Properties

Label 10.22.a.b.1.1
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{157921}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 39480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(199.196\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -174616. q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} +1.78807e8 q^{6} -9.60462e8 q^{7} -1.07374e9 q^{8} +2.00304e10 q^{9} +1.00000e10 q^{10} +8.54661e10 q^{11} -1.83098e11 q^{12} +9.74302e11 q^{13} +9.83513e11 q^{14} +1.70523e12 q^{15} +1.09951e12 q^{16} -1.17045e13 q^{17} -2.05111e13 q^{18} +1.41128e13 q^{19} -1.02400e13 q^{20} +1.67712e14 q^{21} -8.75173e13 q^{22} -2.65862e13 q^{23} +1.87492e14 q^{24} +9.53674e13 q^{25} -9.97685e14 q^{26} -1.67108e15 q^{27} -1.00712e15 q^{28} +1.45075e15 q^{29} -1.74616e15 q^{30} -7.63253e15 q^{31} -1.12590e15 q^{32} -1.49237e16 q^{33} +1.19854e16 q^{34} +9.37951e15 q^{35} +2.10034e16 q^{36} +1.09912e16 q^{37} -1.44515e16 q^{38} -1.70129e17 q^{39} +1.04858e16 q^{40} +7.93639e16 q^{41} -1.71737e17 q^{42} -8.36751e16 q^{43} +8.96177e16 q^{44} -1.95609e17 q^{45} +2.72243e16 q^{46} +3.57127e17 q^{47} -1.91992e17 q^{48} +3.63941e17 q^{49} -9.76562e16 q^{50} +2.04379e18 q^{51} +1.02163e18 q^{52} -8.17016e17 q^{53} +1.71119e18 q^{54} -8.34630e17 q^{55} +1.03129e18 q^{56} -2.46432e18 q^{57} -1.48557e18 q^{58} +8.21011e17 q^{59} +1.78807e18 q^{60} +4.53683e18 q^{61} +7.81571e18 q^{62} -1.92384e19 q^{63} +1.15292e18 q^{64} -9.51466e18 q^{65} +1.52819e19 q^{66} +8.02826e18 q^{67} -1.22730e19 q^{68} +4.64238e18 q^{69} -9.60462e18 q^{70} -5.25728e19 q^{71} -2.15075e19 q^{72} +9.28698e18 q^{73} -1.12549e19 q^{74} -1.66527e19 q^{75} +1.47984e19 q^{76} -8.20869e19 q^{77} +1.74212e20 q^{78} +1.08861e19 q^{79} -1.07374e19 q^{80} +8.22724e19 q^{81} -8.12687e19 q^{82} -4.79994e19 q^{83} +1.75859e20 q^{84} +1.14302e20 q^{85} +8.56833e19 q^{86} -2.53325e20 q^{87} -9.17685e19 q^{88} +2.35270e20 q^{89} +2.00304e20 q^{90} -9.35779e20 q^{91} -2.78777e19 q^{92} +1.33276e21 q^{93} -3.65698e20 q^{94} -1.37820e20 q^{95} +1.96600e20 q^{96} +7.25967e20 q^{97} -3.72675e20 q^{98} +1.71192e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} - 126692 q^{3} + 2097152 q^{4} - 19531250 q^{5} + 129732608 q^{6} - 292598684 q^{7} - 2147483648 q^{8} + 11866737826 q^{9} + 20000000000 q^{10} - 41326831776 q^{11} - 132846190592 q^{12}+ \cdots + 27\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −174616. −1.70730 −0.853652 0.520844i \(-0.825617\pi\)
−0.853652 + 0.520844i \(0.825617\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −9.76562e6 −0.447214
\(6\) 1.78807e8 1.20725
\(7\) −9.60462e8 −1.28514 −0.642570 0.766227i \(-0.722132\pi\)
−0.642570 + 0.766227i \(0.722132\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 2.00304e10 1.91489
\(10\) 1.00000e10 0.316228
\(11\) 8.54661e10 0.993507 0.496753 0.867892i \(-0.334525\pi\)
0.496753 + 0.867892i \(0.334525\pi\)
\(12\) −1.83098e11 −0.853652
\(13\) 9.74302e11 1.96015 0.980073 0.198640i \(-0.0636523\pi\)
0.980073 + 0.198640i \(0.0636523\pi\)
\(14\) 9.83513e11 0.908732
\(15\) 1.70523e12 0.763529
\(16\) 1.09951e12 0.250000
\(17\) −1.17045e13 −1.40812 −0.704058 0.710143i \(-0.748630\pi\)
−0.704058 + 0.710143i \(0.748630\pi\)
\(18\) −2.05111e13 −1.35403
\(19\) 1.41128e13 0.528081 0.264041 0.964512i \(-0.414945\pi\)
0.264041 + 0.964512i \(0.414945\pi\)
\(20\) −1.02400e13 −0.223607
\(21\) 1.67712e14 2.19413
\(22\) −8.75173e13 −0.702515
\(23\) −2.65862e13 −0.133818 −0.0669090 0.997759i \(-0.521314\pi\)
−0.0669090 + 0.997759i \(0.521314\pi\)
\(24\) 1.87492e14 0.603623
\(25\) 9.53674e13 0.200000
\(26\) −9.97685e14 −1.38603
\(27\) −1.67108e15 −1.56199
\(28\) −1.00712e15 −0.642570
\(29\) 1.45075e15 0.640346 0.320173 0.947359i \(-0.396259\pi\)
0.320173 + 0.947359i \(0.396259\pi\)
\(30\) −1.74616e15 −0.539897
\(31\) −7.63253e15 −1.67252 −0.836259 0.548335i \(-0.815262\pi\)
−0.836259 + 0.548335i \(0.815262\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) −1.49237e16 −1.69622
\(34\) 1.19854e16 0.995688
\(35\) 9.37951e15 0.574732
\(36\) 2.10034e16 0.957443
\(37\) 1.09912e16 0.375773 0.187886 0.982191i \(-0.439836\pi\)
0.187886 + 0.982191i \(0.439836\pi\)
\(38\) −1.44515e16 −0.373410
\(39\) −1.70129e17 −3.34656
\(40\) 1.04858e16 0.158114
\(41\) 7.93639e16 0.923406 0.461703 0.887035i \(-0.347239\pi\)
0.461703 + 0.887035i \(0.347239\pi\)
\(42\) −1.71737e17 −1.55148
\(43\) −8.36751e16 −0.590442 −0.295221 0.955429i \(-0.595393\pi\)
−0.295221 + 0.955429i \(0.595393\pi\)
\(44\) 8.96177e16 0.496753
\(45\) −1.95609e17 −0.856363
\(46\) 2.72243e16 0.0946236
\(47\) 3.57127e17 0.990365 0.495183 0.868789i \(-0.335101\pi\)
0.495183 + 0.868789i \(0.335101\pi\)
\(48\) −1.91992e17 −0.426826
\(49\) 3.63941e17 0.651586
\(50\) −9.76562e16 −0.141421
\(51\) 2.04379e18 2.40408
\(52\) 1.02163e18 0.980073
\(53\) −8.17016e17 −0.641702 −0.320851 0.947130i \(-0.603969\pi\)
−0.320851 + 0.947130i \(0.603969\pi\)
\(54\) 1.71119e18 1.10449
\(55\) −8.34630e17 −0.444310
\(56\) 1.03129e18 0.454366
\(57\) −2.46432e18 −0.901595
\(58\) −1.48557e18 −0.452793
\(59\) 8.21011e17 0.209124 0.104562 0.994518i \(-0.466656\pi\)
0.104562 + 0.994518i \(0.466656\pi\)
\(60\) 1.78807e18 0.381765
\(61\) 4.53683e18 0.814310 0.407155 0.913359i \(-0.366521\pi\)
0.407155 + 0.913359i \(0.366521\pi\)
\(62\) 7.81571e18 1.18265
\(63\) −1.92384e19 −2.46090
\(64\) 1.15292e18 0.125000
\(65\) −9.51466e18 −0.876604
\(66\) 1.52819e19 1.19941
\(67\) 8.02826e18 0.538066 0.269033 0.963131i \(-0.413296\pi\)
