Properties

Label 3.28.a.b.1.2
Level $3$
Weight $28$
Character 3.1
Self dual yes
Analytic conductor $13.856$
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] = [3,28,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, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 28 \)
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: \(13.8556672451\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{30001}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 7500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-86.1040\) 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)
 
\(f(q)\) \(=\) \(q+21703.1 q^{2} -1.59432e6 q^{3} +3.36807e8 q^{4} -6.93920e8 q^{5} -3.46018e10 q^{6} +3.81056e11 q^{7} +4.39681e12 q^{8} +2.54187e12 q^{9} +O(q^{10})\) \(q+21703.1 q^{2} -1.59432e6 q^{3} +3.36807e8 q^{4} -6.93920e8 q^{5} -3.46018e10 q^{6} +3.81056e11 q^{7} +4.39681e12 q^{8} +2.54187e12 q^{9} -1.50602e13 q^{10} +1.26337e14 q^{11} -5.36979e14 q^{12} -1.31606e15 q^{13} +8.27009e15 q^{14} +1.10633e15 q^{15} +5.02190e16 q^{16} -3.53536e16 q^{17} +5.51664e16 q^{18} +8.65178e16 q^{19} -2.33717e17 q^{20} -6.07526e17 q^{21} +2.74191e18 q^{22} +5.10955e17 q^{23} -7.00994e18 q^{24} -6.96906e18 q^{25} -2.85626e19 q^{26} -4.05256e18 q^{27} +1.28342e20 q^{28} -2.74961e19 q^{29} +2.40109e19 q^{30} -9.77996e19 q^{31} +4.99779e20 q^{32} -2.01422e20 q^{33} -7.67282e20 q^{34} -2.64422e20 q^{35} +8.56118e20 q^{36} -1.80293e21 q^{37} +1.87770e21 q^{38} +2.09822e21 q^{39} -3.05104e21 q^{40} -5.17966e21 q^{41} -1.31852e22 q^{42} +8.74613e21 q^{43} +4.25513e22 q^{44} -1.76385e21 q^{45} +1.10893e22 q^{46} +2.51817e22 q^{47} -8.00654e22 q^{48} +7.94910e22 q^{49} -1.51250e23 q^{50} +5.63650e22 q^{51} -4.43258e23 q^{52} +2.11957e23 q^{53} -8.79530e22 q^{54} -8.76680e22 q^{55} +1.67543e24 q^{56} -1.37937e23 q^{57} -5.96750e23 q^{58} -7.00844e23 q^{59} +3.72620e23 q^{60} +1.08505e23 q^{61} -2.12256e24 q^{62} +9.68592e23 q^{63} +4.10646e24 q^{64} +9.13240e23 q^{65} -4.37149e24 q^{66} +2.46540e24 q^{67} -1.19073e25 q^{68} -8.14627e23 q^{69} -5.73878e24 q^{70} +1.29459e25 q^{71} +1.11761e25 q^{72} +7.46465e24 q^{73} -3.91291e25 q^{74} +1.11109e25 q^{75} +2.91398e25 q^{76} +4.81415e25 q^{77} +4.55380e25 q^{78} -4.94644e25 q^{79} -3.48480e25 q^{80} +6.46108e24 q^{81} -1.12415e26 q^{82} -5.54332e25 q^{83} -2.04619e26 q^{84} +2.45325e25 q^{85} +1.89818e26 q^{86} +4.38376e25 q^{87} +5.55481e26 q^{88} -1.32973e26 q^{89} -3.82811e25 q^{90} -5.01492e26 q^{91} +1.72093e26 q^{92} +1.55924e26 q^{93} +5.46521e26 q^{94} -6.00364e25 q^{95} -7.96808e26 q^{96} -1.06222e27 q^{97} +1.72520e27 q^{98} +3.21132e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 21582 q^{2} - 3188646 q^{3} + 202603844 q^{4} - 1771946100 q^{5} - 34408678986 q^{6} + 369665199904 q^{7} + 4429319872824 q^{8} + 5083731656658 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 21582 q^{2} - 3188646 q^{3} + 202603844 q^{4} - 1771946100 q^{5} - 34408678986 q^{6} + 369665199904 q^{7} + 4429319872824 q^{8} + 5083731656658 q^{9} - 14929666656300 q^{10} + 75762335668248 q^{11} - 323015968377612 q^{12} - 103021079177588 q^{13} + 82\!\cdots\!36 q^{14}+ \cdots + 19\!\cdots\!92 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21703.1 1.87334 0.936671 0.350212i \(-0.113890\pi\)
0.936671 + 0.350212i \(0.113890\pi\)
\(3\) −1.59432e6 −0.577350
\(4\) 3.36807e8 2.50941
\(5\) −6.93920e8 −0.254223 −0.127111 0.991888i \(-0.540571\pi\)
−0.127111 + 0.991888i \(0.540571\pi\)
\(6\) −3.46018e10 −1.08157
\(7\) 3.81056e11 1.48650 0.743250 0.669014i \(-0.233283\pi\)
0.743250 + 0.669014i \(0.233283\pi\)
\(8\) 4.39681e12 2.82763
\(9\) 2.54187e12 0.333333
\(10\) −1.50602e13 −0.476246
\(11\) 1.26337e14 1.10339 0.551697 0.834045i \(-0.313981\pi\)
0.551697 + 0.834045i \(0.313981\pi\)
\(12\) −5.36979e14 −1.44881
\(13\) −1.31606e15 −1.20515 −0.602574 0.798063i \(-0.705858\pi\)
−0.602574 + 0.798063i \(0.705858\pi\)
\(14\) 8.27009e15 2.78472
\(15\) 1.10633e15 0.146776
\(16\) 5.02190e16 2.78772
\(17\) −3.53536e16 −0.865711 −0.432855 0.901463i \(-0.642494\pi\)
−0.432855 + 0.901463i \(0.642494\pi\)
\(18\) 5.51664e16 0.624447
\(19\) 8.65178e16 0.471989 0.235994 0.971754i \(-0.424165\pi\)
0.235994 + 0.971754i \(0.424165\pi\)
\(20\) −2.33717e17 −0.637948
\(21\) −6.07526e17 −0.858231
\(22\) 2.74191e18 2.06703
\(23\) 5.10955e17 0.211376 0.105688 0.994399i \(-0.466295\pi\)
0.105688 + 0.994399i \(0.466295\pi\)
\(24\) −7.00994e18 −1.63254
\(25\) −6.96906e18 −0.935371
\(26\) −2.85626e19 −2.25765
\(27\) −4.05256e18 −0.192450
\(28\) 1.28342e20 3.73023
\(29\) −2.74961e19 −0.497620 −0.248810 0.968552i \(-0.580039\pi\)
−0.248810 + 0.968552i \(0.580039\pi\)
\(30\) 2.40109e19 0.274961
\(31\) −9.77996e19 −0.719373 −0.359687 0.933073i \(-0.617116\pi\)
−0.359687 + 0.933073i \(0.617116\pi\)
\(32\) 4.99779e20 2.39471
\(33\) −2.01422e20 −0.637044
\(34\) −7.67282e20 −1.62177
\(35\) −2.64422e20 −0.377902
\(36\) 8.56118e20 0.836469
\(37\) −1.80293e21 −1.21690 −0.608450 0.793592i \(-0.708208\pi\)
−0.608450 + 0.793592i \(0.708208\pi\)
\(38\) 1.87770e21 0.884196
\(39\) 2.09822e21 0.695793
\(40\) −3.05104e21 −0.718849
\(41\) −5.17966e21 −0.874419 −0.437209 0.899360i \(-0.644033\pi\)
−0.437209 + 0.899360i \(0.644033\pi\)
\(42\) −1.31852e22 −1.60776
\(43\) 8.74613e21 0.776233 0.388116 0.921610i \(-0.373126\pi\)
0.388116 + 0.921610i \(0.373126\pi\)
\(44\) 4.25513e22 2.76886
\(45\) −1.76385e21 −0.0847409
\(46\) 1.10893e22 0.395980
\(47\) 2.51817e22 0.672611 0.336306 0.941753i \(-0.390823\pi\)
0.336306 + 0.941753i \(0.390823\pi\)
\(48\) −8.00654e22 −1.60949
\(49\) 7.94910e22 1.20968
