Properties

Label 10.22.a.d.1.1
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
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] = [10,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
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: \(27.9477344287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1179649}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 294912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(543.558\) 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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{2} -114848. q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} -1.17604e8 q^{6} -2.03949e8 q^{7} +1.07374e9 q^{8} +2.72970e9 q^{9} -1.00000e10 q^{10} -4.40105e10 q^{11} -1.20427e11 q^{12} -1.80893e11 q^{13} -2.08843e11 q^{14} +1.12156e12 q^{15} +1.09951e12 q^{16} +1.29311e13 q^{17} +2.79522e12 q^{18} +3.50261e13 q^{19} -1.02400e13 q^{20} +2.34231e13 q^{21} -4.50668e13 q^{22} -1.00465e14 q^{23} -1.23317e14 q^{24} +9.53674e13 q^{25} -1.85235e14 q^{26} +8.87849e14 q^{27} -2.13856e14 q^{28} +2.65416e15 q^{29} +1.14848e15 q^{30} +1.12512e15 q^{31} +1.12590e15 q^{32} +5.05452e15 q^{33} +1.32415e16 q^{34} +1.99169e15 q^{35} +2.86230e15 q^{36} +2.62794e16 q^{37} +3.58667e16 q^{38} +2.07752e16 q^{39} -1.04858e16 q^{40} -2.80940e16 q^{41} +2.39852e16 q^{42} +1.48308e17 q^{43} -4.61484e16 q^{44} -2.66573e16 q^{45} -1.02876e17 q^{46} -9.51528e16 q^{47} -1.26277e17 q^{48} -5.16951e17 q^{49} +9.76562e16 q^{50} -1.48511e18 q^{51} -1.89680e17 q^{52} +1.03378e18 q^{53} +9.09158e17 q^{54} +4.29790e17 q^{55} -2.18988e17 q^{56} -4.02268e18 q^{57} +2.71786e18 q^{58} -1.03132e18 q^{59} +1.17604e18 q^{60} -8.22371e18 q^{61} +1.15212e18 q^{62} -5.56719e17 q^{63} +1.15292e18 q^{64} +1.76653e18 q^{65} +5.17583e18 q^{66} -1.74709e19 q^{67} +1.35593e19 q^{68} +1.15382e19 q^{69} +2.03949e18 q^{70} +2.67322e19 q^{71} +2.93100e18 q^{72} +5.60167e19 q^{73} +2.69101e19 q^{74} -1.09528e19 q^{75} +3.67275e19 q^{76} +8.97588e18 q^{77} +2.12738e19 q^{78} -4.33055e19 q^{79} -1.07374e19 q^{80} -1.30521e20 q^{81} -2.87683e19 q^{82} -2.59402e20 q^{83} +2.45609e19 q^{84} -1.26280e20 q^{85} +1.51867e20 q^{86} -3.04825e20 q^{87} -4.72559e19 q^{88} +5.09045e20 q^{89} -2.72970e19 q^{90} +3.68929e19 q^{91} -1.05345e20 q^{92} -1.29218e20 q^{93} -9.74364e19 q^{94} -3.42052e20 q^{95} -1.29307e20 q^{96} +6.58070e20 q^{97} -5.29358e20 q^{98} -1.20136e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2048 q^{2} + 30972 q^{3} + 2097152 q^{4} - 19531250 q^{5} + 31715328 q^{6} - 439959356 q^{7} + 2147483648 q^{8} + 13532817186 q^{9} - 20000000000 q^{10} + 105191777184 q^{11} + 32476495872 q^{12}+ \cdots + 14\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1024.00 0.707107
\(3\) −114848. −1.12292 −0.561462 0.827503i \(-0.689761\pi\)
−0.561462 + 0.827503i \(0.689761\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −9.76562e6 −0.447214
\(6\) −1.17604e8 −0.794027
\(7\) −2.03949e8 −0.272892 −0.136446 0.990647i \(-0.543568\pi\)
−0.136446 + 0.990647i \(0.543568\pi\)
\(8\) 1.07374e9 0.353553
\(9\) 2.72970e9 0.260957
\(10\) −1.00000e10 −0.316228
\(11\) −4.40105e10 −0.511603 −0.255802 0.966729i \(-0.582339\pi\)
−0.255802 + 0.966729i \(0.582339\pi\)
\(12\) −1.20427e11 −0.561462
\(13\) −1.80893e11 −0.363929 −0.181965 0.983305i \(-0.558246\pi\)
−0.181965 + 0.983305i \(0.558246\pi\)
\(14\) −2.08843e11 −0.192964
\(15\) 1.12156e12 0.502187
\(16\) 1.09951e12 0.250000
\(17\) 1.29311e13 1.55569 0.777844 0.628458i \(-0.216314\pi\)
0.777844 + 0.628458i \(0.216314\pi\)
\(18\) 2.79522e12 0.184525
\(19\) 3.50261e13 1.31063 0.655313 0.755357i \(-0.272537\pi\)
0.655313 + 0.755357i \(0.272537\pi\)
\(20\) −1.02400e13 −0.223607
\(21\) 2.34231e13 0.306437
\(22\) −4.50668e13 −0.361758
\(23\) −1.00465e14 −0.505677 −0.252838 0.967509i \(-0.581364\pi\)
−0.252838 + 0.967509i \(0.581364\pi\)
\(24\) −1.23317e14 −0.397013
\(25\) 9.53674e13 0.200000
\(26\) −1.85235e14 −0.257337
\(27\) 8.87849e14 0.829889
\(28\) −2.13856e14 −0.136446
\(29\) 2.65416e15 1.17152 0.585758 0.810486i \(-0.300797\pi\)
0.585758 + 0.810486i \(0.300797\pi\)
\(30\) 1.14848e15 0.355100
\(31\) 1.12512e15 0.246548 0.123274 0.992373i \(-0.460661\pi\)
0.123274 + 0.992373i \(0.460661\pi\)
\(32\) 1.12590e15 0.176777
\(33\) 5.05452e15 0.574491
\(34\) 1.32415e16 1.10004
\(35\) 1.99169e15 0.122041
\(36\) 2.86230e15 0.130479
\(37\) 2.62794e16 0.898458 0.449229 0.893417i \(-0.351699\pi\)
0.449229 + 0.893417i \(0.351699\pi\)
\(38\) 3.58667e16 0.926753
\(39\) 2.07752e16 0.408665
\(40\) −1.04858e16 −0.158114
\(41\) −2.80940e16 −0.326876 −0.163438 0.986554i \(-0.552258\pi\)
−0.163438 + 0.986554i \(0.552258\pi\)
\(42\) 2.39852e16 0.216684
\(43\) 1.48308e17 1.04651 0.523257 0.852175i \(-0.324717\pi\)
0.523257 + 0.852175i \(0.324717\pi\)
\(44\) −4.61484e16 −0.255802
\(45\) −2.66573e16 −0.116704
\(46\) −1.02876e17 −0.357567
\(47\) −9.51528e16 −0.263872 −0.131936 0.991258i \(-0.542119\pi\)
−0.131936 + 0.991258i \(0.542119\pi\)
\(48\) −1.26277e17 −0.280731
\(49\) −5.16951e17 −0.925530
\(50\) 9.76562e16 0.141421
\(51\) −1.48511e18 −1.74692
\(52\) −1.89680e17 −0.181965
\(53\) 1.03378e18 0.811955 0.405978 0.913883i \(-0.366931\pi\)
0.405978 + 0.913883i \(0.366931\pi\)
\(54\) 9.09158e17 0.586820
\(55\) 4.29790e17 0.228796
\(56\) −2.18988e17 −0.0964820
\(57\) −4.02268e18 −1.47173
\(58\) 2.71786e18 0.828387
\(59\) −1.03132e18 −0.262692 −0.131346 0.991337i \(-0.541930\pi\)
−0.131346 + 0.991337i \(0.541930\pi\)
\(60\) 1.17604e18 0.251093
\(61\) −8.22371e18 −1.47606 −0.738031 0.674767i \(-0.764244\pi\)
