Properties

Label 10.12.a.d.1.1
Level $10$
Weight $12$
Character 10.1
Self dual yes
Analytic conductor $7.683$
Analytic rank $0$
Dimension $2$
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] = [10,12,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(7.68343180560\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1969}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 492 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(22.6867\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} -141.734 q^{3} +1024.00 q^{4} +3125.00 q^{5} -4535.49 q^{6} +85586.9 q^{7} +32768.0 q^{8} -157058. q^{9} +100000. q^{10} +767235. q^{11} -145136. q^{12} +220960. q^{13} +2.73878e6 q^{14} -442919. q^{15} +1.04858e6 q^{16} -930719. q^{17} -5.02587e6 q^{18} -1.77341e7 q^{19} +3.20000e6 q^{20} -1.21306e7 q^{21} +2.45515e7 q^{22} -3.99596e7 q^{23} -4.64434e6 q^{24} +9.76562e6 q^{25} +7.07072e6 q^{26} +4.73683e7 q^{27} +8.76410e7 q^{28} +7.68554e7 q^{29} -1.41734e7 q^{30} -2.96314e7 q^{31} +3.35544e7 q^{32} -1.08743e8 q^{33} -2.97830e7 q^{34} +2.67459e8 q^{35} -1.60828e8 q^{36} +5.40911e7 q^{37} -5.67491e8 q^{38} -3.13176e7 q^{39} +1.02400e8 q^{40} +1.26006e8 q^{41} -3.88179e8 q^{42} -2.88676e8 q^{43} +7.85648e8 q^{44} -4.90808e8 q^{45} -1.27871e9 q^{46} -1.57008e9 q^{47} -1.48619e8 q^{48} +5.34780e9 q^{49} +3.12500e8 q^{50} +1.31915e8 q^{51} +2.26263e8 q^{52} -4.09006e9 q^{53} +1.51579e9 q^{54} +2.39761e9 q^{55} +2.80451e9 q^{56} +2.51353e9 q^{57} +2.45937e9 q^{58} +3.77882e9 q^{59} -4.53549e8 q^{60} -9.64103e9 q^{61} -9.48205e8 q^{62} -1.34422e10 q^{63} +1.07374e9 q^{64} +6.90500e8 q^{65} -3.47979e9 q^{66} -1.63819e10 q^{67} -9.53056e8 q^{68} +5.66364e9 q^{69} +8.55869e9 q^{70} +1.03471e10 q^{71} -5.14649e9 q^{72} +4.27149e9 q^{73} +1.73091e9 q^{74} -1.38412e9 q^{75} -1.81597e10 q^{76} +6.56653e10 q^{77} -1.00216e9 q^{78} -1.96636e10 q^{79} +3.27680e9 q^{80} +2.11087e10 q^{81} +4.03220e9 q^{82} -1.35791e10 q^{83} -1.24217e10 q^{84} -2.90850e9 q^{85} -9.23762e9 q^{86} -1.08930e10 q^{87} +2.51407e10 q^{88} +2.25058e10 q^{89} -1.57058e10 q^{90} +1.89113e10 q^{91} -4.09186e10 q^{92} +4.19978e9 q^{93} -5.02424e10 q^{94} -5.54191e10 q^{95} -4.75581e9 q^{96} -1.08976e11 q^{97} +1.71130e11 q^{98} -1.20501e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{2} + 604 q^{3} + 2048 q^{4} + 6250 q^{5} + 19328 q^{6} + 14092 q^{7} + 65536 q^{8} + 221914 q^{9} + 200000 q^{10} + 421584 q^{11} + 618496 q^{12} + 1730524 q^{13} + 450944 q^{14} + 1887500 q^{15}+ \cdots - 251492724912 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 0.707107
\(3\) −141.734 −0.336750 −0.168375 0.985723i \(-0.553852\pi\)
−0.168375 + 0.985723i \(0.553852\pi\)
\(4\) 1024.00 0.500000
\(5\) 3125.00 0.447214
\(6\) −4535.49 −0.238118
\(7\) 85586.9 1.92472 0.962362 0.271772i \(-0.0876097\pi\)
0.962362 + 0.271772i \(0.0876097\pi\)
\(8\) 32768.0 0.353553
\(9\) −157058. −0.886599
\(10\) 100000. 0.316228
\(11\) 767235. 1.43638 0.718188 0.695849i \(-0.244972\pi\)
0.718188 + 0.695849i \(0.244972\pi\)
\(12\) −145136. −0.168375
\(13\) 220960. 0.165054 0.0825268 0.996589i \(-0.473701\pi\)
0.0825268 + 0.996589i \(0.473701\pi\)
\(14\) 2.73878e6 1.36098
\(15\) −442919. −0.150599
\(16\) 1.04858e6 0.250000
\(17\) −930719. −0.158983 −0.0794913 0.996836i \(-0.525330\pi\)
−0.0794913 + 0.996836i \(0.525330\pi\)
\(18\) −5.02587e6 −0.626920
\(19\) −1.77341e7 −1.64310 −0.821551 0.570135i \(-0.806891\pi\)
−0.821551 + 0.570135i \(0.806891\pi\)
\(20\) 3.20000e6 0.223607
\(21\) −1.21306e7 −0.648151
\(22\) 2.45515e7 1.01567
\(23\) −3.99596e7 −1.29455 −0.647274 0.762257i \(-0.724091\pi\)
−0.647274 + 0.762257i \(0.724091\pi\)
\(24\) −4.64434e6 −0.119059
\(25\) 9.76562e6 0.200000
\(26\) 7.07072e6 0.116711
\(27\) 4.73683e7 0.635312
\(28\) 8.76410e7 0.962362
\(29\) 7.68554e7 0.695801 0.347901 0.937531i \(-0.386895\pi\)
0.347901 + 0.937531i \(0.386895\pi\)
\(30\) −1.41734e7 −0.106490
\(31\) −2.96314e7 −0.185893 −0.0929465 0.995671i \(-0.529629\pi\)
−0.0929465 + 0.995671i \(0.529629\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −1.08743e8 −0.483700
\(34\) −2.97830e7 −0.112418
\(35\) 2.67459e8 0.860762
\(36\) −1.60828e8 −0.443300
\(37\) 5.40911e7 0.128238 0.0641189 0.997942i \(-0.479576\pi\)
0.0641189 + 0.997942i \(0.479576\pi\)
\(38\) −5.67491e8 −1.16185
\(39\) −3.13176e7 −0.0555818
\(40\) 1.02400e8 0.158114
\(41\) 1.26006e8 0.169856 0.0849280 0.996387i \(-0.472934\pi\)
0.0849280 + 0.996387i \(0.472934\pi\)
\(42\) −3.88179e8 −0.458312
\(43\) −2.88676e8 −0.299457 −0.149728 0.988727i \(-0.547840\pi\)
−0.149728 + 0.988727i \(0.547840\pi\)
\(44\) 7.85648e8 0.718188
\(45\) −4.90808e8 −0.396499
\(46\) −1.27871e9 −0.915384
\(47\) −1.57008e9 −0.998579 −0.499290 0.866435i \(-0.666406\pi\)
−0.499290 + 0.866435i \(0.666406\pi\)
\(48\) −1.48619e8 −0.0841875
\(49\) 5.34780e9 2.70456
\(50\) 3.12500e8 0.141421
\(51\) 1.31915e8 0.0535374
\(52\) 2.26263e8 0.0825268
\(53\) −4.09006e9 −1.34342 −0.671711 0.740813i \(-0.734440\pi\)
−0.671711 + 0.740813i \(0.734440\pi\)
\(54\) 1.51579e9 0.449234
\(55\) 2.39761e9 0.642367
\(56\) 2.80451e9 0.680492
\(57\) 2.51353e9 0.553315
\(58\) 2.45937e9 0.492006
\(59\) 3.77882e9 0.688129 0.344065 0.938946i \(-0.388196\pi\)
0.344065 + 0.938946i \(0.388196\pi\)
\(60\) −4.53549e8 −0.0752996
\(61\) −9.64103e9 −1.46154 −0.730768 0.682626i \(-0.760838\pi\)
−0.730768 + 0.682626i \(0.760838\pi\)
\(62\) −9.48205e8 −0.131446
\(63\) −1.34422e10 −1.70646
\(64\) 1.07374e9 0.125000
\(65\) 6.90500e8 0.0738143
\(66\) −3.47979e9 −0.342028