0.269033 + 0.963131i \(0.413296\pi\)
\(68\) −1.22730e19 −0.704058
\(69\) 4.64238e18 0.228468
\(70\) −9.60462e18 −0.406397
\(71\) −5.25728e19 −1.91668 −0.958338 0.285638i \(-0.907794\pi\)
−0.958338 + 0.285638i \(0.907794\pi\)
\(72\) −2.15075e19 −0.677014
\(73\) 9.28698e18 0.252921 0.126460 0.991972i \(-0.459638\pi\)
0.126460 + 0.991972i \(0.459638\pi\)
\(74\) −1.12549e19 −0.265711
\(75\) −1.66527e19 −0.341461
\(76\) 1.47984e19 0.264041
\(77\) −8.20869e19 −1.27680
\(78\) 1.74212e20 2.36638
\(79\) 1.08861e19 0.129356 0.0646781 0.997906i \(-0.479398\pi\)
0.0646781 + 0.997906i \(0.479398\pi\)
\(80\) −1.07374e19 −0.111803
\(81\) 8.22724e19 0.751902
\(82\) −8.12687e19 −0.652947
\(83\) −4.79994e19 −0.339560 −0.169780 0.985482i \(-0.554306\pi\)
−0.169780 + 0.985482i \(0.554306\pi\)
\(84\) 1.75859e20 1.09706
\(85\) 1.14302e20 0.629728
\(86\) 8.56833e19 0.417506
\(87\) −2.53325e20 −1.09327
\(88\) −9.17685e19 −0.351258
\(89\) 2.35270e20 0.799783 0.399891 0.916563i \(-0.369048\pi\)
0.399891 + 0.916563i \(0.369048\pi\)
\(90\) 2.00304e20 0.605540
\(91\) −9.35779e20 −2.51906
\(92\) −2.78777e19 −0.0669090
\(93\) 1.33276e21 2.85549
\(94\) −3.65698e20 −0.700294
\(95\) −1.37820e20 −0.236165
\(96\) 1.96600e20 0.301812
\(97\) 7.25967e20 0.999571 0.499786 0.866149i \(-0.333412\pi\)
0.499786 + 0.866149i \(0.333412\pi\)
\(98\) −3.72675e20 −0.460741
\(99\) 1.71192e21 1.90245
\(100\) 1.00000e20 0.100000
\(101\) 3.23940e20 0.291803 0.145902 0.989299i \(-0.453392\pi\)
0.145902 + 0.989299i \(0.453392\pi\)
\(102\) −2.09284e21 −1.69994
\(103\) 5.76074e20 0.422364 0.211182 0.977447i \(-0.432269\pi\)
0.211182 + 0.977447i \(0.432269\pi\)
\(104\) −1.04615e21 −0.693016
\(105\) −1.63781e21 −0.981243
\(106\) 8.36624e20 0.453752
\(107\) −3.09648e21 −1.52174 −0.760868 0.648907i \(-0.775226\pi\)
−0.760868 + 0.648907i \(0.775226\pi\)
\(108\) −1.75225e21 −0.780994
\(109\) −2.84500e21 −1.15108 −0.575539 0.817775i \(-0.695208\pi\)
−0.575539 + 0.817775i \(0.695208\pi\)
\(110\) 8.54661e20 0.314174
\(111\) −1.91923e21 −0.641558
\(112\) −1.05604e21 −0.321285
\(113\) −5.61097e21 −1.55494 −0.777471 0.628919i \(-0.783498\pi\)
−0.777471 + 0.628919i \(0.783498\pi\)
\(114\) 2.52347e21 0.637524
\(115\) 2.59631e20 0.0598452
\(116\) 1.52123e21 0.320173
\(117\) 1.95156e22 3.75345
\(118\) −8.40715e20 −0.147873
\(119\) 1.12417e22 1.80963
\(120\) −1.83098e21 −0.269948
\(121\) −9.57949e19 −0.0129448
\(122\) −4.64572e21 −0.575804
\(123\) −1.38582e22 −1.57653
\(124\) −8.00329e21 −0.836259
\(125\) −9.31323e20 −0.0894427
\(126\) 1.97001e22 1.74012
\(127\) 1.60366e22 1.30368 0.651842 0.758355i \(-0.273997\pi\)
0.651842 + 0.758355i \(0.273997\pi\)
\(128\) −1.18059e21 −0.0883883
\(129\) 1.46110e22 1.00806
\(130\) 9.74302e21 0.619852
\(131\) −1.39854e22 −0.820969 −0.410485 0.911868i \(-0.634640\pi\)
−0.410485 + 0.911868i \(0.634640\pi\)
\(132\) −1.56487e22 −0.848109
\(133\) −1.35548e22 −0.678659
\(134\) −8.22093e21 −0.380470
\(135\) 1.63191e22 0.698542
\(136\) 1.25676e22 0.497844
\(137\) −3.49305e22 −1.28127 −0.640633 0.767847i \(-0.721328\pi\)
−0.640633 + 0.767847i \(0.721328\pi\)
\(138\) −4.75380e21 −0.161551
\(139\) −1.16733e22 −0.367737 −0.183868 0.982951i \(-0.558862\pi\)
−0.183868 + 0.982951i \(0.558862\pi\)
\(140\) 9.83513e21 0.287366
\(141\) −6.23601e22 −1.69085
\(142\) 5.38346e22 1.35529
\(143\) 8.32698e22 1.94742
\(144\) 2.20236e22 0.478722
\(145\) −1.41675e22 −0.286371
\(146\) −9.50986e21 −0.178842
\(147\) −6.35499e22 −1.11246
\(148\) 1.15251e22 0.187886
\(149\) −2.99897e22 −0.455530 −0.227765 0.973716i \(-0.573142\pi\)
−0.227765 + 0.973716i \(0.573142\pi\)
\(150\) 1.70523e22 0.241449
\(151\) −8.80845e22 −1.16317 −0.581583 0.813487i \(-0.697566\pi\)
−0.581583 + 0.813487i \(0.697566\pi\)
\(152\) −1.51535e22 −0.186705
\(153\) −2.34445e23 −2.69638
\(154\) 8.40570e22 0.902831
\(155\) 7.45364e22 0.747972
\(156\) −1.78393e23 −1.67328
\(157\) 1.76504e23 1.54813 0.774067 0.633104i \(-0.218219\pi\)
0.774067 + 0.633104i \(0.218219\pi\)
\(158\) −1.11473e22 −0.0914686
\(159\) 1.42664e23 1.09558
\(160\) 1.09951e22 0.0790569
\(161\) 2.55351e22 0.171975
\(162\) −8.42469e22 −0.531675
\(163\) 9.36633e22 0.554115 0.277057 0.960853i \(-0.410641\pi\)
0.277057 + 0.960853i \(0.410641\pi\)
\(164\) 8.32191e22 0.461703
\(165\) 1.45740e23 0.758571
\(166\) 4.91514e22 0.240105
\(167\) −2.77263e23 −1.27166 −0.635828 0.771831i \(-0.719341\pi\)
−0.635828 + 0.771831i \(0.719341\pi\)
\(168\) −1.80079e23 −0.775740
\(169\) 7.02199e23 2.84217
\(170\) −1.17045e23 −0.445285
\(171\) 2.82685e23 1.01122
\(172\) −8.77397e22 −0.295221
\(173\) −3.84494e23 −1.21732 −0.608661 0.793431i \(-0.708293\pi\)
−0.608661 + 0.793431i \(0.708293\pi\)
\(174\) 2.59405e23 0.773055
\(175\) −9.15968e22 −0.257028
\(176\) 9.39710e22 0.248377
\(177\) −1.43362e23 −0.357037
\(178\) −2.40917e23 −0.565532
\(179\) 4.54367e22 0.100566 0.0502828 0.998735i \(-0.483988\pi\)
0.0502828 + 0.998735i \(0.483988\pi\)
\(180\) −2.05111e23 −0.428182
\(181\) 8.61960e23 1.69770 0.848852 0.528630i \(-0.177294\pi\)
0.848852 + 0.528630i \(0.177294\pi\)
\(182\) 9.58238e23 1.78125
\(183\) −7.92204e23 −1.39027
\(184\) 2.85467e22 0.0473118
\(185\) −1.07336e23 −0.168051
\(186\) −1.36475e24 −2.01914
\(187\) −1.00034e24 −1.39897
\(188\) 3.74475e23 0.495183
\(189\) 1.60501e24 2.00737
\(190\) 1.41128e23 0.166994
\(191\) −2.31575e23 −0.259323 −0.129661 0.991558i \(-0.541389\pi\)
−0.129661 + 0.991558i \(0.541389\pi\)
\(192\) −2.01319e23 −0.213413
\(193\) −1.13487e24 −1.13919 −0.569595 0.821926i \(-0.692900\pi\)
−0.569595 + 0.821926i \(0.692900\pi\)