\(50\) −1.51250e23 −1.75227
\(51\) 5.63650e22 0.499818
\(52\) −4.43258e23 −3.02421
\(53\) 2.11957e23 1.11821 0.559106 0.829096i \(-0.311144\pi\)
0.559106 + 0.829096i \(0.311144\pi\)
\(54\) −8.79530e22 −0.360525
\(55\) −8.76680e22 −0.280508
\(56\) 1.67543e24 4.20328
\(57\) −1.37937e23 −0.272503
\(58\) −5.96750e23 −0.932212
\(59\) −7.00844e23 −0.869199 −0.434600 0.900624i \(-0.643110\pi\)
−0.434600 + 0.900624i \(0.643110\pi\)
\(60\) 3.72620e23 0.368320
\(61\) 1.08505e23 0.0858020 0.0429010 0.999079i \(-0.486340\pi\)
0.0429010 + 0.999079i \(0.486340\pi\)
\(62\) −2.12256e24 −1.34763
\(63\) 9.68592e23 0.495500
\(64\) 4.10646e24 1.69839
\(65\) 9.13240e23 0.306376
\(66\) −4.37149e24 −1.19340
\(67\) 2.46540e24 0.549387 0.274693 0.961532i \(-0.411424\pi\)
0.274693 + 0.961532i \(0.411424\pi\)
\(68\) −1.19073e25 −2.17242
\(69\) −8.14627e23 −0.122038
\(70\) −5.73878e24 −0.707939
\(71\) 1.29459e25 1.31870 0.659349 0.751837i \(-0.270832\pi\)
0.659349 + 0.751837i \(0.270832\pi\)
\(72\) 1.11761e25 0.942545
\(73\) 7.46465e24 0.522577 0.261289 0.965261i \(-0.415853\pi\)
0.261289 + 0.965261i \(0.415853\pi\)
\(74\) −3.91291e25 −2.27967
\(75\) 1.11109e25 0.540037
\(76\) 2.91398e25 1.18441
\(77\) 4.81415e25 1.64019
\(78\) 4.55380e25 1.30346
\(79\) −4.94644e25 −1.19214 −0.596070 0.802933i \(-0.703272\pi\)
−0.596070 + 0.802933i \(0.703272\pi\)
\(80\) −3.48480e25 −0.708701
\(81\) 6.46108e24 0.111111
\(82\) −1.12415e26 −1.63808
\(83\) −5.54332e25 −0.685828 −0.342914 0.939367i \(-0.611414\pi\)
−0.342914 + 0.939367i \(0.611414\pi\)
\(84\) −2.04619e26 −2.15365
\(85\) 2.45325e25 0.220083
\(86\) 1.89818e26 1.45415
\(87\) 4.38376e25 0.287301
\(88\) 5.55481e26 3.11999
\(89\) −1.32973e26 −0.641206 −0.320603 0.947214i \(-0.603886\pi\)
−0.320603 + 0.947214i \(0.603886\pi\)
\(90\) −3.82811e25 −0.158749
\(91\) −5.01492e26 −1.79145
\(92\) 1.72093e26 0.530430
\(93\) 1.55924e26 0.415330
\(94\) 5.46521e26 1.26003
\(95\) −6.00364e25 −0.119990
\(96\) −7.96808e26 −1.38259
\(97\) −1.06222e27 −1.60249 −0.801243 0.598339i \(-0.795827\pi\)
−0.801243 + 0.598339i \(0.795827\pi\)
\(98\) 1.72520e27 2.26615
\(99\) 3.21132e26 0.367798
\(100\) −2.34723e27 −2.34723
\(101\) 1.74011e27 1.52138 0.760691 0.649114i \(-0.224860\pi\)
0.760691 + 0.649114i \(0.224860\pi\)
\(102\) 1.22330e27 0.936330
\(103\) 4.99563e26 0.335187 0.167594 0.985856i \(-0.446400\pi\)
0.167594 + 0.985856i \(0.446400\pi\)
\(104\) −5.78647e27 −3.40772
\(105\) 4.21574e26 0.218182
\(106\) 4.60013e27 2.09479
\(107\) −1.30395e27 −0.523096 −0.261548 0.965190i \(-0.584233\pi\)
−0.261548 + 0.965190i \(0.584233\pi\)
\(108\) −1.36493e27 −0.482936
\(109\) −5.74920e27 −1.79618 −0.898089 0.439814i \(-0.855044\pi\)
−0.898089 + 0.439814i \(0.855044\pi\)
\(110\) −1.90267e27 −0.525486
\(111\) 2.87445e27 0.702578
\(112\) 1.91362e28 4.14394
\(113\) 8.29240e27 1.59265 0.796326 0.604867i \(-0.206774\pi\)
0.796326 + 0.604867i \(0.206774\pi\)
\(114\) −2.99367e27 −0.510491
\(115\) −3.54562e26 −0.0537367
\(116\) −9.26087e27 −1.24873
\(117\) −3.34525e27 −0.401716
\(118\) −1.52105e28 −1.62831
\(119\) −1.34717e28 −1.28688
\(120\) 4.86434e27 0.415028
\(121\) 2.85111e27 0.217476
\(122\) 2.35490e27 0.160736
\(123\) 8.25805e27 0.504846
\(124\) −3.29396e28 −1.80520
\(125\) 1.00061e28 0.492015
\(126\) 2.10215e28 0.928240
\(127\) 2.16098e27 0.0857631 0.0428815 0.999080i \(-0.486346\pi\)
0.0428815 + 0.999080i \(0.486346\pi\)
\(128\) 2.20438e28 0.786957
\(129\) −1.39442e28 −0.448158
\(130\) 1.98202e28 0.573947
\(131\) 7.19824e28 1.87959 0.939797 0.341734i \(-0.111014\pi\)
0.939797 + 0.341734i \(0.111014\pi\)
\(132\) −6.78405e28 −1.59860
\(133\) 3.29681e28 0.701611
\(134\) 5.35069e28 1.02919
\(135\) 2.81215e27 0.0489252
\(136\) −1.55443e29 −2.44791
\(137\) −8.23477e28 −1.17469 −0.587346 0.809336i \(-0.699827\pi\)
−0.587346 + 0.809336i \(0.699827\pi\)
\(138\) −1.76799e28 −0.228619
\(139\) 1.17233e29 1.37515 0.687574 0.726114i \(-0.258676\pi\)
0.687574 + 0.726114i \(0.258676\pi\)
\(140\) −8.90592e28 −0.948310
\(141\) −4.01478e28 −0.388332
\(142\) 2.80967e29 2.47037
\(143\) −1.66267e29 −1.32975
\(144\) 1.27650e29 0.929239
\(145\) 1.90801e28 0.126506
\(146\) 1.62006e29 0.978966
\(147\) −1.26734e29 −0.698410
\(148\) −6.07239e29 −3.05370
\(149\) −2.66795e28 −0.122508 −0.0612538 0.998122i \(-0.519510\pi\)
−0.0612538 + 0.998122i \(0.519510\pi\)
\(150\) 2.41142e29 1.01167
\(151\) −4.49422e29 −1.72372 −0.861859 0.507149i \(-0.830699\pi\)
−0.861859 + 0.507149i \(0.830699\pi\)
\(152\) 3.80403e29 1.33461
\(153\) −8.98640e28 −0.288570
\(154\) 1.04482e30 3.07264
\(155\) 6.78651e28 0.182881
\(156\) 7.06697e29 1.74603
\(157\) 2.00427e29 0.454268 0.227134 0.973864i \(-0.427065\pi\)
0.227134 + 0.973864i \(0.427065\pi\)
\(158\) −1.07353e30 −2.23328
\(159\) −3.37928e29 −0.645600
\(160\) −3.46806e29 −0.608790
\(161\) 1.94702e29 0.314211
\(162\) 1.40226e29 0.208149
\(163\) 6.62774e29 0.905384 0.452692 0.891667i \(-0.350464\pi\)
0.452692 + 0.891667i \(0.350464\pi\)
\(164\) −1.74455e30 −2.19427
\(165\) 1.39771e29 0.161951
\(166\) −1.20307e30 −1.28479
\(167\) 1.07500e30 1.05861 0.529307 0.848431i \(-0.322452\pi\)
0.529307 + 0.848431i \(0.322452\pi\)
\(168\) −2.67118e30 −2.42676
\(169\) 5.39481e29 0.452382
\(170\) 5.32432e29 0.412291
\(171\) 2.19917e29 0.157330
\(172\) 2.94576e30 1.94788
\(173\) 8.13189e29 0.497244 0.248622 0.968601i \(-0.420022\pi\)
0.248622 + 0.968601i \(0.420022\pi\)
\(174\) 9.51413e29 0.538213
\(175\) −2.65560e30 −1.39043
\(176\) 6.34454e30 3.07595
\(177\) 1.11737e30 0.501832
\(178\) −2.88592e30 −1.20120
\(179\) 1.37936e30 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(180\) −5.94077e29 −0.212649