\(67\) −1.63819e10 −1.48236 −0.741178 0.671308i \(-0.765733\pi\)
−0.741178 + 0.671308i \(0.765733\pi\)
\(68\) −9.53056e8 −0.0794913
\(69\) 5.66364e9 0.435939
\(70\) 8.55869e9 0.608651
\(71\) 1.03471e10 0.680609 0.340304 0.940315i \(-0.389470\pi\)
0.340304 + 0.940315i \(0.389470\pi\)
\(72\) −5.14649e9 −0.313460
\(73\) 4.27149e9 0.241159 0.120580 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(74\) 1.73091e9 0.0906778
\(75\) −1.38412e9 −0.0673500
\(76\) −1.81597e10 −0.821551
\(77\) 6.56653e10 2.76463
\(78\) −1.00216e9 −0.0393023
\(79\) −1.96636e10 −0.718977 −0.359489 0.933149i \(-0.617049\pi\)
−0.359489 + 0.933149i \(0.617049\pi\)
\(80\) 3.27680e9 0.111803
\(81\) 2.11087e10 0.672658
\(82\) 4.03220e9 0.120106
\(83\) −1.35791e10 −0.378391 −0.189196 0.981939i \(-0.560588\pi\)
−0.189196 + 0.981939i \(0.560588\pi\)
\(84\) −1.24217e10 −0.324075
\(85\) −2.90850e9 −0.0710991
\(86\) −9.23762e9 −0.211748
\(87\) −1.08930e10 −0.234311
\(88\) 2.51407e10 0.507836
\(89\) 2.25058e10 0.427218 0.213609 0.976919i \(-0.431478\pi\)
0.213609 + 0.976919i \(0.431478\pi\)
\(90\) −1.57058e10 −0.280367
\(91\) 1.89113e10 0.317683
\(92\) −4.09186e10 −0.647274
\(93\) 4.19978e9 0.0625994
\(94\) −5.02424e10 −0.706102
\(95\) −5.54191e10 −0.734818
\(96\) −4.75581e9 −0.0595296
\(97\) −1.08976e11 −1.28851 −0.644255 0.764811i \(-0.722832\pi\)
−0.644255 + 0.764811i \(0.722832\pi\)
\(98\) 1.71130e11 1.91241
\(99\) −1.20501e11 −1.27349
\(100\) 1.00000e10 0.100000
\(101\) 1.63516e11 1.54807 0.774037 0.633140i \(-0.218234\pi\)
0.774037 + 0.633140i \(0.218234\pi\)
\(102\) 4.22127e9 0.0378566
\(103\) −7.69876e10 −0.654359 −0.327180 0.944962i \(-0.606098\pi\)
−0.327180 + 0.944962i \(0.606098\pi\)
\(104\) 7.24042e9 0.0583553
\(105\) −3.79081e10 −0.289862
\(106\) −1.30882e11 −0.949943
\(107\) 1.19891e11 0.826376 0.413188 0.910646i \(-0.364415\pi\)
0.413188 + 0.910646i \(0.364415\pi\)
\(108\) 4.85052e10 0.317656
\(109\) −7.33426e10 −0.456573 −0.228287 0.973594i \(-0.573312\pi\)
−0.228287 + 0.973594i \(0.573312\pi\)
\(110\) 7.67235e10 0.454222
\(111\) −7.66655e9 −0.0431841
\(112\) 8.97444e10 0.481181
\(113\) 2.18835e11 1.11734 0.558669 0.829391i \(-0.311312\pi\)
0.558669 + 0.829391i \(0.311312\pi\)
\(114\) 8.04329e10 0.391253
\(115\) −1.24874e11 −0.578939
\(116\) 7.86999e10 0.347901
\(117\) −3.47036e10 −0.146337
\(118\) 1.20922e11 0.486581
\(119\) −7.96574e10 −0.305997
\(120\) −1.45136e10 −0.0532449
\(121\) 3.03337e11 1.06318
\(122\) −3.08513e11 −1.03346
\(123\) −1.78594e10 −0.0571990
\(124\) −3.03426e10 −0.0929465
\(125\) 3.05176e10 0.0894427
\(126\) −4.30149e11 −1.20665
\(127\) 3.93053e11 1.05568 0.527838 0.849345i \(-0.323003\pi\)
0.527838 + 0.849345i \(0.323003\pi\)
\(128\) 3.43597e10 0.0883883
\(129\) 4.09152e10 0.100842
\(130\) 2.20960e10 0.0521946
\(131\) −2.63175e11 −0.596008 −0.298004 0.954565i \(-0.596321\pi\)
−0.298004 + 0.954565i \(0.596321\pi\)
\(132\) −1.11353e11 −0.241850
\(133\) −1.51781e12 −3.16252
\(134\) −5.24220e11 −1.04818
\(135\) 1.48026e11 0.284120
\(136\) −3.04978e10 −0.0562088
\(137\) 5.20521e11 0.921457 0.460728 0.887541i \(-0.347588\pi\)
0.460728 + 0.887541i \(0.347588\pi\)
\(138\) 1.81237e11 0.308255
\(139\) 6.82675e11 1.11592 0.557960 0.829868i \(-0.311584\pi\)
0.557960 + 0.829868i \(0.311584\pi\)
\(140\) 2.73878e11 0.430381
\(141\) 2.22533e11 0.336272
\(142\) 3.31107e11 0.481263
\(143\) 1.69528e11 0.237079
\(144\) −1.64688e11 −0.221650
\(145\) 2.40173e11 0.311172
\(146\) 1.36688e11 0.170525
\(147\) −7.57966e11 −0.910761
\(148\) 5.53893e10 0.0641189
\(149\) 1.01566e12 1.13299 0.566493 0.824066i \(-0.308300\pi\)
0.566493 + 0.824066i \(0.308300\pi\)
\(150\) −4.42919e10 −0.0476236
\(151\) 5.34040e11 0.553605 0.276803 0.960927i \(-0.410725\pi\)
0.276803 + 0.960927i \(0.410725\pi\)
\(152\) −5.81111e11 −0.580924
\(153\) 1.46177e11 0.140954
\(154\) 2.10129e12 1.95489
\(155\) −9.25981e10 −0.0831338
\(156\) −3.20692e10 −0.0277909
\(157\) −7.29675e11 −0.610494 −0.305247 0.952273i \(-0.598739\pi\)
−0.305247 + 0.952273i \(0.598739\pi\)
\(158\) −6.29237e11 −0.508394
\(159\) 5.79701e11 0.452397
\(160\) 1.04858e11 0.0790569
\(161\) −3.42002e12 −2.49165
\(162\) 6.75479e11 0.475641
\(163\) −2.12171e11 −0.144429 −0.0722143 0.997389i \(-0.523007\pi\)
−0.0722143 + 0.997389i \(0.523007\pi\)
\(164\) 1.29030e11 0.0849280
\(165\) −3.39823e11 −0.216317
\(166\) −4.34531e11 −0.267563
\(167\) −2.36613e11 −0.140961 −0.0704804 0.997513i \(-0.522453\pi\)
−0.0704804 + 0.997513i \(0.522453\pi\)
\(168\) −3.97495e11 −0.229156
\(169\) −1.74334e12 −0.972757
\(170\) −9.30719e10 −0.0502747
\(171\) 2.78529e12 1.45677
\(172\) −2.95604e11 −0.149728
\(173\) −7.36983e11 −0.361579 −0.180790 0.983522i \(-0.557865\pi\)
−0.180790 + 0.983522i \(0.557865\pi\)
\(174\) −3.48577e11 −0.165683
\(175\) 8.35810e11 0.384945
\(176\) 8.04504e11 0.359094
\(177\) −5.35588e11 −0.231727
\(178\) 7.20185e11 0.302089
\(179\) 1.35344e11 0.0550487 0.0275244 0.999621i \(-0.491238\pi\)
0.0275244 + 0.999621i \(0.491238\pi\)
\(180\) −5.02587e11 −0.198250
\(181\) −1.35881e12 −0.519907 −0.259953 0.965621i \(-0.583707\pi\)
−0.259953 + 0.965621i \(0.583707\pi\)
\(182\) 6.05161e11 0.224636
\(183\) 1.36646e12 0.492172
\(184\) −1.30940e12 −0.457692
\(185\) 1.69035e11 0.0573497
\(186\) 1.34393e11 0.0442645
\(187\) −7.14080e11 −0.228359
\(188\) −1.60776e12 −0.499290
\(189\) 4.05411e12 1.22280
\(190\) −1.77341e12 −0.519594
\(191\) 1.10648e12 0.314964 0.157482 0.987522i \(-0.449662\pi\)
0.157482 + 0.987522i \(0.449662\pi\)
\(192\) −1.52186e11 −0.0420938
\(193\) 6.17025e12 1.65858 0.829292 0.558816i \(-0.188744\pi\)