−0.738031 + 0.674767i \(0.764244\pi\)
\(62\) 1.15212e18 0.174335
\(63\) −5.56719e17 −0.0712132
\(64\) 1.15292e18 0.125000
\(65\) 1.76653e18 0.162754
\(66\) 5.17583e18 0.406227
\(67\) −1.74709e19 −1.17092 −0.585462 0.810700i \(-0.699087\pi\)
−0.585462 + 0.810700i \(0.699087\pi\)
\(68\) 1.35593e19 0.777844
\(69\) 1.15382e19 0.567836
\(70\) 2.03949e18 0.0862961
\(71\) 2.67322e19 0.974591 0.487296 0.873237i \(-0.337983\pi\)
0.487296 + 0.873237i \(0.337983\pi\)
\(72\) 2.93100e18 0.0922623
\(73\) 5.60167e19 1.52555 0.762777 0.646662i \(-0.223836\pi\)
0.762777 + 0.646662i \(0.223836\pi\)
\(74\) 2.69101e19 0.635306
\(75\) −1.09528e19 −0.224585
\(76\) 3.67275e19 0.655313
\(77\) 8.97588e18 0.139613
\(78\) 2.12738e19 0.288970
\(79\) −4.33055e19 −0.514587 −0.257293 0.966333i \(-0.582831\pi\)
−0.257293 + 0.966333i \(0.582831\pi\)
\(80\) −1.07374e19 −0.111803
\(81\) −1.30521e20 −1.19286
\(82\) −2.87683e19 −0.231137
\(83\) −2.59402e20 −1.83507 −0.917535 0.397655i \(-0.869824\pi\)
−0.917535 + 0.397655i \(0.869824\pi\)
\(84\) 2.45609e19 0.153219
\(85\) −1.26280e20 −0.695724
\(86\) 1.51867e20 0.739997
\(87\) −3.04825e20 −1.31552
\(88\) −4.72559e19 −0.180879
\(89\) 5.09045e20 1.73046 0.865230 0.501376i \(-0.167173\pi\)
0.865230 + 0.501376i \(0.167173\pi\)
\(90\) −2.72970e19 −0.0825219
\(91\) 3.68929e19 0.0993135
\(92\) −1.05345e20 −0.252838
\(93\) −1.29218e20 −0.276854
\(94\) −9.74364e19 −0.186586
\(95\) −3.42052e20 −0.586130
\(96\) −1.29307e20 −0.198507
\(97\) 6.58070e20 0.906085 0.453042 0.891489i \(-0.350339\pi\)
0.453042 + 0.891489i \(0.350339\pi\)
\(98\) −5.29358e20 −0.654448
\(99\) −1.20136e20 −0.133507
\(100\) 1.00000e20 0.100000
\(101\) 1.15672e21 1.04197 0.520984 0.853566i \(-0.325565\pi\)
0.520984 + 0.853566i \(0.325565\pi\)
\(102\) −1.52076e21 −1.23526
\(103\) 2.10354e21 1.54226 0.771132 0.636675i \(-0.219691\pi\)
0.771132 + 0.636675i \(0.219691\pi\)
\(104\) −1.94233e20 −0.128668
\(105\) −2.28741e20 −0.137043
\(106\) 1.05859e21 0.574139
\(107\) 9.86090e20 0.484604 0.242302 0.970201i \(-0.422098\pi\)
0.242302 + 0.970201i \(0.422098\pi\)
\(108\) 9.30978e20 0.414944
\(109\) 1.78098e21 0.720576 0.360288 0.932841i \(-0.382678\pi\)
0.360288 + 0.932841i \(0.382678\pi\)
\(110\) 4.40105e20 0.161783
\(111\) −3.01814e21 −1.00890
\(112\) −2.24244e20 −0.0682231
\(113\) 3.61184e21 1.00093 0.500466 0.865756i \(-0.333162\pi\)
0.500466 + 0.865756i \(0.333162\pi\)
\(114\) −4.11922e21 −1.04067
\(115\) 9.81104e20 0.226145
\(116\) 2.78309e21 0.585758
\(117\) −4.93785e20 −0.0949700
\(118\) −1.05607e21 −0.185751
\(119\) −2.63728e21 −0.424535
\(120\) 1.20427e21 0.177550
\(121\) −5.46332e21 −0.738262
\(122\) −8.42108e21 −1.04373
\(123\) 3.22654e21 0.367057
\(124\) 1.17977e21 0.123274
\(125\) −9.31323e20 −0.0894427
\(126\) −5.70081e20 −0.0503553
\(127\) 1.58229e22 1.28632 0.643158 0.765734i \(-0.277624\pi\)
0.643158 + 0.765734i \(0.277624\pi\)
\(128\) 1.18059e21 0.0883883
\(129\) −1.70328e22 −1.17516
\(130\) 1.80893e21 0.115085
\(131\) −1.83709e22 −1.07840 −0.539202 0.842176i \(-0.681274\pi\)
−0.539202 + 0.842176i \(0.681274\pi\)
\(132\) 5.30005e21 0.287246
\(133\) −7.14352e21 −0.357660
\(134\) −1.78902e22 −0.827968
\(135\) −8.67040e21 −0.371137
\(136\) 1.38847e22 0.550019
\(137\) 7.05876e21 0.258918 0.129459 0.991585i \(-0.458676\pi\)
0.129459 + 0.991585i \(0.458676\pi\)
\(138\) 1.18151e22 0.401521
\(139\) −3.37498e22 −1.06320 −0.531601 0.846995i \(-0.678410\pi\)
−0.531601 + 0.846995i \(0.678410\pi\)
\(140\) 2.08843e21 0.0610206
\(141\) 1.09281e22 0.296308
\(142\) 2.73738e22 0.689140
\(143\) 7.96120e21 0.186187
\(144\) 3.00134e21 0.0652393
\(145\) −2.59195e22 −0.523918
\(146\) 5.73611e22 1.07873
\(147\) 5.93708e22 1.03930
\(148\) 2.75560e22 0.449229
\(149\) −5.68451e22 −0.863451 −0.431725 0.902005i \(-0.642095\pi\)
−0.431725 + 0.902005i \(0.642095\pi\)
\(150\) −1.12156e22 −0.158805
\(151\) 1.33443e23 1.76213 0.881064 0.472997i \(-0.156828\pi\)
0.881064 + 0.472997i \(0.156828\pi\)
\(152\) 3.76090e22 0.463376
\(153\) 3.52981e22 0.405968
\(154\) 9.19131e21 0.0987210
\(155\) −1.09875e22 −0.110259
\(156\) 2.17844e22 0.204332
\(157\) −1.56723e22 −0.137463 −0.0687317 0.997635i \(-0.521895\pi\)
−0.0687317 + 0.997635i \(0.521895\pi\)
\(158\) −4.43448e22 −0.363868
\(159\) −1.18728e23 −0.911764
\(160\) −1.09951e22 −0.0790569
\(161\) 2.04897e22 0.137995
\(162\) −1.33654e23 −0.843478
\(163\) 1.20617e23 0.713575 0.356788 0.934185i \(-0.383872\pi\)
0.356788 + 0.934185i \(0.383872\pi\)
\(164\) −2.94587e22 −0.163438
\(165\) −4.93605e22 −0.256920
\(166\) −2.65627e23 −1.29759
\(167\) −3.63531e23 −1.66732 −0.833658 0.552281i \(-0.813758\pi\)
−0.833658 + 0.552281i \(0.813758\pi\)
\(168\) 2.51503e22 0.108342
\(169\) −2.14342e23 −0.867555
\(170\) −1.29311e23 −0.491952
\(171\) 9.56109e22 0.342017
\(172\) 1.55512e23 0.523257
\(173\) 5.25008e23 1.66219 0.831097 0.556127i \(-0.187713\pi\)
0.831097 + 0.556127i \(0.187713\pi\)
\(174\) −3.12141e23 −0.930215
\(175\) −1.94501e22 −0.0545785
\(176\) −4.83901e22 −0.127901
\(177\) 1.18445e23 0.294983
\(178\) 5.21262e23 1.22362
\(179\) −4.28059e23 −0.947428 −0.473714 0.880679i \(-0.657087\pi\)
−0.473714 + 0.880679i \(0.657087\pi\)
\(180\) −2.79522e22 −0.0583518
\(181\) 4.13461e23 0.814346 0.407173 0.913351i \(-0.366515\pi\)
0.407173 + 0.913351i \(0.366515\pi\)
\(182\) 3.77783e22 0.0702252
\(183\) 9.44477e23 1.65750
\(184\) −1.07874e23 −0.178784
\(185\) −2.56635e23 −0.401803
\(186\) −1.32319e23 −0.195765
\(187\) −5.69105e23 −0.795895
\(188\) −9.97749e22 −0.131936
\(189\) −1.81076e23 −0.226470
\(190\) −3.50261e23 −0.414456
\(191\) 7.69492e22 0.0861695 0.0430847 0.999071i \(-0.486281\pi\)
