45 0.442656 0.221328 0.975199i \(-0.428961\pi\)
0.221328 + 0.975199i \(0.428961\pi\)
\(570\) 2.86531e44 0.0966579
\(571\) 2.72708e45 0.893722 0.446861 0.894603i \(-0.352542\pi\)
0.446861 + 0.894603i \(0.352542\pi\)
\(572\) −4.53478e45 −1.44385
\(573\) −3.41910e45 −1.05770
\(574\) −3.85422e45 −1.15848
\(575\) 2.26101e45 0.660363
\(576\) 8.81615e44 0.250212
\(577\) −1.55920e45 −0.430032 −0.215016 0.976611i \(-0.568980\pi\)
−0.215016 + 0.976611i \(0.568980\pi\)
\(578\) −2.44800e45 −0.656149
\(579\) −1.14732e45 −0.298874
\(580\) −1.68983e44 −0.0427841
\(581\) 1.13744e46 2.79912
\(582\) −1.49607e45 −0.357868
\(583\) −8.22796e45 −1.91320
\(584\) −1.41322e45 −0.319446
\(585\) −1.40368e45 −0.308457
\(586\) −4.80038e45 −1.02557
\(587\) 4.86237e45 1.00999 0.504997 0.863121i \(-0.331493\pi\)
0.504997 + 0.863121i \(0.331493\pi\)
\(588\) 3.60084e45 0.727240
\(589\) −7.80772e44 −0.153328
\(590\) −1.20836e45 −0.230749
\(591\) −3.58095e44 −0.0664975
\(592\) 4.09332e44 0.0739210
\(593\) −6.11886e45 −1.07465 −0.537326 0.843375i \(-0.680566\pi\)
−0.537326 + 0.843375i \(0.680566\pi\)
\(594\) 1.28488e45 0.219475
\(595\) 5.81547e44 0.0966173
\(596\) 8.40785e44 0.135869
\(597\) 3.14259e44 0.0493983
\(598\) 6.92896e45 1.05949
\(599\) −7.96637e45 −1.18500 −0.592499 0.805571i \(-0.701859\pi\)
−0.592499 + 0.805571i \(0.701859\pi\)
\(600\) −2.52505e45 −0.365405
\(601\) 1.10227e46 1.55188 0.775942 0.630805i \(-0.217275\pi\)
0.775942 + 0.630805i \(0.217275\pi\)
\(602\) −3.52866e45 −0.483356
\(603\) 7.76290e44 0.103464
\(604\) −4.23536e45 −0.549263
\(605\) 9.74718e45 1.23003
\(606\) 1.14682e45 0.140831
\(607\) 3.64627e45 0.435748 0.217874 0.975977i \(-0.430088\pi\)
0.217874 + 0.975977i \(0.430088\pi\)
\(608\) −3.33686e45 −0.388086
\(609\) −1.12044e45 −0.126825
\(610\) −1.12862e45 −0.124339
\(611\) 9.78043e45 1.04876
\(612\) −1.55103e44 −0.0161890
\(613\) −1.42792e46 −1.45079 −0.725396 0.688332i \(-0.758343\pi\)
−0.725396 + 0.688332i \(0.758343\pi\)
\(614\) 7.04038e45 0.696335
\(615\) 3.63530e45 0.350026
\(616\) 3.42138e46 3.20716
\(617\) −1.82679e46 −1.66718 −0.833592 0.552381i \(-0.813720\pi\)
−0.833592 + 0.552381i \(0.813720\pi\)
\(618\) 3.69073e45 0.327946
\(619\) −7.53894e45 −0.652250 −0.326125 0.945327i \(-0.605743\pi\)
−0.326125 + 0.945327i \(0.605743\pi\)
\(620\) −1.56770e45 −0.132069
\(621\) 2.52918e45 0.207476
\(622\) −8.12098e45 −0.648735
\(623\) −1.92971e46 −1.50120
\(624\) −1.36011e45 −0.103045
\(625\) −1.66160e44 −0.0122605
\(626\) −8.01495e45 −0.576002
\(627\) −5.78910e45 −0.405225