\(194\) −7.43390e23 −0.706804
\(195\) 1.66141e24 1.49663
\(196\) 3.81620e23 0.325793
\(197\) −6.29771e23 −0.509668 −0.254834 0.966985i \(-0.582021\pi\)
−0.254834 + 0.966985i \(0.582021\pi\)
\(198\) −1.75300e24 −1.34524
\(199\) −7.13426e23 −0.519268 −0.259634 0.965707i \(-0.583602\pi\)
−0.259634 + 0.965707i \(0.583602\pi\)
\(200\) −1.02400e23 −0.0707107
\(201\) −1.40186e24 −0.918642
\(202\) −3.31715e23 −0.206336
\(203\) −1.39339e24 −0.822935
\(204\) 2.14307e24 1.20204
\(205\) −7.75039e23 −0.412960
\(206\) −5.89899e23 −0.298657
\(207\) −5.32532e23 −0.256246
\(208\) 1.07126e24 0.490036
\(209\) 1.20617e24 0.524652
\(210\) 1.67712e24 0.693843
\(211\) −2.57567e24 −1.01374 −0.506868 0.862024i \(-0.669197\pi\)
−0.506868 + 0.862024i \(0.669197\pi\)
\(212\) −8.56703e23 −0.320851
\(213\) 9.18005e24 3.27235
\(214\) 3.17080e24 1.07603
\(215\) 8.17140e23 0.264054
\(216\) 1.79431e24 0.552246
\(217\) 7.33075e24 2.14942
\(218\) 2.91328e24 0.813934
\(219\) −1.62165e24 −0.431812
\(220\) −8.75173e23 −0.222155
\(221\) −1.14037e25 −2.76011
\(222\) 1.96529e24 0.453650
\(223\) −7.24650e24 −1.59561 −0.797806 0.602914i \(-0.794006\pi\)
−0.797806 + 0.602914i \(0.794006\pi\)
\(224\) 1.08138e24 0.227183
\(225\) 1.91025e24 0.382977
\(226\) 5.74563e24 1.09951
\(227\) −1.55126e24 −0.283409 −0.141704 0.989909i \(-0.545258\pi\)
−0.141704 + 0.989909i \(0.545258\pi\)
\(228\) −2.58403e24 −0.450798
\(229\) −9.95437e24 −1.65860 −0.829299 0.558805i \(-0.811260\pi\)
−0.829299 + 0.558805i \(0.811260\pi\)
\(230\) −2.65862e23 −0.0423170
\(231\) 1.43337e25 2.17988
\(232\) −1.55773e24 −0.226396
\(233\) −4.62738e23 −0.0642833 −0.0321417 0.999483i \(-0.510233\pi\)
−0.0321417 + 0.999483i \(0.510233\pi\)
\(234\) −1.99840e25 −2.65409
\(235\) −3.48757e24 −0.442905
\(236\) 8.60892e23 0.104562
\(237\) −1.90088e24 −0.220850
\(238\) −1.15115e25 −1.27960
\(239\) 1.06208e25 1.12975 0.564873 0.825178i \(-0.308925\pi\)
0.564873 + 0.825178i \(0.308925\pi\)
\(240\) 1.87492e24 0.190882
\(241\) 1.67701e24 0.163439 0.0817195 0.996655i \(-0.473959\pi\)
0.0817195 + 0.996655i \(0.473959\pi\)
\(242\) 9.80940e22 0.00915337
\(243\) 3.11401e24 0.278262
\(244\) 4.75722e24 0.407155
\(245\) −3.55411e24 −0.291398
\(246\) 1.41908e25 1.11478
\(247\) 1.37501e25 1.03512
\(248\) 8.19536e24 0.591324
\(249\) 8.38147e24 0.579732
\(250\) 9.53674e23 0.0632456
\(251\) −2.15807e25 −1.37243 −0.686216 0.727398i \(-0.740730\pi\)
−0.686216 + 0.727398i \(0.740730\pi\)
\(252\) −2.01729e25 −1.23045
\(253\) −2.27222e24 −0.132949
\(254\) −1.64214e25 −0.921843
\(255\) −1.99589e25 −1.07514
\(256\) 1.20893e24 0.0625000
\(257\) 7.72467e24 0.383338 0.191669 0.981460i \(-0.438610\pi\)
0.191669 + 0.981460i \(0.438610\pi\)
\(258\) −1.49617e25 −0.712809
\(259\) −1.05566e25 −0.482921
\(260\) −9.97685e24 −0.438302
\(261\) 2.90592e25 1.22619
\(262\) 1.43211e25 0.580513
\(263\) 1.35616e25 0.528174 0.264087 0.964499i \(-0.414929\pi\)
0.264087 + 0.964499i \(0.414929\pi\)
\(264\) 1.60243e25 0.599703
\(265\) 7.97867e24 0.286978
\(266\) 1.38801e25 0.479884
\(267\) −4.10819e25 −1.36547
\(268\) 8.41824e24 0.269033
\(269\) −2.23157e25 −0.685822 −0.342911 0.939368i \(-0.611413\pi\)
−0.342911 + 0.939368i \(0.611413\pi\)
\(270\) −1.67108e25 −0.493944
\(271\) 8.79105e24 0.249956 0.124978 0.992160i \(-0.460114\pi\)
0.124978 + 0.992160i \(0.460114\pi\)
\(272\) −1.28692e25 −0.352029
\(273\) 1.63402e26 4.30080
\(274\) 3.57689e25 0.905992
\(275\) 8.15068e24 0.198701
\(276\) 4.86789e24 0.114234
\(277\) 3.23947e25 0.731874 0.365937 0.930640i \(-0.380749\pi\)
0.365937 + 0.930640i \(0.380749\pi\)
\(278\) 1.19534e25 0.260029
\(279\) −1.52882e26 −3.20268
\(280\) −1.00712e25 −0.203199
\(281\) 1.62934e25 0.316662 0.158331 0.987386i \(-0.449389\pi\)
0.158331 + 0.987386i \(0.449389\pi\)
\(282\) 6.38568e25 1.19561
\(283\) 7.24900e25 1.30774 0.653869 0.756608i \(-0.273145\pi\)
0.653869 + 0.756608i \(0.273145\pi\)
\(284\) −5.51266e25 −0.958338
\(285\) 2.40656e25 0.403206
\(286\) −8.52682e25 −1.37703
\(287\) −7.62260e25 −1.18671
\(288\) −2.25522e25 −0.338507
\(289\) 6.79028e25 0.982790
\(290\) 1.45075e25 0.202495
\(291\) −1.26765e26 −1.70657
\(292\) 9.73810e24 0.126460
\(293\) −7.90881e25 −0.990835 −0.495418 0.868655i \(-0.664985\pi\)
−0.495418 + 0.868655i \(0.664985\pi\)
\(294\) 6.50751e25 0.786625
\(295\) −8.01769e24 −0.0935229
\(296\) −1.18017e25 −0.132856
\(297\) −1.42821e26 −1.55185
\(298\) 3.07095e25 0.322108
\(299\) −2.59030e25 −0.262303
\(300\) −1.74616e25 −0.170730
\(301\) 8.03667e25 0.758801
\(302\) 9.01985e25 0.822482
\(303\) −5.65651e25 −0.498197
\(304\) 1.55172e25 0.132020
\(305\) −4.43050e25 −0.364170
\(306\) 2.40072e26 1.90663
\(307\) −1.19014e26 −0.913363 −0.456682 0.889630i \(-0.650962\pi\)
−0.456682 + 0.889630i \(0.650962\pi\)
\(308\) −8.60744e25 −0.638398
\(309\) −1.00592e26 −0.721104
\(310\) −7.63253e25 −0.528896
\(311\) −1.66399e26 −1.11472 −0.557360 0.830271i \(-0.688186\pi\)
−0.557360 + 0.830271i \(0.688186\pi\)
\(312\) 1.82674e26 1.18319
\(313\) 2.24533e25 0.140626 0.0703128 0.997525i \(-0.477600\pi\)
0.0703128 + 0.997525i \(0.477600\pi\)
\(314\) −1.80740e26 −1.09470
\(315\) 1.87875e26 1.10055
\(316\) 1.14149e25 0.0646781
\(317\) 1.10755e26 0.607076 0.303538 0.952819i \(-0.401832\pi\)
0.303538 + 0.952819i \(0.401832\pi\)
\(318\) −1.46088e26 −0.774693
\(319\) 1.23990e26 0.636188
\(320\) −1.12590e25 −0.0559017
\(321\) 5.40695e26 2.59806
\(322\) −2.61479e25 −0.121605
\(323\) −1.65183e26 −0.743599
\(324\) 8.62689e25 0.375951
\(325\) 9.29166e25 0.392029
\(326\) −9.59112e25 −0.391818