0.829292 + 0.558816i \(0.188744\pi\)
\(194\) −3.48724e12 −0.911114
\(195\) −9.78674e10 −0.0248570
\(196\) 5.47615e12 1.35228
\(197\) 2.00333e12 0.481047 0.240523 0.970643i \(-0.422681\pi\)
0.240523 + 0.970643i \(0.422681\pi\)
\(198\) −3.85602e12 −0.900494
\(199\) −7.73897e12 −1.75789 −0.878944 0.476925i \(-0.841751\pi\)
−0.878944 + 0.476925i \(0.841751\pi\)
\(200\) 3.20000e11 0.0707107
\(201\) 2.32187e12 0.499183
\(202\) 5.23250e12 1.09465
\(203\) 6.57782e12 1.33922
\(204\) 1.35081e11 0.0267687
\(205\) 3.93770e11 0.0759619
\(206\) −2.46360e12 −0.462702
\(207\) 6.27599e12 1.14775
\(208\) 2.31693e11 0.0412634
\(209\) −1.36062e13 −2.36011
\(210\) −1.21306e12 −0.204963
\(211\) −9.01879e12 −1.48455 −0.742275 0.670096i \(-0.766253\pi\)
−0.742275 + 0.670096i \(0.766253\pi\)
\(212\) −4.18822e12 −0.671711
\(213\) −1.46654e12 −0.229195
\(214\) 3.83653e12 0.584336
\(215\) −9.02112e11 −0.133921
\(216\) 1.55217e12 0.224617
\(217\) −2.53606e12 −0.357792
\(218\) −2.34696e12 −0.322846
\(219\) −6.05416e11 −0.0812103
\(220\) 2.45515e12 0.321184
\(221\) −2.05652e11 −0.0262407
\(222\) −2.45330e11 −0.0305358
\(223\) 2.06620e12 0.250897 0.125448 0.992100i \(-0.459963\pi\)
0.125448 + 0.992100i \(0.459963\pi\)
\(224\) 2.87182e12 0.340246
\(225\) −1.53377e12 −0.177320
\(226\) 7.00271e12 0.790077
\(227\) −1.56806e13 −1.72672 −0.863359 0.504591i \(-0.831643\pi\)
−0.863359 + 0.504591i \(0.831643\pi\)
\(228\) 2.57385e12 0.276657
\(229\) 6.02309e12 0.632010 0.316005 0.948758i \(-0.397658\pi\)
0.316005 + 0.948758i \(0.397658\pi\)
\(230\) −3.99596e12 −0.409372
\(231\) −9.30701e12 −0.930989
\(232\) 2.51840e12 0.246003
\(233\) 1.43129e13 1.36544 0.682718 0.730682i \(-0.260798\pi\)
0.682718 + 0.730682i \(0.260798\pi\)
\(234\) −1.11052e12 −0.103476
\(235\) −4.90649e12 −0.446578
\(236\) 3.86951e12 0.344065
\(237\) 2.78701e12 0.242116
\(238\) −2.54904e12 −0.216373
\(239\) 2.22886e13 1.84881 0.924407 0.381407i \(-0.124560\pi\)
0.924407 + 0.381407i \(0.124560\pi\)
\(240\) −4.64434e11 −0.0376498
\(241\) 2.06638e13 1.63726 0.818629 0.574323i \(-0.194735\pi\)
0.818629 + 0.574323i \(0.194735\pi\)
\(242\) 9.70679e12 0.751781
\(243\) −1.13830e13 −0.861830
\(244\) −9.87241e12 −0.730768
\(245\) 1.67119e13 1.20952
\(246\) −5.71501e11 −0.0404458
\(247\) −3.91853e12 −0.271200
\(248\) −9.70962e11 −0.0657231
\(249\) 1.92462e12 0.127423
\(250\) 9.76562e11 0.0632456
\(251\) −1.23384e13 −0.781721 −0.390861 0.920450i \(-0.627823\pi\)
−0.390861 + 0.920450i \(0.627823\pi\)
\(252\) −1.37648e13 −0.853229
\(253\) −3.06584e13 −1.85946
\(254\) 1.25777e13 0.746475
\(255\) 4.12233e11 0.0239426
\(256\) 1.09951e12 0.0625000
\(257\) 1.19160e13 0.662975 0.331487 0.943460i \(-0.392450\pi\)
0.331487 + 0.943460i \(0.392450\pi\)
\(258\) 1.30929e12 0.0713061
\(259\) 4.62949e12 0.246822
\(260\) 7.07072e11 0.0369071
\(261\) −1.20708e13 −0.616897
\(262\) −8.42159e12 −0.421441
\(263\) 1.54645e12 0.0757845 0.0378923 0.999282i \(-0.487936\pi\)
0.0378923 + 0.999282i \(0.487936\pi\)
\(264\) −3.56330e12 −0.171014
\(265\) −1.27814e13 −0.600797
\(266\) −4.85699e13 −2.23624
\(267\) −3.18984e12 −0.143866
\(268\) −1.67751e13 −0.741178
\(269\) 1.43195e13 0.619855 0.309927 0.950760i \(-0.399695\pi\)
0.309927 + 0.950760i \(0.399695\pi\)
\(270\) 4.73683e12 0.200903
\(271\) 2.26776e13 0.942468 0.471234 0.882008i \(-0.343809\pi\)
0.471234 + 0.882008i \(0.343809\pi\)
\(272\) −9.75929e11 −0.0397456
\(273\) −2.68038e12 −0.106980
\(274\) 1.66567e13 0.651568
\(275\) 7.49253e12 0.287275
\(276\) 5.79957e12 0.217970
\(277\) 9.25283e12 0.340907 0.170454 0.985366i \(-0.445477\pi\)
0.170454 + 0.985366i \(0.445477\pi\)
\(278\) 2.18456e13 0.789074
\(279\) 4.65386e12 0.164813
\(280\) 8.76410e12 0.304325
\(281\) −2.26511e12 −0.0771265 −0.0385633 0.999256i \(-0.512278\pi\)
−0.0385633 + 0.999256i \(0.512278\pi\)
\(282\) 7.12107e12 0.237780
\(283\) −3.60409e13 −1.18024 −0.590121 0.807315i \(-0.700920\pi\)
−0.590121 + 0.807315i \(0.700920\pi\)
\(284\) 1.05954e13 0.340304
\(285\) 7.85478e12 0.247450
\(286\) 5.42490e12 0.167640
\(287\) 1.07845e13 0.326926
\(288\) −5.27001e12 −0.156730
\(289\) −3.34057e13 −0.974725
\(290\) 7.68554e12 0.220032
\(291\) 1.54457e13 0.433905
\(292\) 4.37401e12 0.120580
\(293\) 1.63348e13 0.441918 0.220959 0.975283i \(-0.429081\pi\)
0.220959 + 0.975283i \(0.429081\pi\)
\(294\) −2.42549e13 −0.644005
\(295\) 1.18088e13 0.307741
\(296\) 1.77246e12 0.0453389
\(297\) 3.63426e13 0.912548
\(298\) 3.25012e13 0.801142
\(299\) −8.82948e12 −0.213670
\(300\) −1.41734e12 −0.0336750
\(301\) −2.47069e13 −0.576371
\(302\) 1.70893e13 0.391458
\(303\) −2.31758e13 −0.521314
\(304\) −1.85956e13 −0.410776
\(305\) −3.01282e13 −0.653618
\(306\) 4.67767e12 0.0996694
\(307\) −5.61322e13 −1.17476 −0.587382 0.809310i \(-0.699842\pi\)
−0.587382 + 0.809310i \(0.699842\pi\)
\(308\) 6.72412e13 1.38231
\(309\) 1.09118e13 0.220355
\(310\) −2.96314e12 −0.0587845
\(311\) 6.92340e13 1.34939 0.674694 0.738097i \(-0.264275\pi\)
0.674694 + 0.738097i \(0.264275\pi\)
\(312\) −1.02621e12 −0.0196511
\(313\) −3.23822e13 −0.609273 −0.304637 0.952469i \(-0.598535\pi\)
−0.304637 + 0.952469i \(0.598535\pi\)
\(314\) −2.33496e13 −0.431684
\(315\) −4.20067e13 −0.763151
\(316\) −2.01356e13 −0.359489
\(317\) −1.84023e13 −0.322885 −0.161442 0.986882i \(-0.551615\pi\)
−0.161442 + 0.986882i \(0.551615\pi\)
\(318\) 1.85504e13 0.319893
\(319\) 5.89661e13 0.999433
\(320\) 3.35544e12 0.0559017
\(321\) −1.69927e13 −0.278282
\(322\) −1.09441e14 −1.76186
\(323\) 1.65055e13 0.261224
\(324\) 2.16153e13 0.336329
\(325\) 2.15781e12 0.0330107
\(326\) −6.78946e12 −0.102126