\(628\) −2.60255e45 −0.177446
\(629\) −7.99759e44 −0.0531158
\(630\) 3.81466e45 0.246795
\(631\) 1.94441e46 1.22547 0.612735 0.790288i \(-0.290069\pi\)
0.612735 + 0.790288i \(0.290069\pi\)
\(632\) −2.06019e46 −1.26496
\(633\) 9.67772e45 0.578911
\(634\) 8.83156e45 0.514713
\(635\) 2.07193e46 1.17655
\(636\) 6.51509e45 0.360477
\(637\) −6.16939e46 −3.32615
\(638\) −2.65019e45 −0.139231
\(639\) −8.24565e45 −0.422144
\(640\) −5.70982e45 −0.284874
\(641\) −6.84441e45 −0.332796 −0.166398 0.986059i \(-0.553214\pi\)
−0.166398 + 0.986059i \(0.553214\pi\)
\(642\) −8.89110e45 −0.421335
\(643\) 1.53937e46 0.710986 0.355493 0.934679i \(-0.384313\pi\)
0.355493 + 0.934679i \(0.384313\pi\)
\(644\) 2.42584e46 1.09206
\(645\) 3.32823e45 0.146042
\(646\) −5.42454e44 −0.0232020
\(647\) 3.80872e46 1.58803 0.794014 0.607900i \(-0.207988\pi\)
0.794014 + 0.607900i \(0.207988\pi\)
\(648\) −2.82454e45 −0.114805
\(649\) 2.44138e46 0.967383
\(650\) 1.55830e46 0.601979
\(651\) −1.03946e46 −0.391491
\(652\) 2.04507e46 0.750971
\(653\) 1.23512e46 0.442223 0.221111 0.975249i \(-0.429032\pi\)
0.221111 + 0.975249i \(0.429032\pi\)
\(654\) 1.10985e46 0.387463
\(655\) −1.85450e46 −0.631315
\(656\) 3.52247e45 0.116932
\(657\) 3.18349e45 0.103056
\(658\) −2.65795e46 −0.839112
\(659\) −1.48240e46 −0.456412 −0.228206 0.973613i \(-0.573286\pi\)
−0.228206 + 0.973613i \(0.573286\pi\)
\(660\) −1.16238e46 −0.349039
\(661\) 4.84462e46 1.41885 0.709424 0.704782i \(-0.248955\pi\)
0.709424 + 0.704782i \(0.248955\pi\)
\(662\) 1.93900e46 0.553886
\(663\) 2.65740e45 0.0740431
\(664\) −5.91426e46 −1.60741
\(665\) −1.71872e46 −0.455667
\(666\) −5.24603e45 −0.135677
\(667\) −5.21668e45 −0.131619
\(668\) −2.45100e45 −0.0603298
\(669\) −4.15420e46 −0.997600
\(670\) 5.45137e45 0.127724
\(671\) 2.28028e46 0.521274
\(672\) −4.44244e46 −0.990897
\(673\) −2.12559e45 −0.0462626 −0.0231313 0.999732i \(-0.507364\pi\)
−0.0231313 + 0.999732i \(0.507364\pi\)
\(674\) −2.30127e46 −0.488742
\(675\) 5.68804e45 0.117883
\(676\) −3.36841e46 −0.681247
\(677\) −5.47124e46 −1.07987 −0.539937 0.841705i \(-0.681552\pi\)
−0.539937 + 0.841705i \(0.681552\pi\)
\(678\) 1.34823e46 0.259702
\(679\) 8.97396e46 1.68707
\(680\) −3.02383e45 −0.0554831
\(681\) 6.25766e46 1.12069
\(682\) −2.45865e46 −0.429787
\(683\) −6.15868e45 −0.105086 −0.0525431 0.998619i \(-0.516733\pi\)
−0.0525431 + 0.998619i \(0.516733\pi\)
\(684\) 4.58395e45 0.0763508
\(685\) 2.41480e45 0.0392632
\(686\) 9.27245e46 1.47179
\(687\) 3.81291e46 0.590840