\(327\) 4.96783e26 1.96524
\(328\) −8.52164e25 −0.326473
\(329\) −3.43007e26 −1.27276
\(330\) −1.49237e26 −0.536391
\(331\) −2.18220e26 −0.759803 −0.379901 0.925027i \(-0.624042\pi\)
−0.379901 + 0.925027i \(0.624042\pi\)
\(332\) −5.03311e25 −0.169780
\(333\) 2.20157e26 0.719562
\(334\) 2.83918e26 0.899196
\(335\) −7.84009e25 −0.240631
\(336\) 1.84401e26 0.548531
\(337\) 5.33804e26 1.53910 0.769551 0.638585i \(-0.220480\pi\)
0.769551 + 0.638585i \(0.220480\pi\)
\(338\) −7.19052e26 −2.00972
\(339\) 9.79765e26 2.65476
\(340\) 1.19854e26 0.314864
\(341\) −6.52322e26 −1.66166
\(342\) −2.89469e26 −0.715037
\(343\) 1.86911e26 0.447761
\(344\) 8.98455e25 0.208753
\(345\) −4.53357e25 −0.102174
\(346\) 3.93722e26 0.860776
\(347\) −4.09971e26 −0.869548 −0.434774 0.900540i \(-0.643172\pi\)
−0.434774 + 0.900540i \(0.643172\pi\)
\(348\) −2.65630e26 −0.546633
\(349\) 2.06527e26 0.412392 0.206196 0.978511i \(-0.433892\pi\)
0.206196 + 0.978511i \(0.433892\pi\)
\(350\) 9.37951e25 0.181746
\(351\) −1.62814e27 −3.06172
\(352\) −9.62263e25 −0.175629
\(353\) 2.63582e26 0.466961 0.233481 0.972361i \(-0.424988\pi\)
0.233481 + 0.972361i \(0.424988\pi\)
\(354\) 1.46802e26 0.252464
\(355\) 5.13406e26 0.857163
\(356\) 2.46699e26 0.399891
\(357\) −1.96298e27 −3.08958
\(358\) −4.65272e25 −0.0711107
\(359\) 2.72697e26 0.404751 0.202376 0.979308i \(-0.435134\pi\)
0.202376 + 0.979308i \(0.435134\pi\)
\(360\) 2.10034e26 0.302770
\(361\) −5.15038e26 −0.721130
\(362\) −8.82647e26 −1.20046
\(363\) 1.67273e25 0.0221007
\(364\) −9.81236e26 −1.25953
\(365\) −9.06931e25 −0.113110
\(366\) 8.11217e26 0.983072
\(367\) −1.67539e26 −0.197297 −0.0986487 0.995122i \(-0.531452\pi\)
−0.0986487 + 0.995122i \(0.531452\pi\)
\(368\) −2.92319e25 −0.0334545
\(369\) 1.58969e27 1.76822
\(370\) 1.09912e26 0.118830
\(371\) 7.84712e26 0.824678
\(372\) 1.39750e27 1.42775
\(373\) 9.92406e26 0.985704 0.492852 0.870113i \(-0.335954\pi\)
0.492852 + 0.870113i \(0.335954\pi\)
\(374\) 1.02434e27 0.989223
\(375\) 1.62624e26 0.152706
\(376\) −3.83462e26 −0.350147
\(377\) 1.41347e27 1.25517
\(378\) −1.64353e27 −1.41943
\(379\) 1.41427e25 0.0118801 0.00594007 0.999982i \(-0.498109\pi\)
0.00594007 + 0.999982i \(0.498109\pi\)
\(380\) −1.44515e26 −0.118083
\(381\) −2.80024e27 −2.22578
\(382\) 2.37132e26 0.183369
\(383\) 2.51013e26 0.188847 0.0944234 0.995532i \(-0.469899\pi\)
0.0944234 + 0.995532i \(0.469899\pi\)
\(384\) 2.06150e26 0.150906
\(385\) 8.01630e26 0.571000
\(386\) 1.16211e27 0.805529
\(387\) −1.67604e27 −1.13063
\(388\) 7.61232e26 0.499786
\(389\) −2.50259e27 −1.59926 −0.799629 0.600495i \(-0.794970\pi\)
−0.799629 + 0.600495i \(0.794970\pi\)
\(390\) −1.70129e27 −1.05828
\(391\) 3.11178e26 0.188431
\(392\) −3.90779e26 −0.230371
\(393\) 2.44208e27 1.40164
\(394\) 6.44886e26 0.360390
\(395\) −1.06309e26 −0.0578498
\(396\) 1.79508e27 0.951226
\(397\) 3.29042e26 0.169805 0.0849027 0.996389i \(-0.472942\pi\)
0.0849027 + 0.996389i \(0.472942\pi\)
\(398\) 7.30548e26 0.367178
\(399\) 2.36689e27 1.15868
\(400\) 1.04858e26 0.0500000
\(401\) 8.45309e26 0.392645 0.196322 0.980539i \(-0.437100\pi\)
0.196322 + 0.980539i \(0.437100\pi\)
\(402\) 1.43551e27 0.649578
\(403\) −7.43638e27 −3.27838
\(404\) 3.39676e26 0.145902
\(405\) −8.03441e26 −0.336261
\(406\) 1.42683e27 0.581903
\(407\) 9.39371e26 0.373333
\(408\) −2.19450e27 −0.849971
\(409\) −1.58724e27 −0.599167 −0.299583 0.954070i \(-0.596848\pi\)
−0.299583 + 0.954070i \(0.596848\pi\)
\(410\) 7.93639e26 0.292007
\(411\) 6.09943e27 2.18751
\(412\) 6.04057e26 0.211182
\(413\) −7.88550e26 −0.268753
\(414\) 5.45313e26 0.181193
\(415\) 4.68745e26 0.151856
\(416\) −1.09697e27 −0.346508
\(417\) 2.03834e27 0.627839
\(418\) −1.23512e27 −0.370985
\(419\) −1.46560e25 −0.00429308 −0.00214654 0.999998i \(-0.500683\pi\)
−0.00214654 + 0.999998i \(0.500683\pi\)
\(420\) −1.71737e27 −0.490621
\(421\) −2.07639e27 −0.578559 −0.289280 0.957245i \(-0.593416\pi\)
−0.289280 + 0.957245i \(0.593416\pi\)
\(422\) 2.63749e27 0.716819
\(423\) 7.15340e27 1.89644
\(424\) 8.77264e26 0.226876
\(425\) −1.11623e27 −0.281623
\(426\) −9.40037e27 −2.31390
\(427\) −4.35746e27 −1.04650
\(428\) −3.24690e27 −0.760868
\(429\) −1.45402e28 −3.32483
\(430\) −8.36751e26 −0.186714
\(431\) −2.88896e27 −0.629115 −0.314558 0.949238i \(-0.601856\pi\)
−0.314558 + 0.949238i \(0.601856\pi\)
\(432\) −1.83737e27 −0.390497
\(433\) 6.58569e27 1.36609 0.683044 0.730377i \(-0.260656\pi\)
0.683044 + 0.730377i \(0.260656\pi\)
\(434\) −7.50669e27 −1.51987
\(435\) 2.47387e27 0.488923
\(436\) −2.98320e27 −0.575539
\(437\) −3.75206e26 −0.0706668
\(438\) 1.66057e27 0.305337
\(439\) −9.33293e26 −0.167548 −0.0837742 0.996485i \(-0.526697\pi\)
−0.0837742 + 0.996485i \(0.526697\pi\)
\(440\) 8.96177e26 0.157087
\(441\) 7.28987e27 1.24771
\(442\) 1.16774e28 1.95169
\(443\) 9.69019e27 1.58159 0.790793 0.612083i \(-0.209668\pi\)
0.790793 + 0.612083i \(0.209668\pi\)
\(444\) −2.01246e27 −0.320779
\(445\) −2.29756e27 −0.357674
\(446\) 7.42042e27 1.12827
\(447\) 5.23669e27 0.777728
\(448\) −1.10734e27 −0.160643
\(449\) −9.90883e27 −1.40422 −0.702111 0.712067i \(-0.747759\pi\)
−0.702111 + 0.712067i \(0.747759\pi\)
\(450\) −1.95609e27 −0.270806
\(451\) 6.78293e27 0.917410
\(452\) −5.88353e27 −0.777471
\(453\) 1.53810e28 1.98588
\(454\) 1.58849e27 0.200400
\(455\) 9.13847e27 1.12656
\(456\) 2.64605e27 0.318762
\(457\) −1.41062e28 −1.66069 −0.830346 0.557248i \(-0.811857\pi\)
−0.830346 + 0.557248i \(0.811857\pi\)
\(458\) 1.01933e28 1.17281
\(459\) 1.95591e28 2.19946