\(327\) 1.03951e13 0.153751
\(328\) 4.12897e12 0.0600532
\(329\) −1.34378e14 −1.92199
\(330\) −1.08743e13 −0.152959
\(331\) 6.74629e13 0.933277 0.466639 0.884448i \(-0.345465\pi\)
0.466639 + 0.884448i \(0.345465\pi\)
\(332\) −1.39050e13 −0.189196
\(333\) −8.49546e12 −0.113696
\(334\) −7.57163e12 −0.0996744
\(335\) −5.11934e13 −0.662930
\(336\) −1.27198e13 −0.162038
\(337\) 1.07897e14 1.35222 0.676109 0.736802i \(-0.263665\pi\)
0.676109 + 0.736802i \(0.263665\pi\)
\(338\) −5.57868e13 −0.687843
\(339\) −3.10163e13 −0.376264
\(340\) −2.97830e12 −0.0355496
\(341\) −2.27342e13 −0.267012
\(342\) 8.91293e13 1.03009
\(343\) 2.88468e14 3.28081
\(344\) −9.45933e12 −0.105874
\(345\) 1.76989e13 0.194958
\(346\) −2.35834e13 −0.255675
\(347\) −1.29756e13 −0.138458 −0.0692288 0.997601i \(-0.522054\pi\)
−0.0692288 + 0.997601i \(0.522054\pi\)
\(348\) −1.11545e13 −0.117156
\(349\) −7.76358e13 −0.802643 −0.401321 0.915937i \(-0.631449\pi\)
−0.401321 + 0.915937i \(0.631449\pi\)
\(350\) 2.67459e13 0.272197
\(351\) 1.04665e13 0.104861
\(352\) 2.57441e13 0.253918
\(353\) −4.21031e13 −0.408839 −0.204420 0.978883i \(-0.565531\pi\)
−0.204420 + 0.978883i \(0.565531\pi\)
\(354\) −1.71388e13 −0.163856
\(355\) 3.23347e13 0.304377
\(356\) 2.30459e13 0.213609
\(357\) 1.12902e13 0.103045
\(358\) 4.33101e12 0.0389253
\(359\) 1.84963e14 1.63707 0.818533 0.574460i \(-0.194788\pi\)
0.818533 + 0.574460i \(0.194788\pi\)
\(360\) −1.60828e13 −0.140184
\(361\) 1.98008e14 1.69978
\(362\) −4.34818e13 −0.367630
\(363\) −4.29933e13 −0.358025
\(364\) 1.93652e13 0.158841
\(365\) 1.33484e13 0.107850
\(366\) 4.37268e13 0.348018
\(367\) −1.68959e14 −1.32470 −0.662351 0.749194i \(-0.730441\pi\)
−0.662351 + 0.749194i \(0.730441\pi\)
\(368\) −4.19007e13 −0.323637
\(369\) −1.97903e13 −0.150594
\(370\) 5.40911e12 0.0405524
\(371\) −3.50056e14 −2.58572
\(372\) 4.30058e12 0.0312997
\(373\) −1.87187e14 −1.34238 −0.671191 0.741284i \(-0.734217\pi\)
−0.671191 + 0.741284i \(0.734217\pi\)
\(374\) −2.28505e13 −0.161474
\(375\) −4.32538e12 −0.0301198
\(376\) −5.14483e13 −0.353051
\(377\) 1.69820e13 0.114845
\(378\) 1.29732e14 0.864651
\(379\) −7.47613e13 −0.491090 −0.245545 0.969385i \(-0.578967\pi\)
−0.245545 + 0.969385i \(0.578967\pi\)
\(380\) −5.67491e13 −0.367409
\(381\) −5.57090e13 −0.355499
\(382\) 3.54074e13 0.222713
\(383\) −1.55521e14 −0.964264 −0.482132 0.876099i \(-0.660137\pi\)
−0.482132 + 0.876099i \(0.660137\pi\)
\(384\) −4.86995e12 −0.0297648
\(385\) 2.05204e14 1.23638
\(386\) 1.97448e14 1.17280
\(387\) 4.53390e13 0.265498
\(388\) −1.11592e14 −0.644255
\(389\) −8.76448e13 −0.498889 −0.249444 0.968389i \(-0.580248\pi\)
−0.249444 + 0.968389i \(0.580248\pi\)
\(390\) −3.13176e12 −0.0175765
\(391\) 3.71912e13 0.205811
\(392\) 1.75237e14 0.956206
\(393\) 3.73009e13 0.200706
\(394\) 6.41064e13 0.340151
\(395\) −6.14489e13 −0.321536
\(396\) −1.23393e14 −0.636745
\(397\) 5.19767e13 0.264521 0.132261 0.991215i \(-0.457776\pi\)
0.132261 + 0.991215i \(0.457776\pi\)
\(398\) −2.47647e14 −1.24301
\(399\) 2.15125e14 1.06498
\(400\) 1.02400e13 0.0500000
\(401\) −3.40753e14 −1.64114 −0.820570 0.571546i \(-0.806344\pi\)
−0.820570 + 0.571546i \(0.806344\pi\)
\(402\) 7.42999e13 0.352976
\(403\) −6.54735e12 −0.0306823
\(404\) 1.67440e14 0.774037
\(405\) 6.59648e13 0.300822
\(406\) 2.10490e14 0.946975
\(407\) 4.15005e13 0.184198
\(408\) 4.32258e12 0.0189283
\(409\) 3.34635e14 1.44575 0.722875 0.690979i \(-0.242820\pi\)
0.722875 + 0.690979i \(0.242820\pi\)
\(410\) 1.26006e13 0.0537132
\(411\) −7.37756e13 −0.310301
\(412\) −7.88353e13 −0.327180
\(413\) 3.23417e14 1.32446
\(414\) 2.00832e14 0.811579
\(415\) −4.24347e13 −0.169222
\(416\) 7.41419e12 0.0291776
\(417\) −9.67584e13 −0.375786
\(418\) −4.35399e14 −1.66885
\(419\) 1.37830e14 0.521396 0.260698 0.965420i \(-0.416047\pi\)
0.260698 + 0.965420i \(0.416047\pi\)
\(420\) −3.88179e13 −0.144931
\(421\) 4.35453e14 1.60468 0.802342 0.596864i \(-0.203587\pi\)
0.802342 + 0.596864i \(0.203587\pi\)
\(422\) −2.88601e14 −1.04974
\(423\) 2.46594e14 0.885340
\(424\) −1.34023e14 −0.474972
\(425\) −9.08905e12 −0.0317965
\(426\) −4.69292e13 −0.162065
\(427\) −8.25146e14 −2.81305
\(428\) 1.22769e14 0.413188
\(429\) −2.40279e13 −0.0798365
\(430\) −2.88676e13 −0.0946965
\(431\) 1.51185e14 0.489647 0.244824 0.969568i \(-0.421270\pi\)
0.244824 + 0.969568i \(0.421270\pi\)
\(432\) 4.96693e13 0.158828
\(433\) −3.75044e13 −0.118413 −0.0592065 0.998246i \(-0.518857\pi\)
−0.0592065 + 0.998246i \(0.518857\pi\)
\(434\) −8.11539e13 −0.252997
\(435\) −3.40407e13 −0.104787
\(436\) −7.51028e13 −0.228287
\(437\) 7.08648e14 2.12707
\(438\) −1.93733e13 −0.0574244
\(439\) −1.61934e14 −0.474004 −0.237002 0.971509i \(-0.576165\pi\)
−0.237002 + 0.971509i \(0.576165\pi\)
\(440\) 7.85648e13 0.227111
\(441\) −8.39917e14 −2.39786
\(442\) −6.58085e12 −0.0185549
\(443\) −1.29342e14 −0.360179 −0.180090 0.983650i \(-0.557639\pi\)
−0.180090 + 0.983650i \(0.557639\pi\)
\(444\) −7.85055e12 −0.0215920
\(445\) 7.03306e13 0.191058
\(446\) 6.61183e13 0.177411
\(447\) −1.43954e14 −0.381533
\(448\) 9.18983e13 0.240590
\(449\) 1.30255e13 0.0336852 0.0168426 0.999858i \(-0.494639\pi\)
0.0168426 + 0.999858i \(0.494639\pi\)
\(450\) −4.90808e13 −0.125384
\(451\) 9.66764e13 0.243977
\(452\) 2.24087e14 0.558669
\(453\) −7.56916e13 −0.186427
\(454\) −5.01780e14 −1.22097
\(455\) 5.90978e13 0.142072
\(456\) 8.23633e13 0.195626
\(457\) 3.45724e14 0.811318 0.405659 0.914025i \(-0.367042\pi\)
0.405659 + 0.914025i \(0.367042\pi\)
\(458\) 1.92739e14 0.446899
\(459\) −4.40866e13 −0.101004