\(688\) 3.22493e45 0.0487878
\(689\) −1.11624e47 −1.64870
\(690\) 1.77607e46 0.256124
\(691\) −2.57107e46 −0.362015 −0.181007 0.983482i \(-0.557936\pi\)
−0.181007 + 0.983482i \(0.557936\pi\)
\(692\) 2.79333e46 0.384035
\(693\) −7.70717e46 −1.03466
\(694\) −1.65089e46 −0.216415
\(695\) 1.69493e45 0.0216971
\(696\) 5.82589e45 0.0728299
\(697\) −6.88225e45 −0.0840213
\(698\) 1.43188e46 0.170723
\(699\) 8.45304e46 0.984328
\(700\) 5.45564e46 0.620481
\(701\) 1.25438e47 1.39342 0.696709 0.717354i \(-0.254647\pi\)
0.696709 + 0.717354i \(0.254647\pi\)
\(702\) 1.74313e46 0.189133
\(703\) 2.36363e46 0.250505
\(704\) −1.25083e47 −1.29494
\(705\) 2.50698e46 0.253531
\(706\) 3.51516e46 0.347270
\(707\) −6.87905e46 −0.663909
\(708\) −1.93315e46 −0.182270
\(709\) −4.63194e46 −0.426678 −0.213339 0.976978i \(-0.568434\pi\)
−0.213339 + 0.976978i \(0.568434\pi\)
\(710\) −5.79037e46 −0.521127
\(711\) 4.64089e46 0.408087
\(712\) 1.00338e47 0.862073
\(713\) −4.83964e46 −0.406290
\(714\) −7.22182e45 −0.0592415
\(715\) 1.99153e47 1.59638
\(716\) 3.58185e46 0.280571
\(717\) −4.87995e46 −0.373550
\(718\) −9.36880e46 −0.700859
\(719\) −2.01670e47 −1.47440 −0.737200 0.675675i \(-0.763852\pi\)
−0.737200 + 0.675675i \(0.763852\pi\)
\(720\) −3.48632e45 −0.0249104
\(721\) −2.21383e47 −1.54601
\(722\) −8.08206e46 −0.551643
\(723\) 3.77873e46 0.252095
\(724\) −4.84275e46 −0.315796
\(725\) −1.17322e46 −0.0747826
\(726\) −1.21043e47 −0.754200
\(727\) −1.86775e47 −1.13763 −0.568815 0.822465i \(-0.692598\pi\)
−0.568815 + 0.822465i \(0.692598\pi\)
\(728\) 4.64161e47 2.76376
\(729\) 6.36269e45 0.0370370
\(730\) 2.23555e46 0.127221
\(731\) −6.30091e45 −0.0350564
\(732\) −1.80558e46 −0.0982162
\(733\) 2.33399e47 1.24132 0.620660 0.784080i \(-0.286865\pi\)
0.620660 + 0.784080i \(0.286865\pi\)
\(734\) 1.48993e47 0.774786
\(735\) −1.58138e47 −0.804070
\(736\) −2.06836e47 −1.02835
\(737\) −1.10140e47 −0.535464
\(738\) −4.51442e46 −0.214621
\(739\) 1.40613e47 0.653721 0.326861 0.945073i \(-0.394009\pi\)
0.326861 + 0.945073i \(0.394009\pi\)
\(740\) 4.74589e46 0.215772
\(741\) −7.85376e46 −0.349202
\(742\) 3.03353e47 1.31912
\(743\) 1.19862e47 0.509761 0.254881 0.966973i \(-0.417964\pi\)
0.254881 + 0.966973i \(0.417964\pi\)
\(744\) 5.40482e46 0.224816
\(745\) −3.69246e46 −0.150223
\(746\) −1.97500e47 −0.785915
\(747\) 1.33227e47 0.518565
\(748\) 2.20059e46 0.0837844
\(749\) 5.33320e47 1.98627
\(750\) 1.05153e47 0.383099
\(751\) −2.14100e47 −0.763057 −0.381528 0.924357i \(-0.624602\pi\)
−0.381528 + 0.924357i \(0.624602\pi\)