\(460\) 2.72243e26 0.0299226
\(461\) 1.40343e26 0.0150776 0.00753880 0.999972i \(-0.497600\pi\)
0.00753880 + 0.999972i \(0.497600\pi\)
\(462\) −1.46777e28 −1.54141
\(463\) −1.54202e28 −1.58303 −0.791514 0.611150i \(-0.790707\pi\)
−0.791514 + 0.611150i \(0.790707\pi\)
\(464\) 1.59512e27 0.160086
\(465\) −1.30152e28 −1.27702
\(466\) 4.73844e26 0.0454552
\(467\) 1.42166e28 1.33343 0.666713 0.745314i \(-0.267701\pi\)
0.666713 + 0.745314i \(0.267701\pi\)
\(468\) 2.04636e28 1.87673
\(469\) −7.71083e27 −0.691491
\(470\) 3.57127e27 0.313181
\(471\) −3.08204e28 −2.64313
\(472\) −8.81554e26 −0.0739363
\(473\) −7.15138e27 −0.586608
\(474\) 1.94650e27 0.156165
\(475\) 1.34590e27 0.105616
\(476\) 1.17878e28 0.904813
\(477\) −1.63651e28 −1.22879
\(478\) −1.08757e28 −0.798851
\(479\) 1.82659e28 1.31256 0.656280 0.754518i \(-0.272129\pi\)
0.656280 + 0.754518i \(0.272129\pi\)
\(480\) −1.91992e27 −0.134974
\(481\) 1.07087e28 0.736569
\(482\) −1.71725e27 −0.115569
\(483\) −4.45883e27 −0.293613
\(484\) −1.00448e26 −0.00647241
\(485\) −7.08952e27 −0.447022
\(486\) −3.18875e27 −0.196761
\(487\) 3.19696e28 1.93056 0.965280 0.261218i \(-0.0841243\pi\)
0.965280 + 0.261218i \(0.0841243\pi\)
\(488\) −4.87139e27 −0.287902
\(489\) −1.63551e28 −0.946042
\(490\) 3.63941e27 0.206050
\(491\) −1.06661e28 −0.591086 −0.295543 0.955329i \(-0.595501\pi\)
−0.295543 + 0.955329i \(0.595501\pi\)
\(492\) −1.45314e28 −0.788267
\(493\) −1.69803e28 −0.901681
\(494\) −1.40801e28 −0.731937
\(495\) −1.67180e28 −0.850802
\(496\) −8.39205e27 −0.418129
\(497\) 5.04942e28 2.46320
\(498\) −8.58262e27 −0.409932
\(499\) −1.19664e28 −0.559638 −0.279819 0.960053i \(-0.590275\pi\)
−0.279819 + 0.960053i \(0.590275\pi\)
\(500\) −9.76563e26 −0.0447214
\(501\) 4.84146e28 2.17110
\(502\) 2.20986e28 0.970456
\(503\) −3.75074e28 −1.61307 −0.806534 0.591188i \(-0.798659\pi\)
−0.806534 + 0.591188i \(0.798659\pi\)
\(504\) 2.06571e28 0.870059
\(505\) −3.16348e27 −0.130498
\(506\) 2.32675e27 0.0940092
\(507\) −1.22615e29 −4.85245
\(508\) 1.68156e28 0.651842
\(509\) 3.27640e28 1.24411 0.622057 0.782972i \(-0.286297\pi\)
0.622057 + 0.782972i \(0.286297\pi\)
\(510\) 2.04379e28 0.760237
\(511\) −8.91979e27 −0.325039
\(512\) −1.23794e27 −0.0441942
\(513\) −2.35836e28 −0.824857
\(514\) −7.91006e27 −0.271061
\(515\) −5.62572e27 −0.188887
\(516\) 1.53208e28 0.504032
\(517\) 3.05223e28 0.983934
\(518\) 1.08099e28 0.341477
\(519\) 6.71388e28 2.07834
\(520\) 1.02163e28 0.309926
\(521\) −5.49377e27 −0.163333 −0.0816667 0.996660i \(-0.526024\pi\)
−0.0816667 + 0.996660i \(0.526024\pi\)
\(522\) −2.97566e28 −0.867047
\(523\) 9.60628e27 0.274339 0.137170 0.990548i \(-0.456199\pi\)
0.137170 + 0.990548i \(0.456199\pi\)
\(524\) −1.46648e28 −0.410485
\(525\) 1.59943e28 0.438825
\(526\) −1.38871e28 −0.373475
\(527\) 8.93347e28 2.35510
\(528\) −1.64088e28 −0.424054
\(529\) −3.87648e28 −0.982093
\(530\) −8.17016e27 −0.202924
\(531\) 1.64452e28 0.400448
\(532\) −1.42133e28 −0.339329
\(533\) 7.73244e28 1.81001
\(534\) 4.20679e28 0.965534
\(535\) 3.02391e28 0.680541
\(536\) −8.62027e27 −0.190235
\(537\) −7.93397e27 −0.171696
\(538\) 2.28513e28 0.484949
\(539\) 3.11046e28 0.647355
\(540\) 1.71119e28 0.349271
\(541\) −7.34453e28 −1.47026 −0.735128 0.677929i \(-0.762878\pi\)
−0.735128 + 0.677929i \(0.762878\pi\)
\(542\) −9.00204e27 −0.176745
\(543\) −1.50512e29 −2.89850
\(544\) 1.31781e28 0.248922
\(545\) 2.77832e28 0.514777
\(546\) −1.67324e29 −3.04113
\(547\) 4.12363e28 0.735212 0.367606 0.929982i \(-0.380178\pi\)
0.367606 + 0.929982i \(0.380178\pi\)
\(548\) −3.66273e28 −0.640633
\(549\) 9.08745e28 1.55931
\(550\) −8.34630e27 −0.140503
\(551\) 2.04742e28 0.338155
\(552\) −4.98472e27 −0.0807756
\(553\) −1.04557e28 −0.166241
\(554\) −3.31722e28 −0.517513
\(555\) 1.87425e28 0.286914
\(556\) −1.22403e28 −0.183868
\(557\) −8.83426e27 −0.130224 −0.0651119 0.997878i \(-0.520740\pi\)
−0.0651119 + 0.997878i \(0.520740\pi\)
\(558\) 1.56552e29 2.26464
\(559\) −8.15248e28 −1.15735
\(560\) 1.03129e28 0.143683
\(561\) 1.74675e29 2.38847
\(562\) −1.66845e28 −0.223914
\(563\) −1.06900e29 −1.40811 −0.704057 0.710143i \(-0.748630\pi\)
−0.704057 + 0.710143i \(0.748630\pi\)
\(564\) −6.53893e28 −0.845427
\(565\) 5.47946e28 0.695391
\(566\) −7.42298e28 −0.924711
\(567\) −7.90195e28 −0.966300
\(568\) 5.64496e28 0.677647
\(569\) −1.27688e29 −1.50477 −0.752386 0.658723i \(-0.771097\pi\)
−0.752386 + 0.658723i \(0.771097\pi\)
\(570\) −2.46432e28 −0.285109
\(571\) −8.59333e28 −0.976073 −0.488037 0.872823i \(-0.662287\pi\)
−0.488037 + 0.872823i \(0.662287\pi\)
\(572\) 8.73147e28 0.973709
\(573\) 4.04366e28 0.442743
\(574\) 7.80555e28 0.839128
\(575\) −2.53546e27 −0.0267636
\(576\) 2.30935e28 0.239361
\(577\) −1.28637e29 −1.30925 −0.654623 0.755956i \(-0.727172\pi\)
−0.654623 + 0.755956i \(0.727172\pi\)
\(578\) −6.95325e28 −0.694937
\(579\) 1.98167e29 1.94494
\(580\) −1.48557e28 −0.143186
\(581\) 4.61016e28 0.436382
\(582\) 1.29808e29 1.20673
\(583\) −6.98271e28 −0.637536
\(584\) −9.97182e27 −0.0894210
\(585\) −1.90582e29 −1.67860
\(586\) 8.09863e28 0.700626
\(587\) −3.48294e28 −0.295969 −0.147985 0.988990i \(-0.547279\pi\)
−0.147985 + 0.988990i \(0.547279\pi\)
\(588\) −6.66369e28 −0.556228
\(589\) −1.07716e29 −0.883225
\(590\) 8.21011e27 0.0661307
\(591\) 1.09968e29 0.870158
\(592\) 1.20849e28 0.0939432
\(593\) −8.77330e28 −0.670021 −0.335011 0.942214i \(-0.608740\pi\)
−0.335011 + 0.942214i \(0.608740\pi\)
\(594\) 1.46248e29 1.09732
\(595\) −1.09782e29 −0.809290
\(596\) −3.14465e28 −0.227765