\(460\) −1.27871e14 −0.289470
\(461\) 2.96945e14 0.664234 0.332117 0.943238i \(-0.392237\pi\)
0.332117 + 0.943238i \(0.392237\pi\)
\(462\) −2.97824e14 −0.658308
\(463\) 4.55258e14 0.994402 0.497201 0.867635i \(-0.334361\pi\)
0.497201 + 0.867635i \(0.334361\pi\)
\(464\) 8.05887e13 0.173950
\(465\) 1.31243e13 0.0279953
\(466\) 4.58014e14 0.965509
\(467\) −3.66499e14 −0.763537 −0.381768 0.924258i \(-0.624685\pi\)
−0.381768 + 0.924258i \(0.624685\pi\)
\(468\) −3.55365e13 −0.0731683
\(469\) −1.40208e15 −2.85313
\(470\) −1.57008e14 −0.315778
\(471\) 1.03420e14 0.205584
\(472\) 1.23824e14 0.243290
\(473\) −2.21482e14 −0.430132
\(474\) 8.91843e13 0.171202
\(475\) −1.73185e14 −0.328620
\(476\) −8.15691e13 −0.152999
\(477\) 6.42379e14 1.19108
\(478\) 7.13234e14 1.30731
\(479\) 5.78377e14 1.04801 0.524005 0.851715i \(-0.324437\pi\)
0.524005 + 0.851715i \(0.324437\pi\)
\(480\) −1.48619e13 −0.0266224
\(481\) 1.19520e13 0.0211661
\(482\) 6.61242e14 1.15772
\(483\) 4.84734e14 0.839062
\(484\) 3.10617e14 0.531589
\(485\) −3.40551e14 −0.576239
\(486\) −3.64255e14 −0.609406
\(487\) −4.75203e14 −0.786086 −0.393043 0.919520i \(-0.628578\pi\)
−0.393043 + 0.919520i \(0.628578\pi\)
\(488\) −3.15917e14 −0.516731
\(489\) 3.00718e13 0.0486363
\(490\) 5.34780e14 0.855257
\(491\) 7.59641e14 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(492\) −1.82880e13 −0.0285995
\(493\) −7.15307e13 −0.110620
\(494\) −1.25393e14 −0.191767
\(495\) −3.76565e14 −0.569522
\(496\) −3.10708e13 −0.0464732
\(497\) 8.85576e14 1.30998
\(498\) 6.15879e13 0.0901019
\(499\) −4.91039e14 −0.710499 −0.355249 0.934772i \(-0.615604\pi\)
−0.355249 + 0.934772i \(0.615604\pi\)
\(500\) 3.12500e13 0.0447214
\(501\) 3.35362e13 0.0474686
\(502\) −3.94828e14 −0.552760
\(503\) −4.25544e14 −0.589278 −0.294639 0.955609i \(-0.595199\pi\)
−0.294639 + 0.955609i \(0.595199\pi\)
\(504\) −4.40472e14 −0.603324
\(505\) 5.10987e14 0.692320
\(506\) −9.81069e14 −1.31484
\(507\) 2.47090e14 0.327576
\(508\) 4.02486e14 0.527838
\(509\) −1.41466e15 −1.83529 −0.917644 0.397404i \(-0.869911\pi\)
−0.917644 + 0.397404i \(0.869911\pi\)
\(510\) 1.31915e13 0.0169300
\(511\) 3.65584e14 0.464165
\(512\) 3.51844e13 0.0441942
\(513\) −8.40035e14 −1.04388
\(514\) 3.81311e14 0.468794
\(515\) −2.40586e14 −0.292638
\(516\) 4.18972e13 0.0504210
\(517\) −1.20462e15 −1.43434
\(518\) 1.48144e14 0.174530
\(519\) 1.04456e14 0.121762
\(520\) 2.26263e13 0.0260973
\(521\) 6.27267e14 0.715888 0.357944 0.933743i \(-0.383478\pi\)
0.357944 + 0.933743i \(0.383478\pi\)
\(522\) −3.86265e14 −0.436212
\(523\) 2.23774e14 0.250063 0.125032 0.992153i \(-0.460097\pi\)
0.125032 + 0.992153i \(0.460097\pi\)
\(524\) −2.69491e14 −0.298004
\(525\) −1.18463e14 −0.129630
\(526\) 4.94865e13 0.0535877
\(527\) 2.75785e13 0.0295537
\(528\) −1.14026e14 −0.120925
\(529\) 6.43961e14 0.675855
\(530\) −4.09006e14 −0.424827
\(531\) −5.93495e14 −0.610095
\(532\) −1.55424e15 −1.58126
\(533\) 2.78424e13 0.0280354
\(534\) −1.02075e14 −0.101728
\(535\) 3.74661e14 0.369566
\(536\) −5.36802e14 −0.524092
\(537\) −1.91829e13 −0.0185377
\(538\) 4.58224e14 0.438303
\(539\) 4.10302e15 3.88477
\(540\) 1.51579e14 0.142060
\(541\) 5.47756e14 0.508162 0.254081 0.967183i \(-0.418227\pi\)
0.254081 + 0.967183i \(0.418227\pi\)
\(542\) 7.25684e14 0.666425
\(543\) 1.92589e14 0.175079
\(544\) −3.12297e13 −0.0281044
\(545\) −2.29196e14 −0.204186
\(546\) −8.57720e13 −0.0756460
\(547\) 4.33518e14 0.378509 0.189255 0.981928i \(-0.439393\pi\)
0.189255 + 0.981928i \(0.439393\pi\)
\(548\) 5.33013e14 0.460728
\(549\) 1.51420e15 1.29580
\(550\) 2.39761e14 0.203134
\(551\) −1.36296e15 −1.14327
\(552\) 1.85586e14 0.154128
\(553\) −1.68295e15 −1.38383
\(554\) 2.96091e14 0.241058
\(555\) −2.39580e13 −0.0193125
\(556\) 6.99060e14 0.557960
\(557\) −7.36061e14 −0.581715 −0.290857 0.956766i \(-0.593941\pi\)
−0.290857 + 0.956766i \(0.593941\pi\)
\(558\) 1.48924e14 0.116540
\(559\) −6.37858e13 −0.0494264
\(560\) 2.80451e14 0.215191
\(561\) 1.01209e14 0.0768998
\(562\) −7.24834e13 −0.0545367
\(563\) −9.03199e14 −0.672957 −0.336478 0.941691i \(-0.609236\pi\)
−0.336478 + 0.941691i \(0.609236\pi\)
\(564\) 2.27874e14 0.168136
\(565\) 6.83858e14 0.499689
\(566\) −1.15331e15 −0.834557
\(567\) 1.80663e15 1.29468
\(568\) 3.39054e14 0.240632
\(569\) −2.25786e15 −1.58701 −0.793503 0.608566i \(-0.791745\pi\)
−0.793503 + 0.608566i \(0.791745\pi\)
\(570\) 2.51353e14 0.174973
\(571\) 9.15100e14 0.630913 0.315457 0.948940i \(-0.397842\pi\)
0.315457 + 0.948940i \(0.397842\pi\)
\(572\) 1.73597e14 0.118540
\(573\) −1.56826e14 −0.106064
\(574\) 3.45104e14 0.231172
\(575\) −3.90231e14 −0.258910
\(576\) −1.68640e14 −0.110825
\(577\) 2.84484e14 0.185179 0.0925893 0.995704i \(-0.470486\pi\)
0.0925893 + 0.995704i \(0.470486\pi\)
\(578\) −1.06898e15 −0.689234
\(579\) −8.74535e14 −0.558528
\(580\) 2.45937e14 0.155586
\(581\) −1.16219e15 −0.728299
\(582\) 4.94261e14 0.306817
\(583\) −3.13804e15 −1.92966
\(584\) 1.39968e14 0.0852626
\(585\) −1.08449e14 −0.0654437
\(586\) 5.22714e14 0.312484
\(587\) −1.99879e15 −1.18374 −0.591871 0.806032i \(-0.701611\pi\)
−0.591871 + 0.806032i \(0.701611\pi\)
\(588\) −7.76157e14 −0.455380
\(589\) 5.25486e14 0.305441
\(590\) 3.77882e14 0.217606
\(591\) −2.83940e14 −0.161992
\(592\) 5.67186e13 0.0320595
\(593\) 8.52303e14 0.477302 0.238651 0.971105i \(-0.423295\pi\)
0.238651 + 0.971105i \(0.423295\pi\)
\(594\) 1.16296e15 0.645269
\(595\) −2.48929e14 −0.136846
\(596\) 1.04004e15 0.566493
\(597\) 1.09688e15 0.591969
\(598\) −2.82543e14 −0.151087