0.0430847 + 0.999071i \(0.486281\pi\)
\(192\) −1.32411e23 −0.140365
\(193\) 1.25195e24 1.25671 0.628355 0.777926i \(-0.283728\pi\)
0.628355 + 0.777926i \(0.283728\pi\)
\(194\) 6.73864e23 0.640699
\(195\) −2.02883e23 −0.182760
\(196\) −5.42062e23 −0.462765
\(197\) 9.01149e23 0.729292 0.364646 0.931146i \(-0.381190\pi\)
0.364646 + 0.931146i \(0.381190\pi\)
\(198\) −1.23019e23 −0.0944034
\(199\) −9.25260e23 −0.673452 −0.336726 0.941603i \(-0.609320\pi\)
−0.336726 + 0.941603i \(0.609320\pi\)
\(200\) 1.02400e23 0.0707107
\(201\) 2.00649e24 1.31486
\(202\) 1.18448e24 0.736783
\(203\) −5.41312e23 −0.319698
\(204\) −1.55725e24 −0.873459
\(205\) 2.74356e23 0.146184
\(206\) 2.15402e24 1.09055
\(207\) −2.74240e23 −0.131960
\(208\) −1.98894e23 −0.0909823
\(209\) −1.54152e24 −0.670521
\(210\) −2.34231e23 −0.0969040
\(211\) −9.91626e23 −0.390285 −0.195143 0.980775i \(-0.562517\pi\)
−0.195143 + 0.980775i \(0.562517\pi\)
\(212\) 1.08400e24 0.405978
\(213\) −3.07014e24 −1.09439
\(214\) 1.00976e24 0.342667
\(215\) −1.44832e24 −0.468015
\(216\) 9.53321e23 0.293410
\(217\) −2.29466e23 −0.0672809
\(218\) 1.82372e24 0.509524
\(219\) −6.43340e24 −1.71308
\(220\) 4.50668e23 0.114398
\(221\) −2.33915e24 −0.566160
\(222\) −3.09057e24 −0.713400
\(223\) −3.99965e24 −0.880687 −0.440343 0.897829i \(-0.645143\pi\)
−0.440343 + 0.897829i \(0.645143\pi\)
\(224\) −2.29626e23 −0.0482410
\(225\) 2.60325e23 0.0521914
\(226\) 3.69852e24 0.707766
\(227\) 5.28899e24 0.966276 0.483138 0.875544i \(-0.339497\pi\)
0.483138 + 0.875544i \(0.339497\pi\)
\(228\) −4.21808e24 −0.735866
\(229\) −1.05761e24 −0.176219 −0.0881095 0.996111i \(-0.528083\pi\)
−0.0881095 + 0.996111i \(0.528083\pi\)
\(230\) 1.00465e24 0.159909
\(231\) −1.03086e24 −0.156774
\(232\) 2.84988e24 0.414193
\(233\) 1.05916e25 1.47138 0.735691 0.677317i \(-0.236857\pi\)
0.735691 + 0.677317i \(0.236857\pi\)
\(234\) −5.05636e23 −0.0671539
\(235\) 9.29226e23 0.118007
\(236\) −1.08142e24 −0.131346
\(237\) 4.97355e24 0.577841
\(238\) −2.70058e24 −0.300192
\(239\) 1.72062e25 1.83023 0.915115 0.403192i \(-0.132099\pi\)
0.915115 + 0.403192i \(0.132099\pi\)
\(240\) 1.23317e24 0.125547
\(241\) −1.73077e25 −1.68679 −0.843394 0.537295i \(-0.819446\pi\)
−0.843394 + 0.537295i \(0.819446\pi\)
\(242\) −5.59444e24 −0.522030
\(243\) 5.70290e24 0.509600
\(244\) −8.62319e24 −0.738031
\(245\) 5.04835e24 0.413910
\(246\) 3.30398e24 0.259549
\(247\) −6.33598e24 −0.476975
\(248\) 1.20809e24 0.0871677
\(249\) 2.97917e25 2.06064
\(250\) −9.53674e23 −0.0632456
\(251\) −9.29670e24 −0.591227 −0.295614 0.955308i \(-0.595524\pi\)
−0.295614 + 0.955308i \(0.595524\pi\)
\(252\) −5.83763e23 −0.0356066
\(253\) 4.42152e24 0.258706
\(254\) 1.62027e25 0.909562
\(255\) 1.45031e25 0.781245
\(256\) 1.20893e24 0.0625000
\(257\) 9.32271e24 0.462641 0.231321 0.972878i \(-0.425695\pi\)
0.231321 + 0.972878i \(0.425695\pi\)
\(258\) −1.74416e25 −0.830960
\(259\) −5.35965e24 −0.245182
\(260\) 1.85235e24 0.0813771
\(261\) 7.24508e24 0.305716
\(262\) −1.88118e25 −0.762547
\(263\) −2.87230e25 −1.11865 −0.559325 0.828948i \(-0.688940\pi\)
−0.559325 + 0.828948i \(0.688940\pi\)
\(264\) 5.42725e24 0.203113
\(265\) −1.00955e25 −0.363117
\(266\) −7.31497e24 −0.252904
\(267\) −5.84628e25 −1.94317
\(268\) −1.83195e25 −0.585462
\(269\) 2.41759e25 0.742991 0.371495 0.928435i \(-0.378845\pi\)
0.371495 + 0.928435i \(0.378845\pi\)
\(270\) −8.87849e24 −0.262434
\(271\) −4.48847e25 −1.27621 −0.638103 0.769951i \(-0.720280\pi\)
−0.638103 + 0.769951i \(0.720280\pi\)
\(272\) 1.42179e25 0.388922
\(273\) −4.23708e24 −0.111521
\(274\) 7.22817e24 0.183083
\(275\) −4.19717e24 −0.102321
\(276\) 1.20987e25 0.283918
\(277\) −1.57653e25 −0.356176 −0.178088 0.984015i \(-0.556991\pi\)
−0.178088 + 0.984015i \(0.556991\pi\)
\(278\) −3.45598e25 −0.751798
\(279\) 3.07124e24 0.0643384
\(280\) 2.13856e24 0.0431481
\(281\) 9.40593e25 1.82804 0.914019 0.405671i \(-0.132962\pi\)
0.914019 + 0.405671i \(0.132962\pi\)
\(282\) 1.11904e25 0.209522
\(283\) −4.41232e25 −0.795993 −0.397997 0.917387i \(-0.630294\pi\)
−0.397997 + 0.917387i \(0.630294\pi\)
\(284\) 2.80308e25 0.487296
\(285\) 3.92839e25 0.658179
\(286\) 8.15227e24 0.131654
\(287\) 5.72974e24 0.0892021
\(288\) 3.07337e24 0.0461312
\(289\) 9.81218e25 1.42016
\(290\) −2.65416e25 −0.370466
\(291\) −7.55780e25 −1.01746
\(292\) 5.87378e25 0.762777
\(293\) 2.98120e25 0.373492 0.186746 0.982408i \(-0.440206\pi\)
0.186746 + 0.982408i \(0.440206\pi\)
\(294\) 6.07957e25 0.734895
\(295\) 1.00715e25 0.117479
\(296\) 2.82173e25 0.317653
\(297\) −3.90747e25 −0.424574
\(298\) −5.82094e25 −0.610552
\(299\) 1.81734e25 0.184031
\(300\) −1.14848e25 −0.112292
\(301\) −3.02472e25 −0.285586
\(302\) 1.36645e26 1.24601
\(303\) −1.32847e26 −1.17005
\(304\) 3.85116e25 0.327657
\(305\) 8.03097e25 0.660115
\(306\) 3.61453e25 0.287063
\(307\) −4.58514e25 −0.351884 −0.175942 0.984401i \(-0.556297\pi\)
−0.175942 + 0.984401i \(0.556297\pi\)
\(308\) 9.41190e24 0.0698063
\(309\) −2.41587e26 −1.73185
\(310\) −1.12512e25 −0.0779652
\(311\) −1.30419e26 −0.873691 −0.436845 0.899537i \(-0.643904\pi\)
−0.436845 + 0.899537i \(0.643904\pi\)
\(312\) 2.23072e25 0.144485
\(313\) 2.68379e26 1.68087 0.840434 0.541914i \(-0.182300\pi\)
0.840434 + 0.541914i \(0.182300\pi\)
\(314\) −1.60485e25 −0.0972013
\(315\) 5.43671e24 0.0318475
\(316\) −4.54091e25 −0.257293
\(317\) 1.49751e26 0.820820 0.410410 0.911901i \(-0.365386\pi\)
0.410410 + 0.911901i \(0.365386\pi\)
\(318\) −1.21577e26 −0.644714
\(319\) −1.16811e26 −0.599351
\(320\) −1.12590e25 −0.0559017
\(321\) −1.13250e26 −0.544173