\(752\) 2.42917e46 0.0846962
\(753\) 1.89361e47 0.645913
\(754\) −3.59537e46 −0.119982
\(755\) 1.86004e47 0.607290
\(756\) 6.10272e46 0.194946
\(757\) −4.34005e47 −1.35648 −0.678239 0.734842i \(-0.737256\pi\)
−0.678239 + 0.734842i \(0.737256\pi\)
\(758\) −3.33055e46 −0.101853
\(759\) −3.58839e47 −1.07377
\(760\) 8.93671e46 0.261670
\(761\) −7.81557e46 −0.223931 −0.111965 0.993712i \(-0.535715\pi\)
−0.111965 + 0.993712i \(0.535715\pi\)
\(762\) −2.57298e47 −0.721406
\(763\) −6.65726e47 −1.82659
\(764\) −3.84114e47 −1.03138
\(765\) 6.81162e45 0.0178993
\(766\) 3.17901e47 0.817556
\(767\) 3.31209e47 0.833641
\(768\) 2.46832e47 0.608052
\(769\) 5.65995e47 1.36467 0.682336 0.731039i \(-0.260964\pi\)
0.682336 + 0.731039i \(0.260964\pi\)
\(770\) −5.41223e47 −1.27726
\(771\) 2.81276e46 0.0649733
\(772\) −1.28894e47 −0.291439
\(773\) −3.30903e47 −0.732383 −0.366192 0.930539i \(-0.619339\pi\)
−0.366192 + 0.930539i \(0.619339\pi\)
\(774\) −4.13309e46 −0.0895466
\(775\) −1.08842e47 −0.230844
\(776\) −4.66613e47 −0.968810
\(777\) 3.14676e47 0.639612
\(778\) −5.57413e47 −1.10921
\(779\) 2.03400e47 0.396262
\(780\) −1.57694e47 −0.300784
\(781\) 1.16989e48 2.18476
\(782\) −3.36242e46 −0.0614809
\(783\) −1.31237e46 −0.0234956
\(784\) −1.53229e47 −0.268613
\(785\) 1.14296e47 0.196192
\(786\) 2.30297e47 0.387095
\(787\) 8.02705e47 1.32121 0.660606 0.750732i \(-0.270299\pi\)
0.660606 + 0.750732i \(0.270299\pi\)
\(788\) −4.02298e46 −0.0648433
\(789\) −8.48400e46 −0.133915
\(790\) 3.25899e47 0.503774
\(791\) −8.08719e47 −1.22429
\(792\) 4.00745e47 0.594158
\(793\) 3.09353e47 0.449207
\(794\) 6.02034e47 0.856215
\(795\) −2.86122e47 −0.398560
\(796\) 3.53050e46 0.0481694
\(797\) 4.00063e47 0.534646 0.267323 0.963607i \(-0.413861\pi\)
0.267323 + 0.963607i \(0.413861\pi\)
\(798\) 2.13436e47 0.279395
\(799\) −4.74615e46 −0.0608583
\(800\) −4.65168e47 −0.584285
\(801\) −2.26025e47 −0.278113
\(802\) −1.21748e47 −0.146752
\(803\) −4.51672e47 −0.533355
\(804\) 8.72113e46 0.100890
\(805\) −1.06535e48 −1.20743
\(806\) −3.33551e47 −0.370369
\(807\) 6.06946e47 0.660293
\(808\) 3.57686e47 0.381254
\(809\) 1.23605e48 1.29088 0.645438 0.763813i \(-0.276675\pi\)
0.645438 + 0.763813i \(0.276675\pi\)
\(810\) 4.46809e46 0.0457214
\(811\) −1.79503e48 −1.79982 −0.899908 0.436080i \(-0.856367\pi\)
−0.899908 + 0.436080i \(0.856367\pi\)
\(812\) −1.25875e47 −0.123670
\(813\) −1.27328e47 −0.122583
\(814\) 7.44305e47 0.702180
\(815\) −8.98130e47 −0.830307
\(816\) 6.60021e45 0.00597957