\(599\) 1.08984e14 0.0577449 0.0288725 0.999583i \(-0.490808\pi\)
0.0288725 + 0.999583i \(0.490808\pi\)
\(600\) −4.53549e13 −0.0238118
\(601\) 3.21315e14 0.167156 0.0835778 0.996501i \(-0.473365\pi\)
0.0835778 + 0.996501i \(0.473365\pi\)
\(602\) −7.90620e14 −0.407556
\(603\) 2.57291e15 1.31426
\(604\) 5.46856e14 0.276803
\(605\) 9.47929e14 0.475468
\(606\) −7.41624e14 −0.368625
\(607\) −3.08680e15 −1.52044 −0.760222 0.649664i \(-0.774910\pi\)
−0.760222 + 0.649664i \(0.774910\pi\)
\(608\) −5.95058e14 −0.290462
\(609\) −9.32301e14 −0.450984
\(610\) −9.64103e14 −0.462178
\(611\) −3.46924e14 −0.164819
\(612\) 1.49685e14 0.0704769
\(613\) 8.70867e14 0.406368 0.203184 0.979141i \(-0.434871\pi\)
0.203184 + 0.979141i \(0.434871\pi\)
\(614\) −1.79623e15 −0.830684
\(615\) −5.58106e13 −0.0255802
\(616\) 2.15172e15 0.977444
\(617\) 3.06108e15 1.37818 0.689091 0.724675i \(-0.258010\pi\)
0.689091 + 0.724675i \(0.258010\pi\)
\(618\) 3.49177e14 0.155815
\(619\) −1.21317e15 −0.536566 −0.268283 0.963340i \(-0.586456\pi\)
−0.268283 + 0.963340i \(0.586456\pi\)
\(620\) −9.48205e13 −0.0415669
\(621\) −1.89282e15 −0.822442
\(622\) 2.21549e15 0.954162
\(623\) 1.92620e15 0.822276
\(624\) −3.28389e13 −0.0138955
\(625\) 9.53674e13 0.0400000
\(626\) −1.03623e15 −0.430821
\(627\) 1.92847e15 0.794768
\(628\) −7.47187e14 −0.305247
\(629\) −5.03436e13 −0.0203876
\(630\) −1.34422e15 −0.539630
\(631\) 2.95354e15 1.17539 0.587693 0.809084i \(-0.300036\pi\)
0.587693 + 0.809084i \(0.300036\pi\)
\(632\) −6.44338e14 −0.254197
\(633\) 1.27827e15 0.499922
\(634\) −5.88875e14 −0.228314
\(635\) 1.22829e15 0.472112
\(636\) 5.93614e14 0.226199
\(637\) 1.18165e15 0.446398
\(638\) 1.88691e15 0.706706
\(639\) −1.62510e15 −0.603427
\(640\) 1.07374e14 0.0395285
\(641\) −1.24519e15 −0.454483 −0.227241 0.973838i \(-0.572971\pi\)
−0.227241 + 0.973838i \(0.572971\pi\)
\(642\) −5.43767e14 −0.196775
\(643\) 2.82493e14 0.101356 0.0506778 0.998715i \(-0.483862\pi\)
0.0506778 + 0.998715i \(0.483862\pi\)
\(644\) −3.50210e15 −1.24582
\(645\) 1.27860e14 0.0450979
\(646\) 5.28175e14 0.184714
\(647\) −3.45042e15 −1.19646 −0.598230 0.801324i \(-0.704129\pi\)
−0.598230 + 0.801324i \(0.704129\pi\)
\(648\) 6.91691e14 0.237821
\(649\) 2.89924e15 0.988413
\(650\) 6.90500e13 0.0233421
\(651\) 3.59446e14 0.120487
\(652\) −2.17263e14 −0.0722143
\(653\) 3.55303e15 1.17105 0.585527 0.810653i \(-0.300888\pi\)
0.585527 + 0.810653i \(0.300888\pi\)
\(654\) 3.32645e14 0.108718
\(655\) −8.22421e14 −0.266543
\(656\) 1.32127e14 0.0424640
\(657\) −6.70874e14 −0.213812
\(658\) −4.30010e15 −1.35905
\(659\) 1.76602e14 0.0553511 0.0276756 0.999617i \(-0.491189\pi\)
0.0276756 + 0.999617i \(0.491189\pi\)
\(660\) −3.47979e14 −0.108159
\(661\) 3.46137e15 1.06694 0.533470 0.845819i \(-0.320888\pi\)
0.533470 + 0.845819i \(0.320888\pi\)
\(662\) 2.15881e15 0.659927
\(663\) 2.91479e13 0.00883654
\(664\) −4.44960e14 −0.133782
\(665\) −4.74315e15 −1.41432
\(666\) −2.71855e14 −0.0803949
\(667\) −3.07111e15 −0.900748
\(668\) −2.42292e14 −0.0704804
\(669\) −2.92851e14 −0.0844895
\(670\) −1.63819e15 −0.468762
\(671\) −7.39693e15 −2.09932
\(672\) −4.07035e14 −0.114578
\(673\) 6.92295e15 1.93290 0.966448 0.256864i \(-0.0826893\pi\)
0.966448 + 0.256864i \(0.0826893\pi\)
\(674\) 3.45272e15 0.956162
\(675\) 4.62581e14 0.127062
\(676\) −1.78518e15 −0.486379
\(677\) −3.89173e15 −1.05173 −0.525865 0.850568i \(-0.676258\pi\)
−0.525865 + 0.850568i \(0.676258\pi\)
\(678\) −9.92523e14 −0.266058
\(679\) −9.32695e15 −2.48002
\(680\) −9.53056e13 −0.0251373
\(681\) 2.22248e15 0.581472
\(682\) −7.27496e14 −0.188806
\(683\) 4.79007e15 1.23318 0.616592 0.787283i \(-0.288513\pi\)
0.616592 + 0.787283i \(0.288513\pi\)
\(684\) 2.85214e15 0.728387
\(685\) 1.62663e15 0.412088
\(686\) 9.23099e15 2.31988
\(687\) −8.53677e14 −0.212829
\(688\) −3.02698e14 −0.0748641
\(689\) −9.03740e14 −0.221737
\(690\) 5.66364e14 0.137856
\(691\) 1.60990e15 0.388749 0.194374 0.980927i \(-0.437732\pi\)
0.194374 + 0.980927i \(0.437732\pi\)
\(692\) −7.54670e14 −0.180790
\(693\) −1.03133e16 −2.45112
\(694\) −4.15220e14 −0.0979043
\(695\) 2.13336e15 0.499054
\(696\) −3.56943e14 −0.0828415
\(697\) −1.17276e14 −0.0270041
\(698\) −2.48435e15 −0.567554
\(699\) −2.02863e15 −0.459810
\(700\) 8.55869e14 0.192472
\(701\) −2.46356e15 −0.549686 −0.274843 0.961489i \(-0.588626\pi\)
−0.274843 + 0.961489i \(0.588626\pi\)
\(702\) 3.34928e14 0.0741477
\(703\) −9.59257e14 −0.210708
\(704\) 8.23812e14 0.179547
\(705\) 6.95417e14 0.150385
\(706\) −1.34730e15 −0.289093
\(707\) 1.39948e16 2.97962
\(708\) −5.48442e14 −0.115864
\(709\) 6.21323e15 1.30246 0.651228 0.758882i \(-0.274254\pi\)
0.651228 + 0.758882i \(0.274254\pi\)
\(710\) 1.03471e15 0.215227
\(711\) 3.08834e15 0.637445
\(712\) 7.37470e14 0.151044
\(713\) 1.18406e15 0.240647
\(714\) 3.61285e14 0.0728635
\(715\) 5.29776e14 0.106025
\(716\) 1.38592e14 0.0275244
\(717\) −3.15905e15 −0.622588
\(718\) 5.91883e15 1.15758
\(719\) 7.12825e15 1.38348 0.691742 0.722145i \(-0.256844\pi\)
0.691742 + 0.722145i \(0.256844\pi\)
\(720\) −5.14649e14 −0.0991248
\(721\) −6.58914e15 −1.25946
\(722\) 6.33627e15 1.20193
\(723\) −2.92877e15 −0.551346
\(724\) −1.39142e15 −0.259953
\(725\) 7.50541e14 0.139160
\(726\) −1.37578e15 −0.253162
\(727\) 5.59273e15 1.02137 0.510686 0.859767i \(-0.329391\pi\)
0.510686 + 0.859767i \(0.329391\pi\)
\(728\) 6.19685e14 0.112318
\(729\) −2.12599e15 −0.382437
\(730\) 4.27149e14 0.0762612
\(731\) 2.68676e14 0.0476084
\(732\) 1.39926e15 0.246086
\(733\) −2.75558e15 −0.480995 −0.240498 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(734\) −5.40669e15 −0.936705