\(181\) −1.97778e30 −0.656927 −0.328464 0.944517i \(-0.606531\pi\)
−0.328464 + 0.944517i \(0.606531\pi\)
\(182\) −1.08839e31 −3.35600
\(183\) −1.72992e29 −0.0495378
\(184\) 2.24657e30 0.597695
\(185\) 1.25109e30 0.309364
\(186\) 3.38404e30 0.778056
\(187\) −4.46647e30 −0.955219
\(188\) 8.48137e30 1.68785
\(189\) −1.54425e30 −0.286077
\(190\) −1.30298e30 −0.224783
\(191\) 8.23146e30 1.32290 0.661449 0.749990i \(-0.269942\pi\)
0.661449 + 0.749990i \(0.269942\pi\)
\(192\) −6.54702e30 −0.980567
\(193\) −2.63632e30 −0.368107 −0.184054 0.982916i \(-0.558922\pi\)
−0.184054 + 0.982916i \(0.558922\pi\)
\(194\) −2.30534e31 −3.00200
\(195\) −1.45600e30 −0.176886
\(196\) 2.67731e31 3.03558
\(197\) 9.66937e30 1.02354 0.511769 0.859123i \(-0.328990\pi\)
0.511769 + 0.859123i \(0.328990\pi\)
\(198\) 6.96957e30 0.689011
\(199\) −8.33709e30 −0.770012 −0.385006 0.922914i \(-0.625801\pi\)
−0.385006 + 0.922914i \(0.625801\pi\)
\(200\) −3.06416e31 −2.64489
\(201\) −3.93065e30 −0.317189
\(202\) 3.77658e31 2.85007
\(203\) −1.04775e31 −0.739712
\(204\) 1.89841e31 1.25425
\(205\) 3.59427e30 0.222297
\(206\) 1.08421e31 0.627920
\(207\) 1.29878e30 0.0704588
\(208\) −6.60913e31 −3.35961
\(209\) 1.09304e31 0.520789
\(210\) 9.14947e30 0.408729
\(211\) 1.34579e31 0.563850 0.281925 0.959437i \(-0.409027\pi\)
0.281925 + 0.959437i \(0.409027\pi\)
\(212\) 7.13887e31 2.80605
\(213\) −2.06400e31 −0.761351
\(214\) −2.82998e31 −0.979937
\(215\) −6.06912e30 −0.197336
\(216\) −1.78183e31 −0.544178
\(217\) −3.72671e31 −1.06935
\(218\) −1.24775e32 −3.36485
\(219\) −1.19011e31 −0.301710
\(220\) −2.95272e31 −0.703908
\(221\) 4.65274e31 1.04331
\(222\) 6.23845e31 1.31617
\(223\) 7.20166e30 0.142994 0.0714969 0.997441i \(-0.477222\pi\)
0.0714969 + 0.997441i \(0.477222\pi\)
\(224\) 1.90443e32 3.55974
\(225\) −1.77144e31 −0.311790
\(226\) 1.79971e32 2.98358
\(227\) −1.24397e32 −1.94294 −0.971472 0.237153i \(-0.923786\pi\)
−0.971472 + 0.237153i \(0.923786\pi\)
\(228\) −4.64582e31 −0.683821
\(229\) 1.14944e32 1.59481 0.797403 0.603447i \(-0.206206\pi\)
0.797403 + 0.603447i \(0.206206\pi\)
\(230\) −7.69509e30 −0.100667
\(231\) −7.67531e31 −0.946966
\(232\) −1.20895e32 −1.40709
\(233\) 1.39267e32 1.52948 0.764741 0.644337i \(-0.222867\pi\)
0.764741 + 0.644337i \(0.222867\pi\)
\(234\) −7.26023e31 −0.752551
\(235\) −1.74741e31 −0.170993
\(236\) −2.36049e32 −2.18117
\(237\) 7.88622e31 0.688282
\(238\) −2.92377e32 −2.41076
\(239\) 1.18142e32 0.920517 0.460259 0.887785i \(-0.347757\pi\)
0.460259 + 0.887785i \(0.347757\pi\)
\(240\) 5.55590e31 0.409169
\(241\) 1.39602e32 0.971991 0.485996 0.873961i \(-0.338457\pi\)
0.485996 + 0.873961i \(0.338457\pi\)
\(242\) 6.18779e31 0.407407
\(243\) −1.03011e31 −0.0641500
\(244\) 3.65453e31 0.215312
\(245\) −5.51604e31 −0.307528
\(246\) 1.79225e32 0.945748
\(247\) −1.13863e32 −0.568817
\(248\) −4.30007e32 −2.03412
\(249\) 8.83784e31 0.395963
\(250\) 2.17163e32 0.921712
\(251\) 9.88006e31 0.397342 0.198671 0.980066i \(-0.436338\pi\)
0.198671 + 0.980066i \(0.436338\pi\)
\(252\) 3.26229e32 1.24341
\(253\) 6.45526e31 0.233231
\(254\) 4.69000e31 0.160664
\(255\) −3.91128e31 −0.127065
\(256\) −7.27415e31 −0.224152
\(257\) −8.60649e31 −0.251611 −0.125805 0.992055i \(-0.540151\pi\)
−0.125805 + 0.992055i \(0.540151\pi\)
\(258\) −3.02631e32 −0.839553
\(259\) −6.87016e32 −1.80892
\(260\) 3.07586e32 0.768822
\(261\) −6.98913e31 −0.165873
\(262\) 1.56224e33 3.52112
\(263\) −2.80912e32 −0.601405 −0.300703 0.953718i \(-0.597221\pi\)
−0.300703 + 0.953718i \(0.597221\pi\)
\(264\) −8.85617e32 −1.80133
\(265\) −1.47081e32 −0.284275
\(266\) 7.15510e32 1.31436
\(267\) 2.12002e32 0.370200
\(268\) 8.30365e32 1.37864
\(269\) −1.02703e32 −0.162153 −0.0810767 0.996708i \(-0.525836\pi\)
−0.0810767 + 0.996708i \(0.525836\pi\)
\(270\) 6.10324e31 0.0916536
\(271\) −1.27505e33 −1.82155 −0.910775 0.412903i \(-0.864515\pi\)
−0.910775 + 0.412903i \(0.864515\pi\)
\(272\) −1.77542e33 −2.41336
\(273\) 7.99540e32 1.03430
\(274\) −1.78720e33 −2.20060
\(275\) −8.80451e32 −1.03208
\(276\) −2.74372e32 −0.306244
\(277\) 1.09712e33 1.16621 0.583107 0.812396i \(-0.301837\pi\)
0.583107 + 0.812396i \(0.301837\pi\)
\(278\) 2.54432e33 2.57612
\(279\) −2.48594e32 −0.239791
\(280\) −1.16261e33 −1.06857
\(281\) 1.35684e33 1.18849 0.594244 0.804285i \(-0.297451\pi\)
0.594244 + 0.804285i \(0.297451\pi\)
\(282\) −8.71331e32 −0.727479
\(283\) 2.13864e33 1.70224 0.851120 0.524972i \(-0.175924\pi\)
0.851120 + 0.524972i \(0.175924\pi\)
\(284\) 4.36028e33 3.30915
\(285\) 9.57174e31 0.0692764
\(286\) −3.60852e33 −2.49108
\(287\) −1.97374e33 −1.29982
\(288\) 1.27037e33 0.798237
\(289\) −4.17837e32 −0.250545
\(290\) 4.14097e32 0.236989
\(291\) 1.69351e33 0.925195
\(292\) 2.51414e33 1.31136
\(293\) 4.30088e32 0.214212 0.107106 0.994248i \(-0.465842\pi\)
0.107106 + 0.994248i \(0.465842\pi\)
\(294\) −2.75053e33 −1.30836
\(295\) 4.86330e32 0.220970
\(296\) −7.92714e33 −3.44095
\(297\) −5.11989e32 −0.212348
\(298\) −5.79028e32 −0.229499
\(299\) −6.72447e32 −0.254740
\(300\) 3.74224e33 1.35517
\(301\) 3.33276e33 1.15387
\(302\) −9.75386e33 −3.22911
\(303\) −2.77430e33 −0.878370
\(304\) 4.34484e33 1.31577
\(305\) −7.52938e31 −0.0218128
\(306\) −1.95033e33 −0.540591
\(307\) −5.70033e32 −0.151193 −0.0755966 0.997138i \(-0.524086\pi\)
−0.0755966 + 0.997138i \(0.524086\pi\)
\(308\) 1.62144e34 4.11591
\(309\) −7.96464e32 −0.193520
\(310\) 1.47288e33 0.342599
\(311\) −4.65284e33 −1.03622 −0.518111 0.855313i \(-0.673365\pi\)
−0.518111 + 0.855313i \(0.673365\pi\)
\(312\) 9.22550e33 1.96745