\(817\) 1.86219e47 0.165333
\(818\) 2.20524e47 0.191879
\(819\) −1.04559e48 −0.891615
\(820\) 4.08403e47 0.341319
\(821\) 1.17790e48 0.964816 0.482408 0.875947i \(-0.339762\pi\)
0.482408 + 0.875947i \(0.339762\pi\)
\(822\) −2.99876e46 −0.0240745
\(823\) −8.86254e47 −0.697366 −0.348683 0.937241i \(-0.613371\pi\)
−0.348683 + 0.937241i \(0.613371\pi\)
\(824\) 1.15111e48 0.887806
\(825\) −8.07018e47 −0.610089
\(826\) −9.00102e47 −0.666993
\(827\) 1.01884e46 0.00740060 0.00370030 0.999993i \(-0.498822\pi\)
0.00370030 + 0.999993i \(0.498822\pi\)
\(828\) 2.84137e47 0.202315
\(829\) −3.40790e47 −0.237869 −0.118934 0.992902i \(-0.537948\pi\)
−0.118934 + 0.992902i \(0.537948\pi\)
\(830\) 9.35567e47 0.640157
\(831\) −7.93574e47 −0.532317
\(832\) −1.69694e48 −1.11591
\(833\) 2.99382e47 0.193011
\(834\) −2.10481e46 −0.0133037
\(835\) 1.07640e47 0.0667034
\(836\) −6.50369e47 −0.395145
\(837\) −1.21751e47 −0.0725277
\(838\) 2.12565e48 1.24155
\(839\) −7.64580e47 −0.437876 −0.218938 0.975739i \(-0.570259\pi\)
−0.218938 + 0.975739i \(0.570259\pi\)
\(840\) 1.18977e48 0.668118
\(841\) −1.78901e48 −0.985095
\(842\) 1.83308e47 0.0989765
\(843\) 1.35836e48 0.719219
\(844\) 1.08723e48 0.564510
\(845\) 1.47930e48 0.753218
\(846\) −3.11324e47 −0.155454
\(847\) 7.26062e48 3.55547
\(848\) −2.77242e47 −0.133146
\(849\) −1.25564e48 −0.591410
\(850\) −7.56197e46 −0.0349319
\(851\) 1.46510e48 0.663789
\(852\) −9.26347e47 −0.411642
\(853\) −1.99182e48 −0.868142 −0.434071 0.900879i \(-0.642923\pi\)
−0.434071 + 0.900879i \(0.642923\pi\)
\(854\) −8.40704e47 −0.359409
\(855\) −2.01312e47 −0.0844170
\(856\) −2.77307e48 −1.14063
\(857\) 8.90097e47 0.359132 0.179566 0.983746i \(-0.442531\pi\)
0.179566 + 0.983746i \(0.442531\pi\)
\(858\) −2.47314e48 −0.978834
\(859\) −1.46621e48 −0.569259 −0.284629 0.958638i \(-0.591871\pi\)
−0.284629 + 0.958638i \(0.591871\pi\)
\(860\) 3.73905e47 0.142409
\(861\) 2.70791e48 1.01177
\(862\) 2.51571e48 0.922128
\(863\) −7.60640e47 −0.273528 −0.136764 0.990604i \(-0.543670\pi\)
−0.136764 + 0.990604i \(0.543670\pi\)
\(864\) −5.20340e47 −0.183574
\(865\) −1.22674e48 −0.424607
\(866\) −3.58698e48 −1.21810
\(867\) 1.71993e48 0.573054
\(868\) −1.16777e48 −0.381752
\(869\) −6.58447e48 −2.11200
\(870\) −9.21588e46 −0.0290048
\(871\) −1.49421e48 −0.461436
\(872\) 3.46153e48 1.04893
\(873\) 1.05111e48 0.312547
\(874\) 9.93737e47 0.289956
\(875\) −6.30746e48 −1.80602
\(876\) 3.57645e47 0.100493
\(877\) 5.90370e47 0.162791 0.0813956 0.996682i \(-0.474062\pi\)