\(735\) −2.36864e15 −0.407304
\(736\) −1.34082e15 −0.228846
\(737\) −1.25688e16 −2.12922
\(738\) −6.33291e14 −0.106486
\(739\) 1.99004e15 0.332138 0.166069 0.986114i \(-0.446893\pi\)
0.166069 + 0.986114i \(0.446893\pi\)
\(740\) 1.73091e14 0.0286749
\(741\) 5.55389e14 0.0913266
\(742\) −1.12018e16 −1.82838
\(743\) −7.01045e15 −1.13581 −0.567907 0.823093i \(-0.692247\pi\)
−0.567907 + 0.823093i \(0.692247\pi\)
\(744\) 1.37618e14 0.0221322
\(745\) 3.17394e15 0.506687
\(746\) −5.98997e15 −0.949208
\(747\) 2.13271e15 0.335481
\(748\) −7.31218e14 −0.114179
\(749\) 1.02611e16 1.59054
\(750\) −1.38412e14 −0.0212979
\(751\) −9.37589e15 −1.43216 −0.716082 0.698016i \(-0.754066\pi\)
−0.716082 + 0.698016i \(0.754066\pi\)
\(752\) −1.64634e15 −0.249645
\(753\) 1.74877e15 0.263245
\(754\) 5.43423e14 0.0812074
\(755\) 1.66887e15 0.247580
\(756\) 4.15141e15 0.611400
\(757\) 1.88289e15 0.275294 0.137647 0.990481i \(-0.456046\pi\)
0.137647 + 0.990481i \(0.456046\pi\)
\(758\) −2.39236e15 −0.347253
\(759\) 4.34534e15 0.626173
\(760\) −1.81597e15 −0.259797
\(761\) 6.71695e15 0.954018 0.477009 0.878898i \(-0.341721\pi\)
0.477009 + 0.878898i \(0.341721\pi\)
\(762\) −1.78269e15 −0.251375
\(763\) −6.27717e15 −0.878777
\(764\) 1.13304e15 0.157482
\(765\) 4.56804e14 0.0630365
\(766\) −4.97667e15 −0.681837
\(767\) 8.34968e14 0.113578
\(768\) −1.55838e14 −0.0210469
\(769\) −5.72271e15 −0.767374 −0.383687 0.923463i \(-0.625346\pi\)
−0.383687 + 0.923463i \(0.625346\pi\)
\(770\) 6.56653e15 0.874252
\(771\) −1.68890e15 −0.223257
\(772\) 6.31833e15 0.829292
\(773\) −3.24936e15 −0.423458 −0.211729 0.977328i \(-0.567909\pi\)
−0.211729 + 0.977328i \(0.567909\pi\)
\(774\) 1.45085e15 0.187735
\(775\) −2.89369e14 −0.0371786
\(776\) −3.57094e15 −0.455557
\(777\) −6.56157e14 −0.0831174
\(778\) −2.80463e15 −0.352767
\(779\) −2.23461e15 −0.279091
\(780\) −1.00216e14 −0.0124285
\(781\) 7.93865e15 0.977611
\(782\) 1.19012e15 0.145530
\(783\) 3.64051e15 0.442051
\(784\) 5.60757e15 0.676140
\(785\) −2.28023e15 −0.273021
\(786\) 1.19363e15 0.141920
\(787\) 1.29754e16 1.53201 0.766003 0.642838i \(-0.222243\pi\)
0.766003 + 0.642838i \(0.222243\pi\)
\(788\) 2.05140e15 0.240523
\(789\) −2.19185e14 −0.0255204
\(790\) −1.96636e15 −0.227361
\(791\) 1.87294e16 2.15057
\(792\) −3.94857e15 −0.450247
\(793\) −2.13028e15 −0.241232
\(794\) 1.66325e15 0.187045
\(795\) 1.81157e15 0.202318
\(796\) −7.92470e15 −0.878944
\(797\) −1.21467e16 −1.33795 −0.668974 0.743286i \(-0.733266\pi\)
−0.668974 + 0.743286i \(0.733266\pi\)
\(798\) 6.88401e15 0.753053
\(799\) 1.46130e15 0.158757
\(800\) 3.27680e14 0.0353553
\(801\) −3.53473e15 −0.378771
\(802\) −1.09041e16 −1.16046
\(803\) 3.27724e15 0.346395
\(804\) 2.37760e15 0.249592
\(805\) −1.06876e16 −1.11430
\(806\) −2.09515e14 −0.0216957
\(807\) −2.02956e15 −0.208736
\(808\) 5.35808e15 0.547327
\(809\) 3.06008e15 0.310468 0.155234 0.987878i \(-0.450387\pi\)
0.155234 + 0.987878i \(0.450387\pi\)
\(810\) 2.11087e15 0.212713
\(811\) 1.63765e15 0.163910 0.0819552 0.996636i \(-0.473884\pi\)
0.0819552 + 0.996636i \(0.473884\pi\)
\(812\) 6.73568e15 0.669612
\(813\) −3.21419e15 −0.317376
\(814\) 1.32802e15 0.130248
\(815\) −6.63033e14 −0.0645904
\(816\) 1.38323e14 0.0133843
\(817\) 5.11941e15 0.492038
\(818\) 1.07083e16 1.02230
\(819\) −2.97018e15 −0.281657
\(820\) 4.03220e14 0.0379810
\(821\) 1.07044e15 0.100156 0.0500778 0.998745i \(-0.484053\pi\)
0.0500778 + 0.998745i \(0.484053\pi\)
\(822\) −2.36082e15 −0.219416
\(823\) −3.24575e15 −0.299651 −0.149825 0.988712i \(-0.547871\pi\)
−0.149825 + 0.988712i \(0.547871\pi\)
\(824\) −2.52273e15 −0.231351
\(825\) −1.06195e15 −0.0967400
\(826\) 1.03494e16 0.936533
\(827\) −2.05440e16 −1.84674 −0.923369 0.383913i \(-0.874576\pi\)
−0.923369 + 0.383913i \(0.874576\pi\)
\(828\) 6.42662e15 0.573873
\(829\) 5.85872e15 0.519700 0.259850 0.965649i \(-0.416327\pi\)
0.259850 + 0.965649i \(0.416327\pi\)
\(830\) −1.35791e15 −0.119658
\(831\) −1.31144e15 −0.114800
\(832\) 2.37254e14 0.0206317
\(833\) −4.97730e15 −0.429978
\(834\) −3.09627e15 −0.265721
\(835\) −7.39417e14 −0.0630396
\(836\) −1.39328e16 −1.18006
\(837\) −1.40359e15 −0.118100
\(838\) 4.41057e15 0.368683
\(839\) −2.25383e16 −1.87168 −0.935838 0.352430i \(-0.885355\pi\)
−0.935838 + 0.352430i \(0.885355\pi\)
\(840\) −1.24217e15 −0.102482
\(841\) −6.29376e15 −0.515861
\(842\) 1.39345e16 1.13468
\(843\) 3.21043e14 0.0259724
\(844\) −9.23524e15 −0.742275
\(845\) −5.44793e15 −0.435030
\(846\) 7.89100e15 0.626030
\(847\) 2.59617e16 2.04632
\(848\) −4.28874e15 −0.335856
\(849\) 5.10823e15 0.397446
\(850\) −2.90850e14 −0.0224835
\(851\) −2.16146e15 −0.166010
\(852\) −1.50173e15 −0.114597
\(853\) 1.84471e16 1.39865 0.699326 0.714803i \(-0.253484\pi\)
0.699326 + 0.714803i \(0.253484\pi\)
\(854\) −2.64047e16 −1.98913
\(855\) 8.70403e15 0.651489
\(856\) 3.92860e15 0.292168
\(857\) −1.53957e15 −0.113764 −0.0568820 0.998381i \(-0.518116\pi\)
−0.0568820 + 0.998381i \(0.518116\pi\)
\(858\) −7.68894e14 −0.0564529
\(859\) −2.14973e16 −1.56827 −0.784136 0.620589i \(-0.786893\pi\)
−0.784136 + 0.620589i \(0.786893\pi\)
\(860\) −9.23762e14 −0.0669605
\(861\) −1.52853e15 −0.110092
\(862\) 4.83792e15 0.346233
\(863\) −1.16923e16 −0.831462 −0.415731 0.909488i \(-0.636474\pi\)
−0.415731 + 0.909488i \(0.636474\pi\)
\(864\) 1.58942e15 0.112308
\(865\) −2.30307e15 −0.161703
\(866\) −1.20014e15 −0.0837306
\(867\) 4.73472e15 0.328239
\(868\) −2.59693e15 −0.178896
\(869\) −1.50866e16 −1.03272
\(870\) −1.08930e15 −0.0740957