\(313\) 4.33109e33 0.884604 0.442302 0.896866i \(-0.354162\pi\)
0.442302 + 0.896866i \(0.354162\pi\)
\(314\) 4.34990e33 0.850998
\(315\) −6.72126e32 −0.125967
\(316\) −1.66600e34 −2.99156
\(317\) −2.47386e33 −0.425671 −0.212836 0.977088i \(-0.568270\pi\)
−0.212836 + 0.977088i \(0.568270\pi\)
\(318\) −7.33410e33 −1.20943
\(319\) −3.47378e33 −0.549070
\(320\) −2.84955e33 −0.431770
\(321\) 2.07892e33 0.302010
\(322\) 4.22564e33 0.588624
\(323\) −3.05871e33 −0.408606
\(324\) 2.17614e33 0.278823
\(325\) 9.17170e33 1.12726
\(326\) 1.43842e34 1.69609
\(327\) 9.16607e33 1.03702
\(328\) −2.27740e34 −2.47254
\(329\) 9.59563e33 0.999836
\(330\) 3.03347e33 0.303390
\(331\) −2.93911e33 −0.282188 −0.141094 0.989996i \(-0.545062\pi\)
−0.141094 + 0.989996i \(0.545062\pi\)
\(332\) −1.86703e34 −1.72102
\(333\) −4.58280e33 −0.405633
\(334\) 2.33309e34 1.98314
\(335\) −1.71079e33 −0.139667
\(336\) −3.05094e34 −2.39250
\(337\) −1.57199e34 −1.18426 −0.592128 0.805844i \(-0.701712\pi\)
−0.592128 + 0.805844i \(0.701712\pi\)
\(338\) 1.17084e34 0.847466
\(339\) −1.32208e34 −0.919518
\(340\) 8.26273e33 0.552279
\(341\) −1.23557e34 −0.793752
\(342\) 4.77287e33 0.294732
\(343\) 5.25043e33 0.311691
\(344\) 3.84551e34 2.19490
\(345\) 5.65286e32 0.0310249
\(346\) 1.76487e34 0.931507
\(347\) 2.55505e34 1.29703 0.648516 0.761201i \(-0.275390\pi\)
0.648516 + 0.761201i \(0.275390\pi\)
\(348\) 1.47648e34 0.720955
\(349\) −3.50116e34 −1.64463 −0.822316 0.569031i \(-0.807319\pi\)
−0.822316 + 0.569031i \(0.807319\pi\)
\(350\) −5.76347e34 −2.60475
\(351\) 5.33341e33 0.231931
\(352\) 6.31407e34 2.64231
\(353\) −6.64200e33 −0.267510 −0.133755 0.991014i \(-0.542704\pi\)
−0.133755 + 0.991014i \(0.542704\pi\)
\(354\) 2.42504e34 0.940103
\(355\) −8.98343e33 −0.335243
\(356\) −4.47861e34 −1.60905
\(357\) 2.14782e34 0.742980
\(358\) 2.99363e34 0.997188
\(359\) 4.11061e34 1.31865 0.659327 0.751856i \(-0.270841\pi\)
0.659327 + 0.751856i \(0.270841\pi\)
\(360\) −7.75533e33 −0.239616
\(361\) −2.61153e34 −0.777227
\(362\) −4.29240e34 −1.23065
\(363\) −4.54559e33 −0.125560
\(364\) −1.68906e35 −4.49548
\(365\) −5.17987e33 −0.132851
\(366\) −3.75447e33 −0.0928012
\(367\) −3.04370e33 −0.0725121 −0.0362560 0.999343i \(-0.511543\pi\)
−0.0362560 + 0.999343i \(0.511543\pi\)
\(368\) 2.56597e34 0.589258
\(369\) −1.31660e34 −0.291473
\(370\) 2.71525e34 0.579544
\(371\) 8.07675e34 1.66222
\(372\) 5.25164e34 1.04223
\(373\) 4.11966e34 0.788483 0.394242 0.919007i \(-0.371007\pi\)
0.394242 + 0.919007i \(0.371007\pi\)
\(374\) −9.69363e34 −1.78945
\(375\) −1.59529e34 −0.284065
\(376\) 1.10719e35 1.90190
\(377\) 3.61865e34 0.599706
\(378\) −3.35150e34 −0.535920
\(379\) 8.35010e34 1.28843 0.644217 0.764842i \(-0.277183\pi\)
0.644217 + 0.764842i \(0.277183\pi\)
\(380\) −2.02207e34 −0.301104
\(381\) −3.44530e33 −0.0495153
\(382\) 1.78648e35 2.47824
\(383\) −5.20579e34 −0.697113 −0.348557 0.937288i \(-0.613328\pi\)
−0.348557 + 0.937288i \(0.613328\pi\)
\(384\) −3.51449e34 −0.454350
\(385\) −3.34064e34 −0.416974
\(386\) −5.72163e34 −0.689591
\(387\) 2.22315e34 0.258744
\(388\) −3.57761e35 −4.02129
\(389\) −3.57703e34 −0.388332 −0.194166 0.980969i \(-0.562200\pi\)
−0.194166 + 0.980969i \(0.562200\pi\)
\(390\) −3.15997e34 −0.331368
\(391\) −1.80641e34 −0.182991
\(392\) 3.49507e35 3.42054
\(393\) −1.14763e35 −1.08518
\(394\) 2.09855e35 1.91744
\(395\) 3.43243e34 0.303069
\(396\) 1.08160e35 0.922954
\(397\) −7.79828e34 −0.643172 −0.321586 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(398\) −1.80941e35 −1.44250
\(399\) −5.25618e34 −0.405075
\(400\) −3.49979e35 −2.60755
\(401\) −2.28761e34 −0.164791 −0.0823955 0.996600i \(-0.526257\pi\)
−0.0823955 + 0.996600i \(0.526257\pi\)
\(402\) −8.53073e34 −0.594203
\(403\) 1.28710e35 0.866952
\(404\) 5.86081e35 3.81777
\(405\) −4.48347e33 −0.0282470
\(406\) −2.27395e35 −1.38573
\(407\) −2.27777e35 −1.34272
\(408\) 2.47826e35 1.41330
\(409\) −1.15859e34 −0.0639243 −0.0319621 0.999489i \(-0.510176\pi\)
−0.0319621 + 0.999489i \(0.510176\pi\)
\(410\) 7.80068e34 0.416438
\(411\) 1.31289e35 0.678209
\(412\) 1.68256e35 0.841121
\(413\) −2.67061e35 −1.29206
\(414\) 2.81875e34 0.131993
\(415\) 3.84662e34 0.174353
\(416\) −6.57739e35 −2.88598
\(417\) −1.86907e35 −0.793942
\(418\) 2.37224e35 0.975616
\(419\) 6.19060e34 0.246515 0.123258 0.992375i \(-0.460666\pi\)
0.123258 + 0.992375i \(0.460666\pi\)
\(420\) 1.41989e35 0.547507
\(421\) −8.53227e33 −0.0318608 −0.0159304 0.999873i \(-0.505071\pi\)
−0.0159304 + 0.999873i \(0.505071\pi\)
\(422\) 2.92078e35 1.05628
\(423\) 6.40085e34 0.224204
\(424\) 9.31937e35 3.16190
\(425\) 2.46381e35 0.809761
\(426\) −4.47952e35 −1.42627
\(427\) 4.13465e34 0.127545
\(428\) −4.39181e35 −1.31266
\(429\) 2.65084e35 0.767733
\(430\) −1.31719e35 −0.369678
\(431\) 4.08719e35 1.11169 0.555843 0.831288i \(-0.312396\pi\)
0.555843 + 0.831288i \(0.312396\pi\)
\(432\) −2.03515e35 −0.536496
\(433\) 1.44666e35 0.369642 0.184821 0.982772i \(-0.440830\pi\)
0.184821 + 0.982772i \(0.440830\pi\)
\(434\) −8.08812e35 −2.00325
\(435\) −3.04198e34 −0.0730385
\(436\) −1.93637e36 −4.50734
\(437\) 4.42067e34 0.0997673
\(438\) −2.58290e35 −0.565206
\(439\) 2.97115e34 0.0630454 0.0315227 0.999503i \(-0.489964\pi\)
0.0315227 + 0.999503i \(0.489964\pi\)
\(440\) −3.85460e35 −0.793173
\(441\) 2.02055e35 0.403227
\(442\) 1.00979e36 1.95448
\(443\) −1.94735e35 −0.365589 −0.182795 0.983151i \(-0.558514\pi\)
−0.182795 + 0.983151i \(0.558514\pi\)
\(444\) 9.68134e35 1.76305
\(445\) 9.22725e34 0.163009
\(446\) 1.56298e35 0.267876
\(447\) 4.25357e34 0.0707298
\(448\) 1.56479e36 2.52466