\(597\) 1.24576e29 0.886548
\(598\) 2.65247e28 0.185476
\(599\) −1.00705e29 −0.691944 −0.345972 0.938245i \(-0.612451\pi\)
−0.345972 + 0.938245i \(0.612451\pi\)
\(600\) 1.78807e28 0.120725
\(601\) −1.06089e29 −0.703861 −0.351930 0.936026i \(-0.614475\pi\)
−0.351930 + 0.936026i \(0.614475\pi\)
\(602\) −8.22955e28 −0.536554
\(603\) 1.60809e29 1.03034
\(604\) −9.23633e28 −0.581583
\(605\) 9.35497e26 0.00578910
\(606\) 5.79227e28 0.352278
\(607\) 5.05632e28 0.302241 0.151120 0.988515i \(-0.451712\pi\)
0.151120 + 0.988515i \(0.451712\pi\)
\(608\) −1.58896e28 −0.0933525
\(609\) 2.43309e29 1.40500
\(610\) 4.53683e28 0.257507
\(611\) 3.47950e29 1.94126
\(612\) −2.45834e29 −1.34819
\(613\) 1.21126e29 0.652985 0.326493 0.945200i \(-0.394133\pi\)
0.326493 + 0.945200i \(0.394133\pi\)
\(614\) 1.21870e29 0.645845
\(615\) 1.35334e29 0.705048
\(616\) 8.81402e28 0.451415
\(617\) 7.11151e28 0.358070 0.179035 0.983843i \(-0.442702\pi\)
0.179035 + 0.983843i \(0.442702\pi\)
\(618\) 1.03006e29 0.509898
\(619\) −1.87344e28 −0.0911774 −0.0455887 0.998960i \(-0.514516\pi\)
−0.0455887 + 0.998960i \(0.514516\pi\)
\(620\) 7.81571e28 0.373986
\(621\) 4.44277e28 0.209022
\(622\) 1.70392e29 0.788225
\(623\) −2.25968e29 −1.02783
\(624\) −1.87058e29 −0.836641
\(625\) 9.09495e27 0.0400000
\(626\) −2.29922e28 −0.0994374
\(627\) −2.10616e29 −0.895741
\(628\) 1.85078e29 0.774067
\(629\) −1.28646e29 −0.529131
\(630\) −1.92384e29 −0.778204
\(631\) 1.18886e29 0.472956 0.236478 0.971637i \(-0.424007\pi\)
0.236478 + 0.971637i \(0.424007\pi\)
\(632\) −1.16888e28 −0.0457343
\(633\) 4.49753e29 1.73075
\(634\) −1.13414e29 −0.429267
\(635\) −1.56607e29 −0.583025
\(636\) 1.49594e29 0.547790
\(637\) 3.54588e29 1.27720
\(638\) −1.26966e29 −0.449853
\(639\) −1.05305e30 −3.67021
\(640\) 1.15292e28 0.0395285
\(641\) −4.05009e29 −1.36602 −0.683008 0.730411i \(-0.739329\pi\)
−0.683008 + 0.730411i \(0.739329\pi\)
\(642\) −5.53672e29 −1.83711
\(643\) 2.92040e29 0.953296 0.476648 0.879094i \(-0.341852\pi\)
0.476648 + 0.879094i \(0.341852\pi\)
\(644\) 2.67754e28 0.0859875
\(645\) −1.42686e29 −0.450820
\(646\) 1.69147e29 0.525804
\(647\) −1.30263e29 −0.398407 −0.199203 0.979958i \(-0.563835\pi\)
−0.199203 + 0.979958i \(0.563835\pi\)
\(648\) −8.83393e28 −0.265838
\(649\) 7.01686e28 0.207766
\(650\) −9.51466e28 −0.277206
\(651\) −1.28007e30 −3.66971
\(652\) 9.82131e28 0.277057
\(653\) −1.49980e29 −0.416337 −0.208169 0.978093i \(-0.566750\pi\)
−0.208169 + 0.978093i \(0.566750\pi\)
\(654\) −5.08706e29 −1.38963
\(655\) 1.36576e29 0.367149
\(656\) 8.72616e28 0.230852
\(657\) 1.86022e29 0.484314
\(658\) 3.51239e29 0.899976
\(659\) 3.09551e29 0.780613 0.390307 0.920685i \(-0.372369\pi\)
0.390307 + 0.920685i \(0.372369\pi\)
\(660\) 1.52819e29 0.379286
\(661\) −1.06593e29 −0.260383 −0.130191 0.991489i \(-0.541559\pi\)
−0.130191 + 0.991489i \(0.541559\pi\)
\(662\) 2.23457e29 0.537262
\(663\) 1.99127e30 4.71235
\(664\) 5.15390e28 0.120053
\(665\) 1.32371e29 0.303505
\(666\) −2.25441e29 −0.508807
\(667\) −3.85701e28 −0.0856898
\(668\) −2.90732e29 −0.635828
\(669\) 1.26535e30 2.72419
\(670\) 8.02826e28 0.170151
\(671\) 3.87746e29 0.809022
\(672\) −1.88827e29 −0.387870
\(673\) −7.90406e29 −1.59842 −0.799212 0.601049i \(-0.794750\pi\)
−0.799212 + 0.601049i \(0.794750\pi\)
\(674\) −5.46615e29 −1.08831
\(675\) −1.59367e29 −0.312398
\(676\) 7.36309e29 1.42108
\(677\) −3.48611e29 −0.662461 −0.331230 0.943550i \(-0.607464\pi\)
−0.331230 + 0.943550i \(0.607464\pi\)
\(678\) −1.00328e30 −1.87720
\(679\) −6.97264e29 −1.28459
\(680\) −1.22730e29 −0.222643
\(681\) 2.70875e29 0.483865
\(682\) 6.67978e29 1.17497
\(683\) 3.87738e28 0.0671617 0.0335808 0.999436i \(-0.489309\pi\)
0.0335808 + 0.999436i \(0.489309\pi\)
\(684\) 2.96417e29 0.505608
\(685\) 3.41118e29 0.572999
\(686\) −1.91397e29 −0.316615
\(687\) 1.73819e30 2.83173
\(688\) −9.20017e28 −0.147611
\(689\) −7.96020e29 −1.25783
\(690\) 4.64238e28 0.0722479
\(691\) 7.13503e29 1.09364 0.546822 0.837249i \(-0.315837\pi\)
0.546822 + 0.837249i \(0.315837\pi\)
\(692\) −4.03171e29 −0.608661
\(693\) −1.64423e30 −2.44492
\(694\) 4.19810e29 0.614863
\(695\) 1.13997e29 0.164457
\(696\) 2.72005e29 0.386528
\(697\) −9.28913e29 −1.30026
\(698\) −2.11484e29 −0.291605
\(699\) 8.08015e28 0.109751
\(700\) −9.60462e28 −0.128514
\(701\) −8.08128e28 −0.106522 −0.0532612 0.998581i \(-0.516962\pi\)
−0.0532612 + 0.998581i \(0.516962\pi\)
\(702\) 1.66721e30 2.16497
\(703\) 1.55116e29 0.198439
\(704\) 9.85357e28 0.124188
\(705\) 6.08986e29 0.756173
\(706\) −2.69908e29 −0.330191
\(707\) −3.11132e29 −0.375008
\(708\) −1.50326e29 −0.178519
\(709\) 7.67692e28 0.0898259 0.0449129 0.998991i \(-0.485699\pi\)
0.0449129 + 0.998991i \(0.485699\pi\)
\(710\) −5.25728e29 −0.606106
\(711\) 2.18052e29 0.247702
\(712\) −2.52620e29 −0.282766
\(713\) 2.02920e29 0.223813
\(714\) 2.01009e30 2.18466
\(715\) −8.13181e29 −0.870911
\(716\) 4.76438e28 0.0502828
\(717\) −1.85457e30 −1.92882
\(718\) −2.79241e29 −0.286202
\(719\) 4.77523e29 0.482326 0.241163 0.970485i \(-0.422471\pi\)
0.241163 + 0.970485i \(0.422471\pi\)
\(720\) −2.15075e29 −0.214091
\(721\) −5.53297e29 −0.542797
\(722\) 5.27399e29 0.509916
\(723\) −2.92832e29 −0.279040
\(724\) 9.03830e29 0.848852
\(725\) 1.38355e29 0.128069
\(726\) −1.71288e28 −0.0156276
\(727\) −8.17519e29 −0.735168 −0.367584 0.929990i \(-0.619815\pi\)
−0.367584 + 0.929990i \(0.619815\pi\)
\(728\) 1.00479e30 0.890623
\(729\) −1.40435e30 −1.22698
\(730\) 9.28698e28 0.0799805
\(731\) 9.79373e29 0.831411