\(322\) 2.09815e25 0.0975774
\(323\) 4.52926e26 2.03892
\(324\) −1.36862e26 −0.596429
\(325\) −1.72513e25 −0.0727859
\(326\) 1.23512e26 0.504574
\(327\) −2.04541e26 −0.809151
\(328\) −3.01657e25 −0.115568
\(329\) 1.94063e25 0.0720087
\(330\) −5.05452e25 −0.181670
\(331\) −2.76489e25 −0.0962685 −0.0481343 0.998841i \(-0.515328\pi\)
−0.0481343 + 0.998841i \(0.515328\pi\)
\(332\) −2.72002e26 −0.917535
\(333\) 7.17351e25 0.234459
\(334\) −3.72255e26 −1.17897
\(335\) 1.70614e26 0.523653
\(336\) 2.57540e25 0.0766093
\(337\) 2.02980e25 0.0585247 0.0292624 0.999572i \(-0.490684\pi\)
0.0292624 + 0.999572i \(0.490684\pi\)
\(338\) −2.19486e26 −0.613454
\(339\) −4.14812e26 −1.12397
\(340\) −1.32415e26 −0.347862
\(341\) −4.95171e25 −0.126135
\(342\) 9.79055e25 0.241843
\(343\) 2.19346e26 0.525462
\(344\) 1.59244e26 0.369999
\(345\) −1.12678e26 −0.253944
\(346\) 5.37608e26 1.17535
\(347\) 2.67942e26 0.568304 0.284152 0.958779i \(-0.408288\pi\)
0.284152 + 0.958779i \(0.408288\pi\)
\(348\) −3.19632e26 −0.657761
\(349\) −5.63379e26 −1.12495 −0.562475 0.826814i \(-0.690151\pi\)
−0.562475 + 0.826814i \(0.690151\pi\)
\(350\) −1.99169e25 −0.0385928
\(351\) −1.60606e26 −0.302021
\(352\) −4.95514e25 −0.0904395
\(353\) 1.00657e27 1.78324 0.891622 0.452781i \(-0.149568\pi\)
0.891622 + 0.452781i \(0.149568\pi\)
\(354\) 1.21288e26 0.208585
\(355\) −2.61057e26 −0.435850
\(356\) 5.33773e26 0.865230
\(357\) 3.02887e26 0.476720
\(358\) −4.38332e26 −0.669933
\(359\) 1.04937e27 1.55754 0.778768 0.627312i \(-0.215845\pi\)
0.778768 + 0.627312i \(0.215845\pi\)
\(360\) −2.86230e25 −0.0412610
\(361\) 5.12617e26 0.717741
\(362\) 4.23384e26 0.575830
\(363\) 6.27452e26 0.829012
\(364\) 3.86850e25 0.0496568
\(365\) −5.47038e26 −0.682248
\(366\) 9.67144e26 1.17203
\(367\) −5.00538e26 −0.589445 −0.294723 0.955583i \(-0.595227\pi\)
−0.294723 + 0.955583i \(0.595227\pi\)
\(368\) −1.10463e26 −0.126419
\(369\) −7.66884e25 −0.0853008
\(370\) −2.62794e26 −0.284117
\(371\) −2.10838e26 −0.221576
\(372\) −1.35495e26 −0.138427
\(373\) 5.47776e26 0.544077 0.272039 0.962286i \(-0.412302\pi\)
0.272039 + 0.962286i \(0.412302\pi\)
\(374\) −5.82764e26 −0.562783
\(375\) 1.06961e26 0.100437
\(376\) −1.02169e26 −0.0932930
\(377\) −4.80120e26 −0.426349
\(378\) −1.85421e26 −0.160139
\(379\) −1.26713e27 −1.06441 −0.532205 0.846615i \(-0.678637\pi\)
−0.532205 + 0.846615i \(0.678637\pi\)
\(380\) −3.58667e26 −0.293065
\(381\) −1.81723e27 −1.44443
\(382\) 7.87959e25 0.0609310
\(383\) −5.06606e26 −0.381139 −0.190570 0.981674i \(-0.561033\pi\)
−0.190570 + 0.981674i \(0.561033\pi\)
\(384\) −1.35589e26 −0.0992534
\(385\) −8.76551e25 −0.0624367
\(386\) 1.28200e27 0.888629
\(387\) 4.04836e26 0.273095
\(388\) 6.90036e26 0.453042
\(389\) −1.46115e27 −0.933738 −0.466869 0.884327i \(-0.654618\pi\)
−0.466869 + 0.884327i \(0.654618\pi\)
\(390\) −2.07752e26 −0.129231
\(391\) −1.29913e27 −0.786675
\(392\) −5.55072e26 −0.327224
\(393\) 2.10986e27 1.21097
\(394\) 9.22777e26 0.515687
\(395\) 4.22905e26 0.230130
\(396\) −1.25971e26 −0.0667533
\(397\) −3.60855e27 −1.86223 −0.931113 0.364730i \(-0.881161\pi\)
−0.931113 + 0.364730i \(0.881161\pi\)
\(398\) −9.47466e26 −0.476202
\(399\) 8.20419e26 0.401625
\(400\) 1.04858e26 0.0500000
\(401\) −3.67006e25 −0.0170474 −0.00852369 0.999964i \(-0.502713\pi\)
−0.00852369 + 0.999964i \(0.502713\pi\)
\(402\) 2.05465e27 0.929745
\(403\) −2.03526e26 −0.0897259
\(404\) 1.21291e27 0.520984
\(405\) 1.27462e27 0.533463
\(406\) −5.54304e26 −0.226060
\(407\) −1.15657e27 −0.459654
\(408\) −1.59463e27 −0.617629
\(409\) −2.39045e27 −0.902371 −0.451185 0.892430i \(-0.648999\pi\)
−0.451185 + 0.892430i \(0.648999\pi\)
\(410\) 2.80940e26 0.103367
\(411\) −8.10684e26 −0.290745
\(412\) 2.20572e27 0.771132
\(413\) 2.10336e26 0.0716866
\(414\) −2.80822e26 −0.0933098
\(415\) 2.53322e27 0.820668
\(416\) −2.03668e26 −0.0643342
\(417\) 3.87610e27 1.19390
\(418\) −1.57851e27 −0.474130
\(419\) 6.65037e27 1.94804 0.974021 0.226458i \(-0.0727145\pi\)
0.974021 + 0.226458i \(0.0727145\pi\)
\(420\) −2.39852e26 −0.0685214
\(421\) 5.66789e26 0.157928 0.0789641 0.996877i \(-0.474839\pi\)
0.0789641 + 0.996877i \(0.474839\pi\)
\(422\) −1.01543e27 −0.275973
\(423\) −2.59739e26 −0.0688594
\(424\) 1.11001e27 0.287069
\(425\) 1.23321e27 0.311137
\(426\) −3.14383e27 −0.773851
\(427\) 1.67721e27 0.402806
\(428\) 1.03399e27 0.242302
\(429\) −9.14328e26 −0.209074
\(430\) −1.48308e27 −0.330937
\(431\) 3.51896e27 0.766307 0.383153 0.923685i \(-0.374838\pi\)
0.383153 + 0.923685i \(0.374838\pi\)
\(432\) 9.76201e26 0.207472
\(433\) 1.55391e27 0.322332 0.161166 0.986927i \(-0.448474\pi\)
0.161166 + 0.986927i \(0.448474\pi\)
\(434\) −2.34974e26 −0.0475748
\(435\) 2.97681e27 0.588320
\(436\) 1.86749e27 0.360288
\(437\) −3.51890e27 −0.662753
\(438\) −6.58781e27 −1.21133
\(439\) 5.58352e27 1.00238 0.501188 0.865338i \(-0.332897\pi\)
0.501188 + 0.865338i \(0.332897\pi\)
\(440\) 4.61484e26 0.0808916
\(441\) −1.41112e27 −0.241524
\(442\) −2.39529e27 −0.400336
\(443\) −6.81633e27 −1.11253 −0.556265 0.831005i \(-0.687766\pi\)
−0.556265 + 0.831005i \(0.687766\pi\)
\(444\) −3.16475e27 −0.504450
\(445\) −4.97115e27 −0.773885
\(446\) −4.09565e27 −0.622739
\(447\) 6.52855e27 0.969589
\(448\) −2.35137e26 −0.0341115
\(449\) −3.94071e27 −0.558455 −0.279227 0.960225i \(-0.590078\pi\)
−0.279227 + 0.960225i \(0.590078\pi\)
\(450\) 2.66573e26 0.0369049
\(451\) 1.23643e27 0.167231
\(452\) 3.78729e27 0.500466
\(453\) −1.53256e28 −1.97873
\(454\) 5.41593e27 0.683261
\(455\) −3.60282e26 −0.0444143
\(456\) −4.31931e27 −0.520336