\(449\) −3.89051e35 −0.609089 −0.304544 0.952498i \(-0.598504\pi\)
−0.304544 + 0.952498i \(0.598504\pi\)
\(450\) −3.84458e35 −0.584090
\(451\) −6.54384e35 −0.964827
\(452\) 2.79294e36 3.99661
\(453\) 7.16525e35 0.995188
\(454\) −2.69980e36 −3.63980
\(455\) 3.47995e35 0.455428
\(456\) −6.06485e35 −0.770538
\(457\) −7.79952e35 −0.962053 −0.481027 0.876706i \(-0.659736\pi\)
−0.481027 + 0.876706i \(0.659736\pi\)
\(458\) 2.49465e36 2.98762
\(459\) 1.43272e35 0.166606
\(460\) −1.19419e35 −0.134847
\(461\) −1.33828e36 −1.46752 −0.733760 0.679408i \(-0.762236\pi\)
−0.733760 + 0.679408i \(0.762236\pi\)
\(462\) −1.66578e36 −1.77399
\(463\) 3.79932e35 0.392973 0.196486 0.980507i \(-0.437047\pi\)
0.196486 + 0.980507i \(0.437047\pi\)
\(464\) −1.38083e36 −1.38722
\(465\) −1.08199e35 −0.105586
\(466\) 3.02253e36 2.86524
\(467\) −7.20473e35 −0.663499 −0.331749 0.943368i \(-0.607639\pi\)
−0.331749 + 0.943368i \(0.607639\pi\)
\(468\) −1.12670e36 −1.00807
\(469\) 9.39456e35 0.816664
\(470\) −3.79242e35 −0.320328
\(471\) −3.19546e35 −0.262271
\(472\) −3.08148e36 −2.45778
\(473\) 1.10496e36 0.856490
\(474\) 1.71156e36 1.28939
\(475\) −6.02947e35 −0.441485
\(476\) −4.53735e36 −3.22930
\(477\) 5.38767e35 0.372737
\(478\) 2.56404e36 1.72444
\(479\) −4.56900e34 −0.0298739 −0.0149370 0.999888i \(-0.504755\pi\)
−0.0149370 + 0.999888i \(0.504755\pi\)
\(480\) 5.52921e35 0.351485
\(481\) 2.37276e36 1.46655
\(482\) 3.02980e36 1.82087
\(483\) −3.10418e35 −0.181410
\(484\) 9.60274e35 0.545736
\(485\) 7.37092e35 0.407388
\(486\) −2.23565e35 −0.120175
\(487\) −2.25055e36 −1.17665 −0.588326 0.808624i \(-0.700213\pi\)
−0.588326 + 0.808624i \(0.700213\pi\)
\(488\) 4.77077e35 0.242617
\(489\) −1.05668e36 −0.522723
\(490\) −1.19715e36 −0.576106
\(491\) −3.36099e36 −1.57350 −0.786751 0.617271i \(-0.788238\pi\)
−0.786751 + 0.617271i \(0.788238\pi\)
\(492\) 2.78137e36 1.26686
\(493\) 9.72085e35 0.430795
\(494\) −2.47117e36 −1.06559
\(495\) −2.22840e35 −0.0935025
\(496\) −4.91140e36 −2.00541
\(497\) 4.93312e36 1.96024
\(498\) 1.91808e36 0.741774
\(499\) 1.11121e35 0.0418253 0.0209127 0.999781i \(-0.493343\pi\)
0.0209127 + 0.999781i \(0.493343\pi\)
\(500\) 3.37012e36 1.23467
\(501\) −1.71390e36 −0.611191
\(502\) 2.14428e36 0.744357
\(503\) −1.49384e36 −0.504818 −0.252409 0.967621i \(-0.581223\pi\)
−0.252409 + 0.967621i \(0.581223\pi\)
\(504\) 4.25872e36 1.40109
\(505\) −1.20750e36 −0.386770
\(506\) 1.40099e36 0.436922
\(507\) −8.60107e35 −0.261183
\(508\) 7.27833e35 0.215215
\(509\) −5.00593e36 −1.44143 −0.720717 0.693230i \(-0.756187\pi\)
−0.720717 + 0.693230i \(0.756187\pi\)
\(510\) −8.48869e35 −0.238036
\(511\) 2.84445e36 0.776811
\(512\) −4.53738e36 −1.20687
\(513\) −3.50618e35 −0.0908343
\(514\) −1.86788e36 −0.471352
\(515\) −3.46657e35 −0.0852122
\(516\) −4.69649e36 −1.12461
\(517\) 3.18139e36 0.742154
\(518\) −1.49104e37 −3.38873
\(519\) −1.29649e36 −0.287084
\(520\) 4.01535e36 0.866319
\(521\) 3.77205e36 0.792990 0.396495 0.918037i \(-0.370226\pi\)
0.396495 + 0.918037i \(0.370226\pi\)
\(522\) −1.51686e36 −0.310737
\(523\) −7.85298e35 −0.156770 −0.0783848 0.996923i \(-0.524976\pi\)
−0.0783848 + 0.996923i \(0.524976\pi\)
\(524\) 2.42442e37 4.71666
\(525\) 4.23388e36 0.802764
\(526\) −6.09666e36 −1.12664
\(527\) 3.45757e36 0.622769
\(528\) −1.01152e37 −1.77590
\(529\) −5.58214e36 −0.955320
\(530\) −3.19212e36 −0.532544
\(531\) −1.78145e36 −0.289733
\(532\) 1.11039e37 1.76063
\(533\) 6.81675e36 1.05380
\(534\) 4.60109e36 0.693512
\(535\) 9.04840e35 0.132983
\(536\) 1.08399e37 1.55347
\(537\) −2.19914e36 −0.307326
\(538\) −2.22897e36 −0.303769
\(539\) 1.00427e37 1.33475
\(540\) 9.47151e35 0.122773
\(541\) 6.90432e36 0.872887 0.436444 0.899732i \(-0.356238\pi\)
0.436444 + 0.899732i \(0.356238\pi\)
\(542\) −2.76725e37 −3.41238
\(543\) 3.15322e36 0.379277
\(544\) −1.76690e37 −2.07313
\(545\) 3.98948e36 0.456629
\(546\) 1.73525e37 1.93759
\(547\) −4.90777e36 −0.534632 −0.267316 0.963609i \(-0.586137\pi\)
−0.267316 + 0.963609i \(0.586137\pi\)
\(548\) −2.77353e37 −2.94778
\(549\) 2.75805e35 0.0286007
\(550\) −1.91085e37 −1.93344
\(551\) −2.37890e36 −0.234871
\(552\) −3.58176e36 −0.345080
\(553\) −1.88487e37 −1.77211
\(554\) 2.38109e37 2.18471
\(555\) −1.99464e36 −0.178611
\(556\) 3.94848e37 3.45081
\(557\) −8.37169e36 −0.714115 −0.357058 0.934082i \(-0.616220\pi\)
−0.357058 + 0.934082i \(0.616220\pi\)
\(558\) −5.39525e36 −0.449211
\(559\) −1.15104e37 −0.935475
\(560\) −1.32790e37 −1.05348
\(561\) 7.12100e36 0.551496
\(562\) 2.94477e37 2.22644
\(563\) 1.53607e37 1.13383 0.566913 0.823778i \(-0.308138\pi\)
0.566913 + 0.823778i \(0.308138\pi\)
\(564\) −1.35220e37 −0.974484
\(565\) −5.75426e36 −0.404889
\(566\) 4.64151e37 3.18887
\(567\) 2.46203e36 0.165167
\(568\) 5.69208e37 3.72879
\(569\) −2.56048e37 −1.63797 −0.818986 0.573814i \(-0.805463\pi\)
−0.818986 + 0.573814i \(0.805463\pi\)
\(570\) 2.07737e36 0.129778
\(571\) −1.84375e37 −1.12490 −0.562450 0.826831i \(-0.690141\pi\)
−0.562450 + 0.826831i \(0.690141\pi\)
\(572\) −5.60000e37 −3.33689
\(573\) −1.31236e37 −0.763775
\(574\) −4.28363e37 −2.43501
\(575\) −3.56087e36 −0.197715
\(576\) 1.04381e37 0.566131
\(577\) 2.60648e37 1.38096 0.690479 0.723352i \(-0.257400\pi\)
0.690479 + 0.723352i \(0.257400\pi\)
\(578\) −9.06835e36 −0.469356
\(579\) 4.20315e36 0.212527
\(580\) 6.42630e36 0.317456
\(581\) −2.11231e37 −1.01948
\(582\) 3.67545e37 1.73321
\(583\) 2.67781e37 1.23383
\(584\) 3.28207e37 1.47766
\(585\) 2.32133e36 0.102125
\(586\) 9.33423e36 0.401292
\(587\) 1.64116e37 0.689505 0.344752 0.938694i \(-0.387963\pi\)
0.344752 + 0.938694i \(0.387963\pi\)