0.0813956 + 0.996682i \(0.474062\pi\)
\(878\) −1.12483e47 −0.0304389
\(879\) 3.37267e48 0.895689
\(880\) 4.94638e47 0.128921
\(881\) 2.55961e48 0.654742 0.327371 0.944896i \(-0.393837\pi\)
0.327371 + 0.944896i \(0.393837\pi\)
\(882\) 1.96380e48 0.493021
\(883\) −3.14939e48 −0.776022 −0.388011 0.921655i \(-0.626838\pi\)
−0.388011 + 0.921655i \(0.626838\pi\)
\(884\) 2.98543e47 0.0722011
\(885\) 8.48977e47 0.201526
\(886\) −3.31750e48 −0.772955
\(887\) 4.83041e48 1.10470 0.552350 0.833612i \(-0.313731\pi\)
0.552350 + 0.833612i \(0.313731\pi\)
\(888\) −1.63620e48 −0.367301
\(889\) 1.54337e49 3.40087
\(890\) −1.58723e48 −0.343324
\(891\) −9.02736e47 −0.191681
\(892\) −4.66698e48 −0.972783
\(893\) 1.40269e48 0.287020
\(894\) 4.58541e47 0.0921105
\(895\) −1.57304e48 −0.310212
\(896\) −4.25321e48 −0.823446
\(897\) −4.86818e48 −0.925318
\(898\) 4.09054e48 0.763346
\(899\) 2.51124e47 0.0460102
\(900\) 6.39016e47 0.114950
\(901\) 5.41679e47 0.0956716
\(902\) 6.40504e48 1.11074
\(903\) 2.47918e48 0.422143
\(904\) 4.20504e48 0.703058
\(905\) 2.12678e48 0.349158
\(906\) −2.30985e48 −0.372364
\(907\) 1.73953e48 0.275368 0.137684 0.990476i \(-0.456034\pi\)
0.137684 + 0.990476i \(0.456034\pi\)
\(908\) 7.03009e48 1.09281
\(909\) −8.05739e47 −0.122996
\(910\) −7.34249e48 −1.10068
\(911\) −5.97302e48 −0.879308 −0.439654 0.898167i \(-0.644899\pi\)
−0.439654 + 0.898167i \(0.644899\pi\)
\(912\) −1.95064e47 −0.0282009
\(913\) −1.89022e49 −2.68377
\(914\) −3.99644e48 −0.557265
\(915\) 7.92953e47 0.108592
\(916\) 4.28356e48 0.576142
\(917\) −1.38141e49 −1.82485
\(918\) −8.45887e46 −0.0109751
\(919\) 1.16129e49 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(920\) 5.53945e48 0.693373
\(921\) −4.94646e48 −0.608150
\(922\) −5.33540e48 −0.644327
\(923\) 1.58713e49 1.88271
\(924\) −8.65852e48 −1.00892
\(925\) 3.29497e48 0.377149
\(926\) 4.34254e48 0.488273
\(927\) −2.59304e48 −0.286414
\(928\) 1.07325e48 0.116456
\(929\) −6.75124e48 −0.719654 −0.359827 0.933019i \(-0.617164\pi\)
−0.359827 + 0.933019i \(0.617164\pi\)
\(930\) −8.54979e47 −0.0895338
\(931\) −8.84801e48 −0.910281
\(932\) 9.49646e48 0.959841
\(933\) 5.70567e48 0.566578
\(934\) −8.72022e48 −0.850754
\(935\) −9.66431e47 −0.0926359
\(936\) 5.43669e48 0.512015
\(937\) −5.66941e48 −0.524607 −0.262304 0.964985i \(-0.584482\pi\)
−0.262304 + 0.964985i \(0.584482\pi\)
\(938\) 4.06069e48 0.369193
\(939\) 5.63117e48 0.503056
\(940\) 2.81644e48 0.247224
\(941\) −3.24161e48 −0.279596 −0.139798 0.990180i \(-0.544645\pi\)
−0.139798 + 0.990180i \(0.544645\pi\)