\(732\) −8.30686e29 −0.695137
\(733\) −1.25004e30 −1.03118 −0.515588 0.856837i \(-0.672426\pi\)
−0.515588 + 0.856837i \(0.672426\pi\)
\(734\) 1.71560e29 0.139510
\(735\) 6.20604e29 0.497505
\(736\) 2.99334e28 0.0236559
\(737\) 6.86144e29 0.534572
\(738\) −1.62784e30 −1.25032
\(739\) 1.41926e30 1.07472 0.537359 0.843354i \(-0.319422\pi\)
0.537359 + 0.843354i \(0.319422\pi\)
\(740\) −1.12549e29 −0.0840253
\(741\) −2.40099e30 −1.76726
\(742\) −8.03545e29 −0.583135
\(743\) −4.94344e29 −0.353709 −0.176855 0.984237i \(-0.556592\pi\)
−0.176855 + 0.984237i \(0.556592\pi\)
\(744\) −1.43104e30 −1.00957
\(745\) 2.92869e29 0.203719
\(746\) −1.01622e30 −0.696998
\(747\) −9.61447e29 −0.650218
\(748\) −1.04893e30 −0.699486
\(749\) 2.97405e30 1.95564
\(750\) −1.66527e29 −0.107979
\(751\) −1.81607e30 −1.16121 −0.580607 0.814184i \(-0.697185\pi\)
−0.580607 + 0.814184i \(0.697185\pi\)
\(752\) 3.92666e29 0.247591
\(753\) 3.76833e30 2.34316
\(754\) −1.44740e30 −0.887540
\(755\) 8.60200e29 0.520183
\(756\) 1.68297e30 1.00369
\(757\) −3.23429e30 −1.90227 −0.951136 0.308772i \(-0.900082\pi\)
−0.951136 + 0.308772i \(0.900082\pi\)
\(758\) −1.44822e28 −0.00840053
\(759\) 3.96766e29 0.226984
\(760\) 1.47984e29 0.0834970
\(761\) 5.91876e29 0.329376 0.164688 0.986346i \(-0.447338\pi\)
0.164688 + 0.986346i \(0.447338\pi\)
\(762\) 2.86745e30 1.57387
\(763\) 2.73252e30 1.47930
\(764\) −2.42824e29 −0.129661
\(765\) 2.28950e30 1.20586
\(766\) −2.57038e29 −0.133535
\(767\) 7.99912e29 0.409912
\(768\) −2.11098e29 −0.106706
\(769\) 3.65919e30 1.82456 0.912281 0.409565i \(-0.134320\pi\)
0.912281 + 0.409565i \(0.134320\pi\)
\(770\) −8.20869e29 −0.403758
\(771\) −1.34885e30 −0.654475
\(772\) −1.19000e30 −0.569595
\(773\) −3.12612e30 −1.47612 −0.738059 0.674736i \(-0.764257\pi\)
−0.738059 + 0.674736i \(0.764257\pi\)
\(774\) 1.71627e30 0.799476
\(775\) −7.27895e29 −0.334503
\(776\) −7.79501e29 −0.353402
\(777\) 1.84335e30 0.824492
\(778\) 2.56265e30 1.13085
\(779\) 1.12005e30 0.487633
\(780\) 1.74212e30 0.748314
\(781\) −4.49319e30 −1.90423
\(782\) −3.18646e29 −0.133241
\(783\) −2.42433e30 −1.00021
\(784\) 4.00157e29 0.162897
\(785\) −1.72367e30 −0.692346
\(786\) −2.50069e30 −0.991112
\(787\) 1.94010e30 0.758735 0.379368 0.925246i \(-0.376142\pi\)
0.379368 + 0.925246i \(0.376142\pi\)
\(788\) −6.60363e29 −0.254834
\(789\) −2.36808e30 −0.901753
\(790\) 1.08861e29 0.0409060
\(791\) 5.38912e30 1.99832
\(792\) −1.83816e30 −0.672618
\(793\) 4.42024e30 1.59617
\(794\) −3.36939e29 −0.120071
\(795\) −1.39320e30 −0.489959
\(796\) −7.48081e29 −0.259634
\(797\) 1.87837e30 0.643383 0.321691 0.946845i \(-0.395749\pi\)
0.321691 + 0.946845i \(0.395749\pi\)
\(798\) −2.42369e30 −0.819308
\(799\) −4.17999e30 −1.39455
\(800\) −1.07374e29 −0.0353553
\(801\) 4.71255e30 1.53149
\(802\) −8.65597e29 −0.277642
\(803\) 7.93722e29 0.251278
\(804\) −1.46996e30 −0.459321
\(805\) −2.49366e29 −0.0769095
\(806\) 7.61486e30 2.31816
\(807\) 3.89668e30 1.17091
\(808\) −3.47828e29 −0.103168
\(809\) 3.76184e30 1.10139 0.550694 0.834707i \(-0.314363\pi\)
0.550694 + 0.834707i \(0.314363\pi\)
\(810\) 8.22724e29 0.237772
\(811\) −1.39769e30 −0.398741 −0.199370 0.979924i \(-0.563890\pi\)
−0.199370 + 0.979924i \(0.563890\pi\)
\(812\) −1.46108e30 −0.411467
\(813\) −1.53506e30 −0.426751
\(814\) −9.61916e29 −0.263986
\(815\) −9.14681e29 −0.247808
\(816\) 2.24717e30 0.601020
\(817\) −1.18089e30 −0.311801
\(818\) 1.62533e30 0.423675
\(819\) −1.87440e31 −4.82372
\(820\) −8.12687e29 −0.206480
\(821\) 3.06130e30 0.767897 0.383948 0.923355i \(-0.374564\pi\)
0.383948 + 0.923355i \(0.374564\pi\)
\(822\) −6.24581e30 −1.54680
\(823\) −2.80656e30 −0.686239 −0.343120 0.939292i \(-0.611484\pi\)
−0.343120 + 0.939292i \(0.611484\pi\)
\(824\) −6.18554e29 −0.149328
\(825\) −1.42324e30 −0.339243
\(826\) 8.07475e29 0.190037
\(827\) −1.80807e30 −0.420153 −0.210077 0.977685i \(-0.567371\pi\)
−0.210077 + 0.977685i \(0.567371\pi\)
\(828\) −5.58401e29 −0.128123
\(829\) 6.48512e30 1.46925 0.734625 0.678473i \(-0.237358\pi\)
0.734625 + 0.678473i \(0.237358\pi\)
\(830\) −4.79994e29 −0.107378
\(831\) −5.65663e30 −1.24953
\(832\) 1.12329e30 0.245018
\(833\) −4.25974e30 −0.917509
\(834\) −2.08726e30 −0.443949
\(835\) 2.70765e30 0.568702
\(836\) 1.26476e30 0.262326
\(837\) 1.27546e31 2.61245
\(838\) 1.50078e28 0.00303566
\(839\) 5.47358e30 1.09338 0.546690 0.837335i \(-0.315888\pi\)
0.546690 + 0.837335i \(0.315888\pi\)
\(840\) 1.75859e30 0.346922
\(841\) −3.02816e30 −0.589957
\(842\) 2.12623e30 0.409103
\(843\) −2.84509e30 −0.540639
\(844\) −2.70079e30 −0.506868
\(845\) −6.85741e30 −1.27106
\(846\) −7.32508e30 −1.34098
\(847\) 9.20073e28 0.0166359
\(848\) −8.98318e29 −0.160426
\(849\) −1.26579e31 −2.23271
\(850\) 1.14302e30 0.199138
\(851\) −2.92213e29 −0.0502851
\(852\) 9.62598e30 1.63617
\(853\) −7.71396e30 −1.29513 −0.647564 0.762011i \(-0.724212\pi\)
−0.647564 + 0.762011i \(0.724212\pi\)
\(854\) 4.46203e30 0.739989
\(855\) −2.76060e30 −0.452229
\(856\) 3.32482e30 0.538015
\(857\) 1.69666e29 0.0271205 0.0135602 0.999908i \(-0.495684\pi\)
0.0135602 + 0.999908i \(0.495684\pi\)
\(858\) 1.48892e31 2.35101
\(859\) 4.91435e30 0.766545 0.383273 0.923635i \(-0.374797\pi\)
0.383273 + 0.923635i \(0.374797\pi\)
\(860\) 8.56833e29 0.132027
\(861\) 1.33103e31 2.02607
\(862\) 2.95829e30 0.444851
\(863\) 1.11211e31 1.65209 0.826047 0.563602i \(-0.190585\pi\)
0.826047 + 0.563602i \(0.190585\pi\)
\(864\) 1.88147e30 0.276123
\(865\) 3.75482e30 0.544403
\(866\) −6.74375e30 −0.965970
\(867\) −1.18569e31 −1.67792