\(588\) −4.26850e37 −1.75259
\(589\) −8.46141e36 −0.339536
\(590\) 1.05549e37 0.413953
\(591\) −1.54161e37 −0.590940
\(592\) −9.05413e37 −3.39237
\(593\) 6.39232e35 0.0234110 0.0117055 0.999931i \(-0.496274\pi\)
0.0117055 + 0.999931i \(0.496274\pi\)
\(594\) −1.11117e37 −0.397800
\(595\) 9.34827e36 0.327154
\(596\) −8.98584e36 −0.307421
\(597\) 1.32920e37 0.444567
\(598\) −1.45942e37 −0.477215
\(599\) 3.67500e37 1.17488 0.587442 0.809266i \(-0.300135\pi\)
0.587442 + 0.809266i \(0.300135\pi\)
\(600\) 4.88527e37 1.52703
\(601\) 4.35901e36 0.133224 0.0666120 0.997779i \(-0.478781\pi\)
0.0666120 + 0.997779i \(0.478781\pi\)
\(602\) 7.23313e37 2.16159
\(603\) 6.26673e36 0.183129
\(604\) −1.51369e38 −4.32551
\(605\) −1.97844e36 −0.0552874
\(606\) −6.02109e37 −1.64549
\(607\) 4.98438e37 1.33218 0.666092 0.745870i \(-0.267966\pi\)
0.666092 + 0.745870i \(0.267966\pi\)
\(608\) 4.32397e37 1.13028
\(609\) 1.67046e37 0.427073
\(610\) −1.63411e36 −0.0408628
\(611\) −3.31406e37 −0.810596
\(612\) −3.02668e37 −0.724140
\(613\) 2.09778e37 0.490958 0.245479 0.969402i \(-0.421055\pi\)
0.245479 + 0.969402i \(0.421055\pi\)
\(614\) −1.23715e37 −0.283236
\(615\) −5.73043e36 −0.128343
\(616\) 2.11669e38 4.63787
\(617\) −9.28596e37 −1.99057 −0.995284 0.0970039i \(-0.969074\pi\)
−0.995284 + 0.0970039i \(0.969074\pi\)
\(618\) −1.72857e37 −0.362530
\(619\) 4.21148e37 0.864194 0.432097 0.901827i \(-0.357774\pi\)
0.432097 + 0.901827i \(0.357774\pi\)
\(620\) 2.28574e37 0.458923
\(621\) −2.07067e36 −0.0406794
\(622\) −1.00981e38 −1.94120
\(623\) −5.06700e37 −0.953153
\(624\) 1.05371e38 1.93967
\(625\) 4.49801e37 0.810289
\(626\) 9.39981e37 1.65716
\(627\) −1.74266e37 −0.300678
\(628\) 6.75053e37 1.13994
\(629\) 6.37399e37 1.05348
\(630\) −1.45872e37 −0.235980
\(631\) −1.60405e37 −0.253993 −0.126997 0.991903i \(-0.540534\pi\)
−0.126997 + 0.991903i \(0.540534\pi\)
\(632\) −2.17486e38 −3.37093
\(633\) −2.14562e37 −0.325539
\(634\) −5.36903e37 −0.797427
\(635\) −1.49955e36 −0.0218029
\(636\) −1.13817e38 −1.62007
\(637\) −1.04615e38 −1.45784
\(638\) −7.53918e37 −1.02860
\(639\) 3.29068e37 0.439566
\(640\) −1.52966e37 −0.200062
\(641\) −4.27881e37 −0.547948 −0.273974 0.961737i \(-0.588338\pi\)
−0.273974 + 0.961737i \(0.588338\pi\)
\(642\) 4.51191e37 0.565767
\(643\) 6.44875e36 0.0791821 0.0395911 0.999216i \(-0.487394\pi\)
0.0395911 + 0.999216i \(0.487394\pi\)
\(644\) 6.55770e37 0.788483
\(645\) 9.67613e36 0.113932
\(646\) −6.63835e37 −0.765458
\(647\) 1.16912e38 1.32024 0.660118 0.751162i \(-0.270506\pi\)
0.660118 + 0.751162i \(0.270506\pi\)
\(648\) 2.84082e37 0.314182
\(649\) −8.85427e37 −0.959068
\(650\) 1.99054e38 2.11174
\(651\) 5.94158e37 0.617389
\(652\) 2.23227e38 2.27198
\(653\) −7.34409e37 −0.732167 −0.366083 0.930582i \(-0.619301\pi\)
−0.366083 + 0.930582i \(0.619301\pi\)
\(654\) 1.98932e38 1.94270
\(655\) −4.99500e37 −0.477835
\(656\) −2.60118e38 −2.43763
\(657\) 1.89741e37 0.174192
\(658\) 2.08255e38 1.87303
\(659\) 3.42254e37 0.301575 0.150788 0.988566i \(-0.451819\pi\)
0.150788 + 0.988566i \(0.451819\pi\)
\(660\) 4.70759e37 0.406401
\(661\) −1.89312e37 −0.160125 −0.0800625 0.996790i \(-0.525512\pi\)
−0.0800625 + 0.996790i \(0.525512\pi\)
\(662\) −6.37878e37 −0.528634
\(663\) −7.41797e37 −0.602355
\(664\) −2.43729e38 −1.93927
\(665\) −2.28772e37 −0.178366
\(666\) −9.94610e37 −0.759890
\(667\) −1.40493e37 −0.105185
\(668\) 3.62068e38 2.65649
\(669\) −1.14818e37 −0.0825575
\(670\) −3.71295e37 −0.261643
\(671\) 1.37082e37 0.0946733
\(672\) −3.03628e38 −2.05521
\(673\) 2.43532e38 1.61567 0.807835 0.589409i \(-0.200639\pi\)
0.807835 + 0.589409i \(0.200639\pi\)
\(674\) −3.41170e38 −2.21852
\(675\) 2.82425e37 0.180012
\(676\) 1.81701e38 1.13521
\(677\) 2.58654e38 1.58406 0.792032 0.610479i \(-0.209023\pi\)
0.792032 + 0.610479i \(0.209023\pi\)
\(678\) −2.86931e38 −1.72257
\(679\) −4.04763e38 −2.38209
\(680\) 1.07865e38 0.622315
\(681\) 1.98329e38 1.12176
\(682\) −2.68158e38 −1.48697
\(683\) 5.72592e37 0.311290 0.155645 0.987813i \(-0.450254\pi\)
0.155645 + 0.987813i \(0.450254\pi\)
\(684\) 7.40694e37 0.394804
\(685\) 5.71427e37 0.298633
\(686\) 1.13951e38 0.583903
\(687\) −1.83258e38 −0.920762
\(688\) 4.39222e38 2.16392
\(689\) −2.78949e38 −1.34761
\(690\) 1.22685e37 0.0581202
\(691\) −1.45765e38 −0.677175 −0.338588 0.940935i \(-0.609949\pi\)
−0.338588 + 0.940935i \(0.609949\pi\)
\(692\) 2.73888e38 1.24779
\(693\) 1.22369e38 0.546731
\(694\) 5.54524e38 2.42978
\(695\) −8.13502e37 −0.349594
\(696\) 1.92746e38 0.812382
\(697\) 1.83120e38 0.756994
\(698\) −7.59861e38 −3.08096
\(699\) −2.22037e38 −0.883047
\(700\) −8.94424e38 −3.48915
\(701\) −5.69328e37 −0.217856 −0.108928 0.994050i \(-0.534742\pi\)
−0.108928 + 0.994050i \(0.534742\pi\)
\(702\) 1.15751e38 0.434486
\(703\) −1.55985e38 −0.574363
\(704\) 5.18799e38 1.87399
\(705\) 2.78593e37 0.0987229
\(706\) −1.44152e38 −0.501138
\(707\) 6.63079e38 2.26153
\(708\) 3.76339e38 1.25930
\(709\) 2.15586e38 0.707776 0.353888 0.935288i \(-0.384859\pi\)
0.353888 + 0.935288i \(0.384859\pi\)
\(710\) −1.94968e38 −0.628024
\(711\) −1.25732e38 −0.397380
\(712\) −5.84656e38 −1.81310
\(713\) −4.99712e37 −0.152059
\(714\) 4.66144e38 1.39185
\(715\) 1.15376e38 0.338053
\(716\) 4.64577e38 1.33577
\(717\) −1.88356e38 −0.531461
\(718\) 8.92129e38 2.47029
\(719\) 2.25632e38 0.613141 0.306571 0.951848i \(-0.400818\pi\)
0.306571 + 0.951848i \(0.400818\pi\)
\(720\) −8.85789e37 −0.236234
\(721\) 1.90361e38 0.498256
\(722\) −5.66783e38 −1.45601
\(723\) −2.22571e38 −0.561179
\(724\) −6.66130e38 −1.64850
\(725\) 1.91622e38 0.465459
\(726\) −9.86534e37 −0.235217