\(871\) −3.61974e15 −0.244668
\(872\) −2.40329e15 −0.161423
\(873\) 1.71156e16 1.14239
\(874\) 2.26767e16 1.50407
\(875\) 2.61191e15 0.172152
\(876\) −6.19946e14 −0.0406052
\(877\) 1.13573e16 0.739225 0.369613 0.929186i \(-0.379490\pi\)
0.369613 + 0.929186i \(0.379490\pi\)
\(878\) −5.18188e15 −0.335172
\(879\) −2.31520e15 −0.148816
\(880\) 2.51407e15 0.160592
\(881\) 1.12847e16 0.716346 0.358173 0.933655i \(-0.383400\pi\)
0.358173 + 0.933655i \(0.383400\pi\)
\(882\) −2.68773e16 −1.69554
\(883\) −5.36286e15 −0.336211 −0.168106 0.985769i \(-0.553765\pi\)
−0.168106 + 0.985769i \(0.553765\pi\)
\(884\) −2.10587e14 −0.0131203
\(885\) −1.67371e15 −0.103632
\(886\) −4.13894e15 −0.254685
\(887\) 8.12685e15 0.496984 0.248492 0.968634i \(-0.420065\pi\)
0.248492 + 0.968634i \(0.420065\pi\)
\(888\) −2.51218e14 −0.0152679
\(889\) 3.36402e16 2.03188
\(890\) 2.25058e15 0.135098
\(891\) 1.61953e16 0.966190
\(892\) 2.11579e15 0.125448
\(893\) 2.78439e16 1.64077
\(894\) −4.60653e15 −0.269785
\(895\) 4.22950e14 0.0246185
\(896\) 2.94075e15 0.170123
\(897\) 1.25144e15 0.0719534
\(898\) 4.16815e14 0.0238190
\(899\) −2.27733e15 −0.129345
\(900\) −1.57058e15 −0.0886599
\(901\) 3.80670e15 0.213581
\(902\) 3.09364e15 0.172518
\(903\) 3.50181e15 0.194093
\(904\) 7.17077e15 0.395039
\(905\) −4.24627e15 −0.232509
\(906\) −2.42213e15 −0.131823
\(907\) −1.18191e16 −0.639361 −0.319680 0.947525i \(-0.603576\pi\)
−0.319680 + 0.947525i \(0.603576\pi\)
\(908\) −1.60570e16 −0.863359
\(909\) −2.56815e16 −1.37252
\(910\) 1.89113e15 0.100460
\(911\) 1.50443e16 0.794366 0.397183 0.917739i \(-0.369988\pi\)
0.397183 + 0.917739i \(0.369988\pi\)
\(912\) 2.63563e15 0.138329
\(913\) −1.04183e16 −0.543513
\(914\) 1.10632e16 0.573688
\(915\) 4.27020e15 0.220106
\(916\) 6.16764e15 0.316005
\(917\) −2.25243e16 −1.14715
\(918\) −1.41077e15 −0.0714203
\(919\) −3.98208e15 −0.200389 −0.100195 0.994968i \(-0.531947\pi\)
−0.100195 + 0.994968i \(0.531947\pi\)
\(920\) −4.09186e15 −0.204686
\(921\) 7.95585e15 0.395602
\(922\) 9.50225e15 0.469685
\(923\) 2.28629e15 0.112337
\(924\) −9.53038e15 −0.465494
\(925\) 5.28233e14 0.0256476
\(926\) 1.45682e16 0.703148
\(927\) 1.20916e16 0.580154
\(928\) 2.57884e15 0.123001
\(929\) 4.06546e16 1.92763 0.963814 0.266575i \(-0.0858919\pi\)
0.963814 + 0.266575i \(0.0858919\pi\)
\(930\) 4.19978e14 0.0197957
\(931\) −9.48384e16 −4.44387
\(932\) 1.46564e16 0.682718
\(933\) −9.81282e15 −0.454407
\(934\) −1.17280e16 −0.539902
\(935\) −2.23150e15 −0.102125
\(936\) −1.13717e15 −0.0517378
\(937\) −1.18458e15 −0.0535792 −0.0267896 0.999641i \(-0.508528\pi\)
−0.0267896 + 0.999641i \(0.508528\pi\)
\(938\) −4.48664e16 −2.01746
\(939\) 4.58966e15 0.205173
\(940\) −5.02424e15 −0.223289
\(941\) −1.02223e16 −0.451652 −0.225826 0.974168i \(-0.572508\pi\)
−0.225826 + 0.974168i \(0.572508\pi\)
\(942\) 3.30944e15 0.145370
\(943\) −5.03516e15 −0.219887
\(944\) 3.96238e15 0.172032
\(945\) 1.26691e16 0.546853
\(946\) −7.08743e15 −0.304150
\(947\) −1.68932e16 −0.720755 −0.360378 0.932807i \(-0.617352\pi\)
−0.360378 + 0.932807i \(0.617352\pi\)
\(948\) 2.85390e15 0.121058
\(949\) 9.43829e14 0.0398042
\(950\) −5.54191e15 −0.232370
\(951\) 2.60824e15 0.108731
\(952\) −2.61021e15 −0.108186
\(953\) −3.76078e16 −1.54977 −0.774884 0.632103i \(-0.782192\pi\)
−0.774884 + 0.632103i \(0.782192\pi\)
\(954\) 2.05561e16 0.842219
\(955\) 3.45775e15 0.140856
\(956\) 2.28235e16 0.924407
\(957\) −8.35751e15 −0.336559
\(958\) 1.85081e16 0.741055
\(959\) 4.45498e16 1.77355
\(960\) −4.75581e14 −0.0188249
\(961\) −2.45305e16 −0.965444
\(962\) 3.82463e14 0.0149667
\(963\) −1.88300e16 −0.732664
\(964\) 2.11598e16 0.818629
\(965\) 1.92820e16 0.741741
\(966\) 1.55115e16 0.593306
\(967\) −3.52262e16 −1.33974 −0.669869 0.742479i \(-0.733650\pi\)
−0.669869 + 0.742479i \(0.733650\pi\)
\(968\) 9.93976e15 0.375890
\(969\) −2.33939e15 −0.0879674
\(970\) −1.08976e16 −0.407462
\(971\) 3.56726e16 1.32626 0.663131 0.748504i \(-0.269227\pi\)
0.663131 + 0.748504i \(0.269227\pi\)
\(972\) −1.16562e16 −0.430915
\(973\) 5.84281e16 2.14784
\(974\) −1.52065e16 −0.555847
\(975\) −3.05836e14 −0.0111164
\(976\) −1.01094e16 −0.365384
\(977\) −4.79343e16 −1.72277 −0.861383 0.507957i \(-0.830401\pi\)
−0.861383 + 0.507957i \(0.830401\pi\)
\(978\) 9.62298e14 0.0343911
\(979\) 1.72672e16 0.613646
\(980\) 1.71130e16 0.604758
\(981\) 1.15191e16 0.404798
\(982\) 2.43085e16 0.849464
\(983\) 2.05131e16 0.712833 0.356416 0.934327i \(-0.383998\pi\)
0.356416 + 0.934327i \(0.383998\pi\)
\(984\) −5.85217e14 −0.0202229
\(985\) 6.26039e15 0.215131
\(986\) −2.28898e15 −0.0782203
\(987\) 1.90460e16 0.647230
\(988\) −4.01257e15 −0.135600
\(989\) 1.15354e16 0.387661
\(990\) −1.20501e16 −0.402713
\(991\) −2.11314e16 −0.702301 −0.351150 0.936319i \(-0.614209\pi\)
−0.351150 + 0.936319i \(0.614209\pi\)
\(992\) −9.94265e14 −0.0328615
\(993\) −9.56179e15 −0.314281
\(994\) 2.83384e16 0.926298
\(995\) −2.41843e16 −0.786151
\(996\) 1.97081e15 0.0637116
\(997\) 4.57566e16 1.47106 0.735531 0.677491i \(-0.236933\pi\)
0.735531 + 0.677491i \(0.236933\pi\)
\(998\) −1.57133e16 −0.502398
\(999\) 2.56220e15 0.0814711
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 10.12.a.d.1.1 2
3.2 odd 2 90.12.a.l.1.2 2
4.3 odd 2 80.12.a.g.1.2 2
5.2 odd 4 50.12.b.f.49.4 4
5.3 odd 4 50.12.b.f.49.1 4
5.4 even 2 50.12.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.12.a.d.1.1 2 1.1 even 1 trivial
50.12.a.f.1.2 2 5.4 even 2
50.12.b.f.49.1 4 5.3 odd 4
50.12.b.f.49.4 4 5.2 odd 4
80.12.a.g.1.2 2 4.3 odd 2
90.12.a.l.1.2 2 3.2 odd 2