\(457\) −2.47274e26 −0.0291111 −0.0145555 0.999894i \(-0.504633\pi\)
−0.0145555 + 0.999894i \(0.504633\pi\)
\(458\) −1.08299e27 −0.124606
\(459\) 1.14809e28 1.29105
\(460\) 1.02876e27 0.113073
\(461\) −6.40047e27 −0.687625 −0.343813 0.939038i \(-0.611719\pi\)
−0.343813 + 0.939038i \(0.611719\pi\)
\(462\) −1.05560e27 −0.110856
\(463\) −8.60733e26 −0.0883625 −0.0441812 0.999024i \(-0.514068\pi\)
−0.0441812 + 0.999024i \(0.514068\pi\)
\(464\) 2.91828e27 0.292879
\(465\) 1.26189e27 0.123813
\(466\) 1.08458e28 1.04042
\(467\) −1.25054e28 −1.17292 −0.586461 0.809977i \(-0.699479\pi\)
−0.586461 + 0.809977i \(0.699479\pi\)
\(468\) −5.17771e26 −0.0474850
\(469\) 3.56316e27 0.319536
\(470\) 9.51528e26 0.0834438
\(471\) 1.79993e27 0.154361
\(472\) −1.10737e27 −0.0928756
\(473\) −6.52710e27 −0.535400
\(474\) 5.09291e27 0.408596
\(475\) 3.34035e27 0.262125
\(476\) −2.76539e27 −0.212268
\(477\) 2.82192e27 0.211886
\(478\) 1.76191e28 1.29417
\(479\) 9.68180e27 0.695718 0.347859 0.937547i \(-0.386909\pi\)
0.347859 + 0.937547i \(0.386909\pi\)
\(480\) 1.26277e27 0.0887749
\(481\) −4.75377e27 −0.326975
\(482\) −1.77231e28 −1.19274
\(483\) −2.35320e27 −0.154958
\(484\) −5.72871e27 −0.369131
\(485\) −6.42646e27 −0.405213
\(486\) 5.83977e27 0.360342
\(487\) −4.62318e27 −0.279182 −0.139591 0.990209i \(-0.544579\pi\)
−0.139591 + 0.990209i \(0.544579\pi\)
\(488\) −8.83014e27 −0.521867
\(489\) −1.38527e28 −0.801291
\(490\) 5.16951e27 0.292678
\(491\) −2.14634e28 −1.18944 −0.594719 0.803934i \(-0.702737\pi\)
−0.594719 + 0.803934i \(0.702737\pi\)
\(492\) 3.38328e27 0.183529
\(493\) 3.43213e28 1.82251
\(494\) −6.48804e27 −0.337272
\(495\) 1.17320e27 0.0597060
\(496\) 1.23708e27 0.0616369
\(497\) −5.45200e27 −0.265958
\(498\) 3.05067e28 1.45709
\(499\) 2.68376e28 1.25513 0.627564 0.778565i \(-0.284052\pi\)
0.627564 + 0.778565i \(0.284052\pi\)
\(500\) −9.76563e26 −0.0447214
\(501\) 4.17508e28 1.87227
\(502\) −9.51982e27 −0.418061
\(503\) −1.25016e27 −0.0537651 −0.0268826 0.999639i \(-0.508558\pi\)
−0.0268826 + 0.999639i \(0.508558\pi\)
\(504\) −5.97773e26 −0.0251777
\(505\) −1.12961e28 −0.465983
\(506\) 4.52764e27 0.182933
\(507\) 2.46168e28 0.974198
\(508\) 1.65915e28 0.643158
\(509\) −2.39131e28 −0.908029 −0.454014 0.890994i \(-0.650009\pi\)
−0.454014 + 0.890994i \(0.650009\pi\)
\(510\) 1.48511e28 0.552424
\(511\) −1.14245e28 −0.416312
\(512\) 1.23794e27 0.0441942
\(513\) 3.10979e28 1.08767
\(514\) 9.54645e27 0.327137
\(515\) −2.05423e28 −0.689722
\(516\) −1.78602e28 −0.587578
\(517\) 4.18772e27 0.134998
\(518\) −5.48828e27 −0.173370
\(519\) −6.02961e28 −1.86652
\(520\) 1.89680e27 0.0575423
\(521\) 1.24898e28 0.371328 0.185664 0.982613i \(-0.440556\pi\)
0.185664 + 0.982613i \(0.440556\pi\)
\(522\) 7.41896e27 0.216174
\(523\) −6.85611e28 −1.95799 −0.978995 0.203886i \(-0.934643\pi\)
−0.978995 + 0.203886i \(0.934643\pi\)
\(524\) −1.92633e28 −0.539202
\(525\) 2.23380e27 0.0612874
\(526\) −2.94123e28 −0.791005
\(527\) 1.45490e28 0.383551
\(528\) 5.55750e27 0.143623
\(529\) −2.93783e28 −0.744291
\(530\) −1.03378e28 −0.256763
\(531\) −2.81520e27 −0.0685514
\(532\) −7.49052e27 −0.178830
\(533\) 5.08202e27 0.118960
\(534\) −5.98659e28 −1.37403
\(535\) −9.62978e27 −0.216721
\(536\) −1.87592e28 −0.413984
\(537\) 4.91617e28 1.06389
\(538\) 2.47561e28 0.525374
\(539\) 2.27513e28 0.473504
\(540\) −9.09158e27 −0.185569
\(541\) 6.77012e28 1.35527 0.677634 0.735399i \(-0.263005\pi\)
0.677634 + 0.735399i \(0.263005\pi\)
\(542\) −4.59619e28 −0.902413
\(543\) −4.74851e28 −0.914449
\(544\) 1.45591e28 0.275009
\(545\) −1.73923e28 −0.322251
\(546\) −4.33877e27 −0.0788576
\(547\) −1.66829e28 −0.297443 −0.148721 0.988879i \(-0.547516\pi\)
−0.148721 + 0.988879i \(0.547516\pi\)
\(548\) 7.40164e27 0.129459
\(549\) −2.24483e28 −0.385189
\(550\) −4.29790e27 −0.0723516
\(551\) 9.29649e28 1.53542
\(552\) 1.23891e28 0.200760
\(553\) 8.83209e27 0.140427
\(554\) −1.61437e28 −0.251854
\(555\) 2.94740e28 0.451194
\(556\) −3.53893e28 −0.531601
\(557\) 9.39997e28 1.38563 0.692815 0.721116i \(-0.256370\pi\)
0.692815 + 0.721116i \(0.256370\pi\)
\(558\) 3.14495e27 0.0454941
\(559\) −2.68279e28 −0.380857
\(560\) 2.18988e27 0.0305103
\(561\) 6.53606e28 0.893729
\(562\) 9.63167e28 1.29262
\(563\) 5.21769e28 0.687290 0.343645 0.939100i \(-0.388338\pi\)
0.343645 + 0.939100i \(0.388338\pi\)
\(564\) 1.14589e28 0.148154
\(565\) −3.52718e28 −0.447630
\(566\) −4.51821e28 −0.562852
\(567\) 2.66197e28 0.325522
\(568\) 2.87035e28 0.344570
\(569\) 9.80844e27 0.115590 0.0577951 0.998328i \(-0.481593\pi\)
0.0577951 + 0.998328i \(0.481593\pi\)
\(570\) 4.02268e28 0.465403
\(571\) 4.43638e28 0.503906 0.251953 0.967740i \(-0.418927\pi\)
0.251953 + 0.967740i \(0.418927\pi\)
\(572\) 8.34793e27 0.0930937
\(573\) −8.83745e27 −0.0967617
\(574\) 5.86725e27 0.0630754
\(575\) −9.58110e27 −0.101135
\(576\) 3.14714e27 0.0326197
\(577\) −1.56720e29 −1.59507 −0.797533 0.603276i \(-0.793862\pi\)
−0.797533 + 0.603276i \(0.793862\pi\)
\(578\) 1.00477e29 1.00421
\(579\) −1.43784e29 −1.41119
\(580\) −2.71786e28 −0.261959
\(581\) 5.29046e28 0.500776
\(582\) −7.73919e28 −0.719456
\(583\) −4.54973e28 −0.415399
\(584\) 6.01475e28 0.539365
\(585\) 4.82212e27 0.0424719
\(586\) 3.05275e28 0.264099
\(587\) −1.20878e29 −1.02718 −0.513592 0.858035i \(-0.671685\pi\)
−0.513592 + 0.858035i \(0.671685\pi\)
\(588\) 6.22548e28 0.519650
\(589\) 3.94085e28 0.323132
\(590\) 1.03132e28 0.0830705
\(591\) −1.03495e29 −0.818939
\(592\) 2.88945e28 0.224614
\(593\) −3.14301e28 −0.240033 −0.120017 0.992772i \(-0.538295\pi\)
−0.120017 + 0.992772i \(0.538295\pi\)
\(594\) −4.00125e28 −0.300219