\(727\) 3.26448e38 0.764010 0.382005 0.924160i \(-0.375234\pi\)
0.382005 + 0.924160i \(0.375234\pi\)
\(728\) −2.20497e39 −5.06557
\(729\) 1.64232e37 0.0370370
\(730\) −1.12419e38 −0.248875
\(731\) −3.09207e38 −0.671993
\(732\) −5.82649e37 −0.124310
\(733\) 3.51091e38 0.735387 0.367694 0.929947i \(-0.380147\pi\)
0.367694 + 0.929947i \(0.380147\pi\)
\(734\) −6.60578e37 −0.135840
\(735\) 8.79435e37 0.177552
\(736\) 2.55364e38 0.506185
\(737\) 3.11472e38 0.606190
\(738\) −2.85743e38 −0.546028
\(739\) −5.12467e38 −0.961537 −0.480768 0.876848i \(-0.659642\pi\)
−0.480768 + 0.876848i \(0.659642\pi\)
\(740\) 4.21375e38 0.776320
\(741\) 1.81534e38 0.328406
\(742\) 1.75291e39 3.11391
\(743\) −6.94450e38 −1.21141 −0.605706 0.795688i \(-0.707109\pi\)
−0.605706 + 0.795688i \(0.707109\pi\)
\(744\) 6.85570e38 1.17440
\(745\) 1.85134e37 0.0311442
\(746\) 8.94095e38 1.47710
\(747\) −1.40904e38 −0.228609
\(748\) −1.50434e39 −2.39703
\(749\) −4.96879e38 −0.777582
\(750\) −3.46228e38 −0.532151
\(751\) 5.87337e38 0.886642 0.443321 0.896363i \(-0.353800\pi\)
0.443321 + 0.896363i \(0.353800\pi\)
\(752\) 1.26460e39 1.87505
\(753\) −1.57520e38 −0.229405
\(754\) 7.85359e38 1.12345
\(755\) 3.11863e38 0.438208
\(756\) −5.20114e38 −0.717884
\(757\) 1.01202e38 0.137212 0.0686062 0.997644i \(-0.478145\pi\)
0.0686062 + 0.997644i \(0.478145\pi\)
\(758\) 1.81223e39 2.41368
\(759\) −1.02918e38 −0.134656
\(760\) −2.63969e38 −0.339289
\(761\) −1.48838e39 −1.87941 −0.939705 0.341987i \(-0.888900\pi\)
−0.939705 + 0.341987i \(0.888900\pi\)
\(762\) −7.47737e37 −0.0927591
\(763\) −2.19076e39 −2.67002
\(764\) 2.77241e39 3.31969
\(765\) 6.23585e37 0.0733611
\(766\) −1.12982e39 −1.30593
\(767\) 9.22353e38 1.04751
\(768\) 1.15973e38 0.129414
\(769\) −1.63286e39 −1.79037 −0.895186 0.445692i \(-0.852958\pi\)
−0.895186 + 0.445692i \(0.852958\pi\)
\(770\) −7.25022e38 −0.781135
\(771\) 1.37215e38 0.145267
\(772\) −8.87931e38 −0.923731
\(773\) 9.47969e36 0.00969105 0.00484552 0.999988i \(-0.498458\pi\)
0.00484552 + 0.999988i \(0.498458\pi\)
\(774\) 4.82492e38 0.484716
\(775\) 6.81571e38 0.672881
\(776\) −4.67036e39 −4.53124
\(777\) 1.09532e39 1.04438
\(778\) −7.76327e38 −0.727479
\(779\) −4.48133e38 −0.412716
\(780\) −4.90391e38 −0.443880
\(781\) 1.63555e39 1.45504
\(782\) −3.92046e38 −0.342804
\(783\) 1.11429e38 0.0957670
\(784\) 3.99196e39 3.37225
\(785\) −1.39081e38 −0.115485
\(786\) −2.49072e39 −2.03292
\(787\) 1.01258e39 0.812401 0.406201 0.913784i \(-0.366853\pi\)
0.406201 + 0.913784i \(0.366853\pi\)
\(788\) 3.25671e39 2.56847
\(789\) 4.47864e38 0.347222
\(790\) 7.44945e38 0.567752
\(791\) 3.15986e39 2.36748
\(792\) 1.41196e39 1.04000
\(793\) −1.42799e38 −0.103404
\(794\) −1.69247e39 −1.20488
\(795\) 2.34495e38 0.164126
\(796\) −2.80799e39 −1.93227
\(797\) −1.64542e39 −1.11324 −0.556620 0.830767i \(-0.687902\pi\)
−0.556620 + 0.830767i \(0.687902\pi\)
\(798\) −1.14075e39 −0.758844
\(799\) −8.90263e38 −0.582287
\(800\) −3.48298e39 −2.23994
\(801\) −3.37999e38 −0.213735
\(802\) −4.96483e38 −0.308710
\(803\) 9.43063e38 0.576608
\(804\) −1.32387e39 −0.795956
\(805\) −1.35108e38 −0.0798796
\(806\) 2.79341e39 1.62410
\(807\) 1.63741e38 0.0936193
\(808\) 7.65094e39 4.30191
\(809\) −3.76453e38 −0.208164 −0.104082 0.994569i \(-0.533190\pi\)
−0.104082 + 0.994569i \(0.533190\pi\)
\(810\) −9.73053e37 −0.0529162
\(811\) 1.84807e39 0.988408 0.494204 0.869346i \(-0.335460\pi\)
0.494204 + 0.869346i \(0.335460\pi\)
\(812\) −3.52891e39 −1.85624
\(813\) 2.03284e39 1.05167
\(814\) −4.94347e39 −2.51537
\(815\) −4.59912e38 −0.230169
\(816\) 2.83060e39 1.39335
\(817\) 7.56696e38 0.366373
\(818\) −2.51450e38 −0.119752
\(819\) −1.27473e39 −0.597151
\(820\) 1.21058e39 0.557834
\(821\) −4.29041e39 −1.94476 −0.972380 0.233401i \(-0.925014\pi\)
−0.972380 + 0.233401i \(0.925014\pi\)
\(822\) 2.84938e39 1.27052
\(823\) −1.74258e39 −0.764355 −0.382177 0.924089i \(-0.624826\pi\)
−0.382177 + 0.924089i \(0.624826\pi\)
\(824\) 2.19648e39 0.947787
\(825\) 1.40372e39 0.595873
\(826\) −5.79604e39 −2.42048
\(827\) 3.90760e39 1.60541 0.802704 0.596378i \(-0.203394\pi\)
0.802704 + 0.596378i \(0.203394\pi\)
\(828\) 4.37438e38 0.176810
\(829\) 1.05175e39 0.418243 0.209121 0.977890i \(-0.432940\pi\)
0.209121 + 0.977890i \(0.432940\pi\)
\(830\) 8.34836e38 0.326623
\(831\) −1.74917e39 −0.673313
\(832\) −5.40435e39 −2.04681
\(833\) −2.81029e39 −1.04723
\(834\) −4.05646e39 −1.48732
\(835\) −7.45966e38 −0.269124
\(836\) 3.68144e39 1.30687
\(837\) 3.96338e38 0.138443
\(838\) 1.34355e39 0.461807
\(839\) −9.84534e38 −0.333000 −0.166500 0.986041i \(-0.553247\pi\)
−0.166500 + 0.986041i \(0.553247\pi\)
\(840\) 1.85358e39 0.616938
\(841\) −2.29710e39 −0.752374
\(842\) −1.85177e38 −0.0596861
\(843\) −2.16325e39 −0.686173
\(844\) 4.53270e39 1.41493
\(845\) −3.74357e38 −0.115006
\(846\) 1.38918e39 0.420010
\(847\) 1.08643e39 0.323278
\(848\) 1.06443e40 3.11726
\(849\) −3.40968e39 −0.982788
\(850\) 5.34723e39 1.51696
\(851\) −9.21214e38 −0.257224
\(852\) −6.95169e39 −1.91054
\(853\) 2.30350e39 0.623125 0.311563 0.950226i \(-0.399148\pi\)
0.311563 + 0.950226i \(0.399148\pi\)
\(854\) 8.97347e38 0.238935
\(855\) −1.52605e38 −0.0399968
\(856\) −5.73324e39 −1.47912
\(857\) −1.77038e39 −0.449600 −0.224800 0.974405i \(-0.572173\pi\)
−0.224800 + 0.974405i \(0.572173\pi\)
\(858\) 5.75314e39 1.43823
\(859\) 5.46881e39 1.34581 0.672907 0.739727i \(-0.265046\pi\)
0.672907 + 0.739727i \(0.265046\pi\)
\(860\) −2.04412e39 −0.495196
\(861\) 3.14678e39 0.750453
\(862\) 8.87047e39 2.08257
\(863\) 5.53394e39 1.27905 0.639526 0.768769i \(-0.279130\pi\)