\(868\) 7.68685e30 1.07471
\(869\) 9.30391e29 0.128516
\(870\) −2.53325e30 −0.345721
\(871\) 7.82194e30 1.05469
\(872\) 3.05480e30 0.406967
\(873\) 1.45414e31 1.91406
\(874\) 3.84211e29 0.0499689
\(875\) 8.94500e29 0.114946
\(876\) −1.70043e30 −0.215906
\(877\) −1.05245e31 −1.32041 −0.660203 0.751088i \(-0.729530\pi\)
−0.660203 + 0.751088i \(0.729530\pi\)
\(878\) 9.55692e29 0.118475
\(879\) 1.38101e31 1.69166
\(880\) −9.17685e29 −0.111077
\(881\) −8.70976e30 −1.04174 −0.520870 0.853636i \(-0.674392\pi\)
−0.520870 + 0.853636i \(0.674392\pi\)
\(882\) −7.46483e30 −0.882267
\(883\) 7.95603e30 0.929199 0.464600 0.885521i \(-0.346198\pi\)
0.464600 + 0.885521i \(0.346198\pi\)
\(884\) −1.19576e31 −1.38006
\(885\) 1.40002e30 0.159672
\(886\) −9.92275e30 −1.11835
\(887\) −1.08949e31 −1.21346 −0.606729 0.794909i \(-0.707519\pi\)
−0.606729 + 0.794909i \(0.707519\pi\)
\(888\) 2.06076e30 0.226825
\(889\) −1.54025e31 −1.67542
\(890\) 2.35270e30 0.252913
\(891\) 7.03150e30 0.747020
\(892\) −7.59851e30 −0.797806
\(893\) 5.04007e30 0.522993
\(894\) −5.36237e30 −0.549937
\(895\) −4.43718e29 −0.0449743
\(896\) 1.13391e30 0.113591
\(897\) 4.52308e30 0.447830
\(898\) 1.01466e31 0.992935
\(899\) −1.10729e31 −1.07099
\(900\) 2.00304e30 0.191489
\(901\) 9.56274e30 0.903591
\(902\) −6.94572e30 −0.648707
\(903\) −1.40333e31 −1.29550
\(904\) 6.02473e30 0.549755
\(905\) −8.41758e30 −0.759236
\(906\) −1.57501e31 −1.40423
\(907\) 7.14477e30 0.629668 0.314834 0.949147i \(-0.398051\pi\)
0.314834 + 0.949147i \(0.398051\pi\)
\(908\) −1.62661e30 −0.141704
\(909\) 6.48864e30 0.558770
\(910\) −9.35779e30 −0.796597
\(911\) −9.25953e30 −0.779194 −0.389597 0.920985i \(-0.627386\pi\)
−0.389597 + 0.920985i \(0.627386\pi\)
\(912\) −2.70955e30 −0.225399
\(913\) −4.10233e30 −0.337355
\(914\) 1.44447e31 1.17429
\(915\) 7.73636e30 0.621749
\(916\) −1.04379e31 −0.829299
\(917\) 1.34325e31 1.05506
\(918\) −2.00285e31 −1.55525
\(919\) 2.16799e31 1.66435 0.832174 0.554515i \(-0.187096\pi\)
0.832174 + 0.554515i \(0.187096\pi\)
\(920\) −2.78777e29 −0.0211585
\(921\) 2.07817e31 1.55939
\(922\) −1.43712e29 −0.0106615
\(923\) −5.12218e31 −3.75696
\(924\) 1.50300e31 1.08994
\(925\) 1.04820e30 0.0751545
\(926\) 1.57903e31 1.11937
\(927\) 1.15390e31 0.808779
\(928\) −1.63340e30 −0.113198
\(929\) −1.54089e31 −1.05586 −0.527929 0.849289i \(-0.677031\pi\)
−0.527929 + 0.849289i \(0.677031\pi\)
\(930\) 1.33276e31 0.902987
\(931\) 5.13623e30 0.344091
\(932\) −4.85216e29 −0.0321417
\(933\) 2.90558e31 1.90316
\(934\) −1.45578e31 −0.942875
\(935\) 9.76891e30 0.625639
\(936\) −2.09548e31 −1.32705
\(937\) −2.18719e31 −1.36969 −0.684843 0.728691i \(-0.740129\pi\)
−0.684843 + 0.728691i \(0.740129\pi\)
\(938\) 7.89589e30 0.488958
\(939\) −3.92070e30 −0.240091
\(940\) −3.65698e30 −0.221452
\(941\) −5.71240e29 −0.0342080 −0.0171040 0.999854i \(-0.505445\pi\)
−0.0171040 + 0.999854i \(0.505445\pi\)
\(942\) 3.15601e31 1.86898
\(943\) −2.10999e30 −0.123568
\(944\) 9.02711e29 0.0522809
\(945\) −1.56739e31 −0.897725
\(946\) 7.32302e30 0.414795
\(947\) 6.38251e29 0.0357534 0.0178767 0.999840i \(-0.494309\pi\)
0.0178767 + 0.999840i \(0.494309\pi\)
\(948\) −1.99322e30 −0.110425
\(949\) 9.04832e30 0.495761
\(950\) −1.37820e30 −0.0746820
\(951\) −1.93397e31 −1.03646
\(952\) −1.20707e31 −0.639800
\(953\) −2.06863e31 −1.08444 −0.542222 0.840235i \(-0.682417\pi\)
−0.542222 + 0.840235i \(0.682417\pi\)
\(954\) 1.67579e31 0.868884
\(955\) 2.26147e30 0.115973
\(956\) 1.11368e31 0.564873
\(957\) −2.16507e31 −1.08617
\(958\) −1.87043e31 −0.928120
\(959\) 3.35494e31 1.64661
\(960\) 1.96600e30 0.0954412
\(961\) 3.74300e31 1.79731
\(962\) −1.09657e31 −0.520833
\(963\) −6.20238e31 −2.91395
\(964\) 1.75847e30 0.0817195
\(965\) 1.10828e31 0.509461
\(966\) 4.56584e30 0.207616
\(967\) 2.49286e31 1.12129 0.560647 0.828055i \(-0.310552\pi\)
0.560647 + 0.828055i \(0.310552\pi\)
\(968\) 1.02859e29 0.00457668
\(969\) 2.88436e31 1.26955
\(970\) 7.25967e30 0.316092
\(971\) 4.80972e30 0.207166 0.103583 0.994621i \(-0.466969\pi\)
0.103583 + 0.994621i \(0.466969\pi\)
\(972\) 3.26528e30 0.139131
\(973\) 1.12117e31 0.472594
\(974\) −3.27369e31 −1.36511
\(975\) −1.62247e31 −0.669313
\(976\) 4.98830e30 0.203577
\(977\) 1.25924e31 0.508411 0.254205 0.967150i \(-0.418186\pi\)
0.254205 + 0.967150i \(0.418186\pi\)
\(978\) 1.67476e31 0.668953
\(979\) 2.01076e31 0.794589
\(980\) −3.72675e30 −0.145699
\(981\) −5.69865e31 −2.20418
\(982\) 1.09221e31 0.417961
\(983\) 1.94974e31 0.738185 0.369093 0.929393i \(-0.379669\pi\)
0.369093 + 0.929393i \(0.379669\pi\)
\(984\) 1.48801e31 0.557389
\(985\) 6.15011e30 0.227930
\(986\) 1.73878e31 0.637585
\(987\) 5.98945e31 2.17299
\(988\) 1.44181e31 0.517558
\(989\) 2.22461e30 0.0790118
\(990\) 1.71192e31 0.601608
\(991\) −1.67935e31 −0.583941 −0.291971 0.956427i \(-0.594311\pi\)
−0.291971 + 0.956427i \(0.594311\pi\)
\(992\) 8.59346e30 0.295662
\(993\) 3.81047e31 1.29721
\(994\) −5.17060e31 −1.74174
\(995\) 6.96705e30 0.232224
\(996\) 8.78861e30 0.289866
\(997\) 2.24576e31 0.732932 0.366466 0.930431i \(-0.380568\pi\)
0.366466 + 0.930431i \(0.380568\pi\)
\(998\) 1.22536e31 0.395724
\(999\) −1.83671e31 −0.586953
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.b.1.1 2
4.3 odd 2 80.22.a.d.1.2 2
5.2 odd 4 50.22.b.f.49.2 4
5.3 odd 4 50.22.b.f.49.3 4
5.4 even 2 50.22.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.b.1.1 2 1.1 even 1 trivial
50.22.a.f.1.2 2 5.4 even 2
50.22.b.f.49.2 4 5.2 odd 4
50.22.b.f.49.3 4 5.3 odd 4
80.22.a.d.1.2 2 4.3 odd 2