\(595\) 2.57547e28 0.189858
\(596\) −5.96064e28 −0.431725
\(597\) 1.06264e29 0.756235
\(598\) 1.86096e28 0.130129
\(599\) 1.87438e29 1.28788 0.643942 0.765075i \(-0.277298\pi\)
0.643942 + 0.765075i \(0.277298\pi\)
\(600\) −1.17604e28 −0.0794027
\(601\) −2.65601e29 −1.76217 −0.881085 0.472958i \(-0.843186\pi\)
−0.881085 + 0.472958i \(0.843186\pi\)
\(602\) −3.09731e28 −0.201940
\(603\) −4.76903e28 −0.305561
\(604\) 1.39925e29 0.881064
\(605\) 5.33528e28 0.330161
\(606\) −1.36036e29 −0.827351
\(607\) −2.33120e29 −1.39347 −0.696737 0.717326i \(-0.745366\pi\)
−0.696737 + 0.717326i \(0.745366\pi\)
\(608\) 3.94359e28 0.231688
\(609\) 6.21686e28 0.358996
\(610\) 8.22371e28 0.466772
\(611\) 1.72125e28 0.0960309
\(612\) 3.70128e28 0.202984
\(613\) −5.29871e28 −0.285651 −0.142825 0.989748i \(-0.545619\pi\)
−0.142825 + 0.989748i \(0.545619\pi\)
\(614\) −4.69519e28 −0.248820
\(615\) −3.15092e28 −0.164153
\(616\) 9.63778e27 0.0493605
\(617\) −9.06542e28 −0.456450 −0.228225 0.973608i \(-0.573292\pi\)
−0.228225 + 0.973608i \(0.573292\pi\)
\(618\) −2.47385e29 −1.22460
\(619\) 1.52550e29 0.742441 0.371220 0.928545i \(-0.378939\pi\)
0.371220 + 0.928545i \(0.378939\pi\)
\(620\) −1.15212e28 −0.0551297
\(621\) −8.91979e28 −0.419655
\(622\) −1.33549e29 −0.617793
\(623\) −1.03819e29 −0.472229
\(624\) 2.28426e28 0.102166
\(625\) 9.09495e27 0.0400000
\(626\) 2.74820e29 1.18855
\(627\) 1.77040e29 0.752943
\(628\) −1.64336e28 −0.0687317
\(629\) 3.39822e29 1.39772
\(630\) 5.56719e27 0.0225196
\(631\) −3.07474e29 −1.22321 −0.611604 0.791164i \(-0.709475\pi\)
−0.611604 + 0.791164i \(0.709475\pi\)
\(632\) −4.64989e28 −0.181934
\(633\) 1.13886e29 0.438260
\(634\) 1.53345e29 0.580407
\(635\) −1.54521e29 −0.575258
\(636\) −1.24495e29 −0.455882
\(637\) 9.35129e28 0.336827
\(638\) −1.19614e29 −0.423806
\(639\) 7.29711e28 0.254327
\(640\) −1.15292e28 −0.0395285
\(641\) 4.26974e29 1.44010 0.720049 0.693923i \(-0.244119\pi\)
0.720049 + 0.693923i \(0.244119\pi\)
\(642\) −1.15968e29 −0.384788
\(643\) −1.96406e29 −0.641120 −0.320560 0.947228i \(-0.603871\pi\)
−0.320560 + 0.947228i \(0.603871\pi\)
\(644\) 2.14850e28 0.0689976
\(645\) 1.66336e29 0.525545
\(646\) 4.63797e29 1.44174
\(647\) 4.70139e29 1.43791 0.718955 0.695057i \(-0.244621\pi\)
0.718955 + 0.695057i \(0.244621\pi\)
\(648\) −1.40146e29 −0.421739
\(649\) 4.53889e28 0.134394
\(650\) −1.76653e28 −0.0514674
\(651\) 2.63538e28 0.0755513
\(652\) 1.26476e29 0.356788
\(653\) −3.28351e29 −0.911485 −0.455742 0.890112i \(-0.650626\pi\)
−0.455742 + 0.890112i \(0.650626\pi\)
\(654\) −2.09450e29 −0.572156
\(655\) 1.79403e29 0.482277
\(656\) −3.08897e28 −0.0817191
\(657\) 1.52909e29 0.398104
\(658\) 1.98720e28 0.0509179
\(659\) −5.89725e28 −0.148714 −0.0743571 0.997232i \(-0.523690\pi\)
−0.0743571 + 0.997232i \(0.523690\pi\)
\(660\) −5.17583e28 −0.128460
\(661\) −2.99109e29 −0.730657 −0.365328 0.930879i \(-0.619043\pi\)
−0.365328 + 0.930879i \(0.619043\pi\)
\(662\) −2.83125e28 −0.0680721
\(663\) 2.68647e29 0.635755
\(664\) −2.78530e29 −0.648795
\(665\) 6.97609e28 0.159950
\(666\) 7.34567e28 0.165788
\(667\) −2.66651e29 −0.592408
\(668\) −3.81190e29 −0.833658
\(669\) 4.59352e29 0.988944
\(670\) 1.74709e29 0.370279
\(671\) 3.61930e29 0.755158
\(672\) 2.63720e28 0.0541710
\(673\) −4.08136e29 −0.825366 −0.412683 0.910875i \(-0.635408\pi\)
−0.412683 + 0.910875i \(0.635408\pi\)
\(674\) 2.07852e28 0.0413832
\(675\) 8.46719e28 0.165978
\(676\) −2.24754e29 −0.433778
\(677\) −6.17392e28 −0.117322 −0.0586611 0.998278i \(-0.518683\pi\)
−0.0586611 + 0.998278i \(0.518683\pi\)
\(678\) −4.24768e29 −0.794767
\(679\) −1.34212e29 −0.247264
\(680\) −1.35593e29 −0.245976
\(681\) −6.07430e29 −1.08505
\(682\) −5.07055e28 −0.0891906
\(683\) 8.86519e29 1.53557 0.767787 0.640706i \(-0.221358\pi\)
0.767787 + 0.640706i \(0.221358\pi\)
\(684\) 1.00255e29 0.171009
\(685\) −6.89332e28 −0.115792
\(686\) 2.24610e29 0.371558
\(687\) 1.21464e29 0.197880
\(688\) 1.63066e29 0.261629
\(689\) −1.87004e29 −0.295494
\(690\) −1.15382e29 −0.179566
\(691\) −3.58171e29 −0.548998 −0.274499 0.961587i \(-0.588512\pi\)
−0.274499 + 0.961587i \(0.588512\pi\)
\(692\) 5.50511e29 0.831097
\(693\) 2.45015e28 0.0364329
\(694\) 2.74372e29 0.401851
\(695\) 3.29588e29 0.475479
\(696\) −3.27303e29 −0.465108
\(697\) −3.63287e29 −0.508518
\(698\) −5.76900e29 −0.795460
\(699\) −1.21643e30 −1.65225
\(700\) −2.03949e28 −0.0272892
\(701\) −9.30503e29 −1.22653 −0.613265 0.789877i \(-0.710144\pi\)
−0.613265 + 0.789877i \(0.710144\pi\)
\(702\) −1.64460e29 −0.213561
\(703\) 9.20465e29 1.17754
\(704\) −5.07407e28 −0.0639504
\(705\) −1.06720e29 −0.132513
\(706\) 1.03073e30 1.26094
\(707\) −2.35912e29 −0.284345
\(708\) 1.24198e29 0.147492
\(709\) −1.29300e30 −1.51290 −0.756452 0.654049i \(-0.773069\pi\)
−0.756452 + 0.654049i \(0.773069\pi\)
\(710\) −2.67322e29 −0.308193
\(711\) −1.18211e29 −0.134285
\(712\) 5.46583e29 0.611810
\(713\) −1.13035e29 −0.124673
\(714\) 3.10156e29 0.337092
\(715\) −7.77461e28 −0.0832655
\(716\) −4.48852e29 −0.473714
\(717\) −1.97609e30 −2.05521
\(718\) 1.07456e30 1.10134
\(719\) −3.29773e29 −0.333090 −0.166545 0.986034i \(-0.553261\pi\)
−0.166545 + 0.986034i \(0.553261\pi\)
\(720\) −2.93100e28 −0.0291759
\(721\) −4.29013e29 −0.420872
\(722\) 5.24920e29 0.507519
\(723\) 1.98776e30 1.89413
\(724\) 4.33545e29 0.407173
\(725\) 2.53121e29 0.234303
\(726\) 6.42511e29 0.586200
\(727\) 1.06829e30 0.960677 0.480338 0.877083i \(-0.340514\pi\)
0.480338 + 0.877083i \(0.340514\pi\)
\(728\) 3.96135e28 0.0351126
\(729\) 7.10333e29 0.620616
\(730\) −5.60167e29 −0.482422