\(942\) −1.41936e48 −0.120296
\(943\) 1.26078e49 1.05002
\(944\) 8.22627e47 0.0673232
\(945\) −2.68012e48 −0.215541
\(946\) 5.86401e48 0.463438
\(947\) −7.43561e48 −0.577486 −0.288743 0.957407i \(-0.593237\pi\)
−0.288743 + 0.957407i \(0.593237\pi\)
\(948\) 5.21374e48 0.397935
\(949\) −6.12759e48 −0.459618
\(950\) 2.23488e48 0.164746
\(951\) −6.20492e48 −0.449529
\(952\) −2.25243e48 −0.160377
\(953\) 1.08448e49 0.758910 0.379455 0.925210i \(-0.376111\pi\)
0.379455 + 0.925210i \(0.376111\pi\)
\(954\) 3.55315e48 0.244380
\(955\) 1.68691e49 1.14034
\(956\) −5.48232e48 −0.364258
\(957\) 1.86198e48 0.121599
\(958\) 2.24313e48 0.143987
\(959\) 1.79877e48 0.113493
\(960\) −4.34969e48 −0.269764
\(961\) −1.40737e49 −0.857973
\(962\) 1.00976e49 0.605103
\(963\) 6.24674e48 0.367976
\(964\) 4.24517e48 0.245824
\(965\) 5.66063e48 0.322229
\(966\) 1.32299e49 0.740343
\(967\) −8.75090e48 −0.481411 −0.240705 0.970598i \(-0.577379\pi\)
−0.240705 + 0.970598i \(0.577379\pi\)
\(968\) −3.77526e49 −2.04175
\(969\) 3.81119e47 0.0202637
\(970\) 7.38128e48 0.385832
\(971\) −9.33158e48 −0.479554 −0.239777 0.970828i \(-0.577074\pi\)
−0.239777 + 0.970828i \(0.577074\pi\)
\(972\) 7.14808e47 0.0361157
\(973\) 1.26254e48 0.0627168
\(974\) −1.81749e49 −0.887666
\(975\) −1.09484e49 −0.525743
\(976\) 7.68342e47 0.0362771
\(977\) −2.29643e49 −1.06609 −0.533044 0.846087i \(-0.678952\pi\)
−0.533044 + 0.846087i \(0.678952\pi\)
\(978\) 1.11532e49 0.509108
\(979\) 3.20684e49 1.43934
\(980\) −1.77658e49 −0.784067
\(981\) −7.79760e48 −0.338394
\(982\) 5.89657e48 0.251629
\(983\) −1.62693e49 −0.682709 −0.341355 0.939935i \(-0.610886\pi\)
−0.341355 + 0.939935i \(0.610886\pi\)
\(984\) −1.40801e49 −0.581016
\(985\) 1.76676e48 0.0716937
\(986\) 1.74473e47 0.00696239
\(987\) 1.86744e49 0.732846
\(988\) −8.82320e48 −0.340515
\(989\) 1.15428e49 0.438100
\(990\) −6.33931e48 −0.236625
\(991\) −2.62407e47 −0.00963297 −0.00481649 0.999988i \(-0.501533\pi\)
−0.00481649 + 0.999988i \(0.501533\pi\)
\(992\) 9.95682e48 0.359483
\(993\) −1.36231e49 −0.483741
\(994\) −4.31321e49 −1.50635
\(995\) −1.55049e48 −0.0532583
\(996\) 1.49672e49 0.505665
\(997\) −8.00569e47 −0.0266029 −0.0133014 0.999912i \(-0.504234\pi\)
−0.0133014 + 0.999912i \(0.504234\pi\)
\(998\) 1.70968e48 0.0558805
\(999\) 3.68578e48 0.118495
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 3.34.a.b.1.2 3
3.2 odd 2 9.34.a.c.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.34.a.b.1.2 3 1.1 even 1 trivial
9.34.a.c.1.2 3 3.2 odd 2