0.639526 + 0.768769i \(0.279130\pi\)
\(864\) −2.02538e39 −0.460862
\(865\) −5.64288e38 −0.126411
\(866\) 3.13971e39 0.692465
\(867\) 6.66166e38 0.144652
\(868\) −1.25518e40 −2.68343
\(869\) −6.24920e39 −1.31540
\(870\) −6.60204e38 −0.136826
\(871\) −3.24462e39 −0.662093
\(872\) −2.52781e40 −5.07893
\(873\) −2.70001e39 −0.534162
\(874\) 9.59422e38 0.186898
\(875\) 3.81287e39 0.731380
\(876\) −4.00836e39 −0.757114
\(877\) 7.58578e39 1.41093 0.705466 0.708744i \(-0.250738\pi\)
0.705466 + 0.708744i \(0.250738\pi\)
\(878\) 6.44831e38 0.118106
\(879\) −6.85699e38 −0.123675
\(880\) −4.40260e39 −0.781976
\(881\) −5.97063e38 −0.104435 −0.0522174 0.998636i \(-0.516629\pi\)
−0.0522174 + 0.998636i \(0.516629\pi\)
\(882\) 4.38523e39 0.755382
\(883\) −7.85839e39 −1.33310 −0.666552 0.745459i \(-0.732230\pi\)
−0.666552 + 0.745459i \(0.732230\pi\)
\(884\) 1.56708e40 2.61809
\(885\) −7.75367e38 −0.127577
\(886\) −4.22635e39 −0.684874
\(887\) 5.04306e39 0.804869 0.402434 0.915449i \(-0.368164\pi\)
0.402434 + 0.915449i \(0.368164\pi\)
\(888\) 1.26384e40 1.98663
\(889\) 8.23453e38 0.127487
\(890\) 2.00260e39 0.305372
\(891\) 8.16275e38 0.122599
\(892\) 2.42557e39 0.358829
\(893\) 2.17866e39 0.317465
\(894\) 9.23157e38 0.132501
\(895\) −9.57163e38 −0.135324
\(896\) 8.39990e39 1.16981
\(897\) 1.07210e39 0.147074
\(898\) −8.44360e39 −1.14103
\(899\) 2.68911e39 0.357975
\(900\) −5.96633e39 −0.782409
\(901\) −7.49345e39 −0.968048
\(902\) −1.42022e40 −1.80745
\(903\) −5.31350e39 −0.666187
\(904\) 3.64601e40 4.50344
\(905\) 1.37242e39 0.167006
\(906\) 1.55508e40 1.86433
\(907\) −7.33863e39 −0.866796 −0.433398 0.901203i \(-0.642686\pi\)
−0.433398 + 0.901203i \(0.642686\pi\)
\(908\) −4.18977e40 −4.87564
\(909\) 4.42313e39 0.507127
\(910\) 7.55258e39 0.853172
\(911\) 7.23054e39 0.804771 0.402386 0.915470i \(-0.368181\pi\)
0.402386 + 0.915470i \(0.368181\pi\)
\(912\) −6.92708e39 −0.759661
\(913\) −7.00327e39 −0.756738
\(914\) −1.69274e40 −1.80225
\(915\) 1.20043e38 0.0125936
\(916\) 3.87140e40 4.00202
\(917\) 2.74293e40 2.79401
\(918\) 3.10945e39 0.312110
\(919\) −1.06263e40 −1.05105 −0.525523 0.850780i \(-0.676130\pi\)
−0.525523 + 0.850780i \(0.676130\pi\)
\(920\) −1.55894e39 −0.151948
\(921\) 9.08817e38 0.0872914
\(922\) −2.90448e40 −2.74917
\(923\) −1.70376e40 −1.58923
\(924\) −2.58510e40 −2.37632
\(925\) 1.25647e40 1.13825
\(926\) 8.24570e39 0.736172
\(927\) 1.26982e39 0.111729
\(928\) −1.37419e40 −1.19166
\(929\) −9.62717e39 −0.822785 −0.411392 0.911458i \(-0.634957\pi\)
−0.411392 + 0.911458i \(0.634957\pi\)
\(930\) −2.34825e39 −0.197799
\(931\) 6.87738e39 0.570956
\(932\) 4.69062e40 3.83809
\(933\) 7.41813e39 0.598263
\(934\) −1.56365e40 −1.24296
\(935\) 3.09938e39 0.242838
\(936\) −1.47084e40 −1.13591
\(937\) 1.21318e40 0.923509 0.461755 0.887008i \(-0.347220\pi\)
0.461755 + 0.887008i \(0.347220\pi\)
\(938\) 2.03891e40 1.52989
\(939\) −6.90516e39 −0.510726
\(940\) −5.88539e39 −0.429091
\(941\) −5.05393e38 −0.0363219 −0.0181610 0.999835i \(-0.505781\pi\)
−0.0181610 + 0.999835i \(0.505781\pi\)
\(942\) −6.93514e39 −0.491324
\(943\) −2.64657e39 −0.184831
\(944\) −3.51957e40 −2.42308
\(945\) 1.07159e39 0.0727273
\(946\) 2.39811e40 1.60450
\(947\) 8.58094e39 0.565992 0.282996 0.959121i \(-0.408672\pi\)
0.282996 + 0.959121i \(0.408672\pi\)
\(948\) 2.65613e40 1.72718
\(949\) −9.82392e39 −0.629783
\(950\) −1.30858e40 −0.827051
\(951\) 3.94413e39 0.245761
\(952\) −5.92324e40 −3.63882
\(953\) −2.53482e39 −0.153530 −0.0767649 0.997049i \(-0.524459\pi\)
−0.0767649 + 0.997049i \(0.524459\pi\)
\(954\) 1.16929e40 0.698264
\(955\) −5.71197e39 −0.336311
\(956\) 3.97910e40 2.30995
\(957\) 5.53833e39 0.317006
\(958\) −9.91616e38 −0.0559640
\(959\) −3.13791e40 −1.74618
\(960\) 4.54311e39 0.249282
\(961\) −8.91794e39 −0.482502
\(962\) 5.14963e40 2.74734
\(963\) −3.31448e39 −0.174365
\(964\) 4.70190e40 2.43912
\(965\) 1.82940e39 0.0935813
\(966\) −6.73704e39 −0.339842
\(967\) −2.17064e39 −0.107977 −0.0539884 0.998542i \(-0.517193\pi\)
−0.0539884 + 0.998542i \(0.517193\pi\)
\(968\) 1.25358e40 0.614943
\(969\) 4.87658e39 0.235909
\(970\) 1.59972e40 0.763177
\(971\) 2.71062e40 1.27529 0.637645 0.770330i \(-0.279909\pi\)
0.637645 + 0.770330i \(0.279909\pi\)
\(972\) −3.46947e39 −0.160979
\(973\) 4.46722e40 2.04416
\(974\) −4.88440e40 −2.20427
\(975\) −1.46226e40 −0.650824
\(976\) 5.44902e39 0.239192
\(977\) −4.41710e40 −1.91232 −0.956161 0.292841i \(-0.905399\pi\)
−0.956161 + 0.292841i \(0.905399\pi\)
\(978\) −2.29331e40 −0.979239
\(979\) −1.67994e40 −0.707502
\(980\) −1.85784e40 −0.771714
\(981\) −1.46137e40 −0.598726
\(982\) −7.29439e40 −2.94770
\(983\) −2.40915e40 −0.960266 −0.480133 0.877196i \(-0.659412\pi\)
−0.480133 + 0.877196i \(0.659412\pi\)
\(984\) 3.63091e40 1.42752
\(985\) −6.70977e39 −0.260207
\(986\) 2.10972e40 0.807026
\(987\) −1.52985e40 −0.577256
\(988\) −3.83497e40 −1.42739
\(989\) 4.46888e39 0.164077
\(990\) −4.83632e39 −0.175162
\(991\) −1.16541e39 −0.0416376 −0.0208188 0.999783i \(-0.506627\pi\)
−0.0208188 + 0.999783i \(0.506627\pi\)
\(992\) −4.88782e40 −1.72269
\(993\) 4.68589e39 0.162921
\(994\) 1.07064e41 3.67221
\(995\) 5.78527e39 0.195755
\(996\) 2.97664e40 0.993633
\(997\) 5.76017e40 1.89693 0.948463 0.316888i \(-0.102638\pi\)
0.948463 + 0.316888i \(0.102638\pi\)
\(998\) 2.41167e39 0.0783531
\(999\) 7.30646e39 0.234193
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 3.28.a.b.1.2 2
3.2 odd 2 9.28.a.b.1.1 2
4.3 odd 2 48.28.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.28.a.b.1.2 2 1.1 even 1 trivial
9.28.a.b.1.1 2 3.2 odd 2
48.28.a.e.1.2 2 4.3 odd 2