\(731\) 1.91778e30 1.62805
\(732\) 9.90356e29 0.828752
\(733\) 1.28132e30 1.05698 0.528490 0.848939i \(-0.322758\pi\)
0.528490 + 0.848939i \(0.322758\pi\)
\(734\) −5.12551e29 −0.416801
\(735\) −5.79793e29 −0.464789
\(736\) −1.13114e29 −0.0893919
\(737\) 7.68901e29 0.599049
\(738\) −7.85289e28 −0.0603168
\(739\) −1.68393e29 −0.127514 −0.0637569 0.997965i \(-0.520308\pi\)
−0.0637569 + 0.997965i \(0.520308\pi\)
\(740\) −2.69101e29 −0.200901
\(741\) 7.27674e29 0.535607
\(742\) −2.15898e29 −0.156678
\(743\) 5.78502e28 0.0413925 0.0206963 0.999786i \(-0.493412\pi\)
0.0206963 + 0.999786i \(0.493412\pi\)
\(744\) −1.38746e29 −0.0978827
\(745\) 5.55128e29 0.386147
\(746\) 5.60923e29 0.384721
\(747\) −7.08090e29 −0.478875
\(748\) −5.96750e29 −0.397947
\(749\) −2.01112e29 −0.132245
\(750\) 1.09528e29 0.0710199
\(751\) 1.16126e30 0.742526 0.371263 0.928528i \(-0.378925\pi\)
0.371263 + 0.928528i \(0.378925\pi\)
\(752\) −1.04622e29 −0.0659681
\(753\) 1.06771e30 0.663903
\(754\) −4.91642e29 −0.301474
\(755\) −1.30315e30 −0.788048
\(756\) −1.89872e29 −0.113235
\(757\) −2.47468e30 −1.45550 −0.727749 0.685843i \(-0.759434\pi\)
−0.727749 + 0.685843i \(0.759434\pi\)
\(758\) −1.29754e30 −0.752652
\(759\) −5.07803e29 −0.290507
\(760\) −3.67275e29 −0.207228
\(761\) 8.23029e29 0.458011 0.229006 0.973425i \(-0.426453\pi\)
0.229006 + 0.973425i \(0.426453\pi\)
\(762\) −1.86084e30 −1.02137
\(763\) −3.63227e29 −0.196640
\(764\) 8.06870e28 0.0430847
\(765\) −3.44708e29 −0.181554
\(766\) −5.18765e29 −0.269506
\(767\) 1.86559e29 0.0956013
\(768\) −1.38843e29 −0.0701827
\(769\) −8.22507e29 −0.410122 −0.205061 0.978749i \(-0.565739\pi\)
−0.205061 + 0.978749i \(0.565739\pi\)
\(770\) −8.97588e28 −0.0441494
\(771\) −1.07069e30 −0.519511
\(772\) 1.31276e30 0.628355
\(773\) −4.08869e30 −1.93063 −0.965316 0.261083i \(-0.915920\pi\)
−0.965316 + 0.261083i \(0.915920\pi\)
\(774\) 4.14552e29 0.193108
\(775\) 1.07300e29 0.0493095
\(776\) 7.06597e29 0.320349
\(777\) 6.15545e29 0.275321
\(778\) −1.49622e30 −0.660252
\(779\) −9.84024e29 −0.428413
\(780\) −2.12738e29 −0.0913802
\(781\) −1.17650e30 −0.498604
\(782\) −1.33030e30 −0.556263
\(783\) 2.35649e30 0.972228
\(784\) −5.68393e29 −0.231382
\(785\) 1.53050e29 0.0614755
\(786\) 2.16050e30 0.856282
\(787\) 2.27937e30 0.891414 0.445707 0.895179i \(-0.352952\pi\)
0.445707 + 0.895179i \(0.352952\pi\)
\(788\) 9.44924e29 0.364646
\(789\) 3.29878e30 1.25616
\(790\) 4.33055e29 0.162727
\(791\) −7.36629e29 −0.273147
\(792\) −1.28995e29 −0.0472017
\(793\) 1.48761e30 0.537182
\(794\) −3.69516e30 −1.31679
\(795\) 1.15945e30 0.407753
\(796\) −9.70205e29 −0.336726
\(797\) 3.26331e30 1.11775 0.558877 0.829250i \(-0.311232\pi\)
0.558877 + 0.829250i \(0.311232\pi\)
\(798\) 8.40109e29 0.283991
\(799\) −1.23043e30 −0.410503
\(800\) 1.07374e29 0.0353553
\(801\) 1.38954e30 0.451576
\(802\) −3.75814e28 −0.0120543
\(803\) −2.46532e30 −0.780478
\(804\) 2.10396e30 0.657429
\(805\) −2.00095e29 −0.0617134
\(806\) −2.08411e29 −0.0634458
\(807\) −2.77655e30 −0.834322
\(808\) 1.24202e30 0.368392
\(809\) 1.06157e30 0.310805 0.155403 0.987851i \(-0.450333\pi\)
0.155403 + 0.987851i \(0.450333\pi\)
\(810\) 1.30521e30 0.377215
\(811\) −5.07809e30 −1.44871 −0.724355 0.689428i \(-0.757862\pi\)
−0.724355 + 0.689428i \(0.757862\pi\)
\(812\) −5.67607e29 −0.159849
\(813\) 5.15491e30 1.43308
\(814\) −1.18433e30 −0.325025
\(815\) −1.17790e30 −0.319121
\(816\) −1.63290e30 −0.436729
\(817\) 5.19464e30 1.37159
\(818\) −2.44782e30 −0.638073
\(819\) 1.00707e29 0.0259166
\(820\) 2.87683e29 0.0730918
\(821\) −3.24366e30 −0.813640 −0.406820 0.913508i \(-0.633362\pi\)
−0.406820 + 0.913508i \(0.633362\pi\)
\(822\) −8.30141e29 −0.205588
\(823\) 1.47953e30 0.361764 0.180882 0.983505i \(-0.442105\pi\)
0.180882 + 0.983505i \(0.442105\pi\)
\(824\) 2.25865e30 0.545273
\(825\) 4.82037e29 0.114898
\(826\) 2.15384e29 0.0506901
\(827\) 1.06680e30 0.247898 0.123949 0.992289i \(-0.460444\pi\)
0.123949 + 0.992289i \(0.460444\pi\)
\(828\) −2.87562e29 −0.0659800
\(829\) 1.37112e30 0.310637 0.155319 0.987864i \(-0.450360\pi\)
0.155319 + 0.987864i \(0.450360\pi\)
\(830\) 2.59402e30 0.580300
\(831\) 1.81061e30 0.399958
\(832\) −2.08556e29 −0.0454912
\(833\) −6.68475e30 −1.43983
\(834\) 3.96913e30 0.844212
\(835\) 3.55010e30 0.745647
\(836\) −1.61640e30 −0.335260
\(837\) 9.98936e29 0.204607
\(838\) 6.80997e30 1.37747
\(839\) 7.78595e29 0.155529 0.0777645 0.996972i \(-0.475222\pi\)
0.0777645 + 0.996972i \(0.475222\pi\)
\(840\) −2.45609e29 −0.0484520
\(841\) 1.91173e30 0.372450
\(842\) 5.80392e29 0.111672
\(843\) −1.08025e31 −2.05275
\(844\) −1.03980e30 −0.195143
\(845\) 2.09319e30 0.387983
\(846\) −2.65973e29 −0.0486909
\(847\) 1.11424e30 0.201466
\(848\) 1.13665e30 0.202989
\(849\) 5.06746e30 0.893839
\(850\) 1.26280e30 0.220007
\(851\) −2.64016e30 −0.454329
\(852\) −3.21928e30 −0.547196
\(853\) −5.07514e30 −0.852086 −0.426043 0.904703i \(-0.640093\pi\)
−0.426043 + 0.904703i \(0.640093\pi\)
\(854\) 1.71747e30 0.284827
\(855\) −9.33700e29 −0.152955
\(856\) 1.05881e30 0.171333
\(857\) −7.65635e30 −1.22384 −0.611918 0.790921i \(-0.709602\pi\)
−0.611918 + 0.790921i \(0.709602\pi\)
\(858\) −9.36272e29 −0.147838
\(859\) 6.42591e30 1.00232 0.501160 0.865355i \(-0.332907\pi\)
0.501160 + 0.865355i \(0.332907\pi\)
\(860\) −1.51867e30 −0.234008
\(861\) −6.58049e29 −0.100167
\(862\) 3.60341e30 0.541861
\(863\) −1.00027e31 −1.48594 −0.742972 0.669322i \(-0.766585\pi\)
−0.742972 + 0.669322i \(0.766585\pi\)
\(864\) 9.99629e29 0.146705
\(865\) −5.12703e30 −0.743356
\(866\) 1.59121e30 0.227923
\(867\) −1.12691e31 −1.59473
\(868\) −2.40613e29 −0.0336405
\(869\) 1.90590e30 0.263264
\(870\) 3.04825e30 0.416005
\(871\) 3.16036e30 0.426134
\(872\) 1.91231e30 0.254762
\(873\) 1.79634e30 0.236449
\(874\) −3.60335e30 −0.468637
\(875\) 1.89942e29 0.0244082
\(876\) −6.74591e30 −0.856540
\(877\) 1.69135e30 0.212197 0.106098 0.994356i \(-0.466164\pi\)
0.106098 + 0.994356i \(0.466164\pi\)
\(878\) 5.71752e30 0.708787
\(879\) −3.42385e30 −0.419403
\(880\) 4.72559e29 0.0571990
\(881\) 7.04503e30 0.842628 0.421314 0.906915i \(-0.361569\pi\)
0.421314 + 0.906915i \(0.361569\pi\)
\(882\) −1.44499e30 −0.170783
\(883\) −7.12748e30 −0.832431 −0.416216 0.909266i \(-0.636644\pi\)
−0.416216 + 0.909266i \(0.636644\pi\)
\(884\) −2.45278e30 −0.283080
\(885\) −1.15669e30 −0.131920
\(886\) −6.97992e30 −0.786677
\(887\) 7.82765e30 0.871834 0.435917 0.899987i \(-0.356424\pi\)
0.435917 + 0.899987i \(0.356424\pi\)
\(888\) −3.24070e30 −0.356700
\(889\) −3.22706e30 −0.351026
\(890\) −5.09045e30 −0.547219
\(891\) 5.74431e30 0.610270
\(892\) −4.19394e30 −0.440343
\(893\) −3.33283e30 −0.345838
\(894\) 6.68523e30 0.685603
\(895\) 4.18026e30 0.423703
\(896\) −2.40780e29 −0.0241205
\(897\) −2.08718e30 −0.206652
\(898\) −4.03529e30 −0.394887
\(899\) 2.98625e30 0.288834
\(900\) 2.72970e29 0.0260957
\(901\) 1.33679e31 1.26315
\(902\) 1.26611e30 0.118250
\(903\) 3.47383e30 0.320691
\(904\) 3.87818e30 0.353883
\(905\) −4.03770e30 −0.364187
\(906\) −1.56935e31 −1.39918
\(907\) 4.42371e30 0.389862 0.194931 0.980817i \(-0.437552\pi\)
0.194931 + 0.980817i \(0.437552\pi\)
\(908\) 5.54591e30 0.483138
\(909\) 3.15751e30 0.271909
\(910\) −3.68929e29 −0.0314057
\(911\) 1.34581e31 1.13251 0.566253 0.824231i \(-0.308392\pi\)
0.566253 + 0.824231i \(0.308392\pi\)
\(912\) −4.42298e30 −0.367933
\(913\) 1.14164e31 0.938828
\(914\) −2.53208e29 −0.0205846
\(915\) −9.22341e30 −0.741259
\(916\) −1.10898e30 −0.0881095
\(917\) 3.74672e30 0.294288
\(918\) 1.17564e31 0.912908
\(919\) 2.07587e31 1.59363 0.796815 0.604223i \(-0.206517\pi\)
0.796815 + 0.604223i \(0.206517\pi\)
\(920\) 1.05345e30 0.0799545
\(921\) 5.26594e30 0.395139
\(922\) −6.55408e30 −0.486224
\(923\) −4.83568e30 −0.354682
\(924\) −1.08094e30 −0.0783871
\(925\) 2.50620e30 0.179692
\(926\) −8.81390e29 −0.0624817
\(927\) 5.74203e30 0.402465
\(928\) 2.98832e30 0.207097
\(929\) −2.58824e30 −0.177353 −0.0886767 0.996060i \(-0.528264\pi\)
−0.0886767 + 0.996060i \(0.528264\pi\)
\(930\) 1.29218e30 0.0875489
\(931\) −1.81068e31 −1.21302
\(932\) 1.11061e31 0.735691
\(933\) 1.49784e31 0.981088
\(934\) −1.28055e31 −0.829381
\(935\) 5.55767e30 0.355935
\(936\) −5.30198e29 −0.0335770
\(937\) −1.82163e31 −1.14076 −0.570381 0.821380i \(-0.693205\pi\)
−0.570381 + 0.821380i \(0.693205\pi\)
\(938\) 3.64867e30 0.225946
\(939\) −3.08228e31 −1.88749
\(940\) 9.74364e29 0.0590037
\(941\) 1.59208e31 0.953394 0.476697 0.879068i \(-0.341834\pi\)
0.476697 + 0.879068i \(0.341834\pi\)
\(942\) 1.84313e30 0.109150
\(943\) 2.82247e30 0.165294
\(944\) −1.13395e30 −0.0656730
\(945\) 1.76832e30 0.101281
\(946\) −6.68375e30 −0.378585
\(947\) 6.90254e30 0.386664 0.193332 0.981133i \(-0.438071\pi\)
0.193332 + 0.981133i \(0.438071\pi\)
\(948\) 5.21514e30 0.288921
\(949\) −1.01330e31 −0.555194
\(950\) 3.42052e30 0.185351
\(951\) −1.71986e31 −0.921718
\(952\) −2.83176e30 −0.150096
\(953\) 8.59868e30 0.450771 0.225386 0.974270i \(-0.427636\pi\)
0.225386 + 0.974270i \(0.427636\pi\)
\(954\) 2.88964e30 0.149826
\(955\) −7.51457e29 −0.0385362
\(956\) 1.80420e31 0.915115
\(957\) 1.34155e31 0.673026
\(958\) 9.91416e30 0.491947
\(959\) −1.43962e30 −0.0706567
\(960\) 1.29307e30 0.0627733
\(961\) −1.95596e31 −0.939214
\(962\) −4.86786e30 −0.231206
\(963\) 2.69173e30 0.126461
\(964\) −1.81484e31 −0.843394
\(965\) −1.22261e31 −0.562018
\(966\) −2.40968e30 −0.109572
\(967\) −2.71571e30 −0.122153 −0.0610766 0.998133i \(-0.519453\pi\)
−0.0610766 + 0.998133i \(0.519453\pi\)
\(968\) −5.86620e30 −0.261015
\(969\) −5.20177e31 −2.28956
\(970\) −6.58070e30 −0.286529
\(971\) −2.52216e30 −0.108635 −0.0543175 0.998524i \(-0.517298\pi\)
−0.0543175 + 0.998524i \(0.517298\pi\)
\(972\) 5.97992e30 0.254800
\(973\) 6.88323e30 0.290140
\(974\) −4.73414e30 −0.197411
\(975\) 1.98128e30 0.0817329
\(976\) −9.04207e30 −0.369016
\(977\) −3.27075e30 −0.132055 −0.0660274 0.997818i \(-0.521032\pi\)
−0.0660274 + 0.997818i \(0.521032\pi\)
\(978\) −1.41851e31 −0.566598
\(979\) −2.24034e31 −0.885309
\(980\) 5.29358e30 0.206955
\(981\) 4.86154e30 0.188039
\(982\) −2.19785e31 −0.841060
\(983\) −1.41284e29 −0.00534910 −0.00267455 0.999996i \(-0.500851\pi\)
−0.00267455 + 0.999996i \(0.500851\pi\)
\(984\) 3.46447e30 0.129774
\(985\) −8.80029e30 −0.326149
\(986\) 3.51450e31 1.28871
\(987\) −2.22877e30 −0.0808603
\(988\) −6.64376e30 −0.238488
\(989\) −1.48998e31 −0.529198
\(990\) 1.20136e30 0.0422185
\(991\) 2.73270e31 0.950209 0.475105 0.879929i \(-0.342410\pi\)
0.475105 + 0.879929i \(0.342410\pi\)
\(992\) 1.26677e30 0.0435839
\(993\) 3.17542e30 0.108102
\(994\) −5.58285e30 −0.188061
\(995\) 9.03574e30 0.301177
\(996\) 3.12389e31 1.03032
\(997\) 2.01068e31 0.656213 0.328106 0.944641i \(-0.393590\pi\)
0.328106 + 0.944641i \(0.393590\pi\)
\(998\) 2.74817e31 0.887509
\(999\) 2.33322e31 0.745620
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 10.22.a.d.1.1 2
4.3 odd 2 80.22.a.c.1.2 2
5.2 odd 4 50.22.b.e.49.4 4
5.3 odd 4 50.22.b.e.49.1 4
5.4 even 2 50.22.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.d.1.1 2 1.1 even 1 trivial
50.22.a.d.1.2 2 5.4 even 2
50.22.b.e.49.1 4 5.3 odd 4
50.22.b.e.49.4 4 5.2 odd 4
80.22.a.c.1.2 2 4.3 odd 2