Properties

Label 10.20.a.d.1.1
Level $10$
Weight $20$
Character 10.1
Self dual yes
Analytic conductor $22.882$
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] = [10,20,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, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
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: \(22.8816696556\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2925852 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1711.01\) 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)
 
\(f(q)\) \(=\) \(q+512.000 q^{2} -17348.2 q^{3} +262144. q^{4} +1.95312e6 q^{5} -8.88230e6 q^{6} +5.57963e7 q^{7} +1.34218e8 q^{8} -8.61300e8 q^{9} +1.00000e9 q^{10} -5.35830e9 q^{11} -4.54774e9 q^{12} +6.91495e10 q^{13} +2.85677e10 q^{14} -3.38833e10 q^{15} +6.87195e10 q^{16} +8.22149e11 q^{17} -4.40986e11 q^{18} -3.19849e11 q^{19} +5.12000e11 q^{20} -9.67968e11 q^{21} -2.74345e12 q^{22} +8.01595e12 q^{23} -2.32844e12 q^{24} +3.81470e12 q^{25} +3.54046e13 q^{26} +3.51052e13 q^{27} +1.46267e13 q^{28} +2.41111e13 q^{29} -1.73482e13 q^{30} +2.07759e14 q^{31} +3.51844e13 q^{32} +9.29571e13 q^{33} +4.20940e14 q^{34} +1.08977e14 q^{35} -2.25785e14 q^{36} -2.90774e14 q^{37} -1.63763e14 q^{38} -1.19962e15 q^{39} +2.62144e14 q^{40} +9.57878e14 q^{41} -4.95600e14 q^{42} -5.00622e15 q^{43} -1.40465e15 q^{44} -1.68223e15 q^{45} +4.10417e15 q^{46} -3.13847e14 q^{47} -1.19216e15 q^{48} -8.28567e15 q^{49} +1.95312e15 q^{50} -1.42628e16 q^{51} +1.81271e16 q^{52} -1.16717e16 q^{53} +1.79739e16 q^{54} -1.04654e16 q^{55} +7.48886e15 q^{56} +5.54882e15 q^{57} +1.23449e16 q^{58} -2.11995e16 q^{59} -8.88230e15 q^{60} +4.20425e16 q^{61} +1.06373e17 q^{62} -4.80574e16 q^{63} +1.80144e16 q^{64} +1.35058e17 q^{65} +4.75940e16 q^{66} +2.45879e17 q^{67} +2.15521e17 q^{68} -1.39063e17 q^{69} +5.57963e16 q^{70} -3.98392e17 q^{71} -1.15602e17 q^{72} -5.53484e17 q^{73} -1.48876e17 q^{74} -6.61783e16 q^{75} -8.38464e16 q^{76} -2.98973e17 q^{77} -6.14207e17 q^{78} -1.76148e18 q^{79} +1.34218e17 q^{80} +3.92041e17 q^{81} +4.90433e17 q^{82} +1.89559e18 q^{83} -2.53747e17 q^{84} +1.60576e18 q^{85} -2.56318e18 q^{86} -4.18286e17 q^{87} -7.19179e17 q^{88} +4.56180e18 q^{89} -8.61300e17 q^{90} +3.85829e18 q^{91} +2.10133e18 q^{92} -3.60426e18 q^{93} -1.60690e17 q^{94} -6.24705e17 q^{95} -6.10387e17 q^{96} +1.22283e19 q^{97} -4.24226e18 q^{98} +4.61510e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1024 q^{2} + 33724 q^{3} + 524288 q^{4} + 3906250 q^{5} + 17266688 q^{6} + 83061292 q^{7} + 268435456 q^{8} + 584812954 q^{9} + 2000000000 q^{10} + 3549480144 q^{11} + 8840544256 q^{12} + 30250225564 q^{13}+ \cdots + 17\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 512.000 0.707107
\(3\) −17348.2 −0.508866 −0.254433 0.967090i \(-0.581889\pi\)
−0.254433 + 0.967090i \(0.581889\pi\)
\(4\) 262144. 0.500000
\(5\) 1.95312e6 0.447214
\(6\) −8.88230e6 −0.359823
\(7\) 5.57963e7 0.522606 0.261303 0.965257i \(-0.415848\pi\)
0.261303 + 0.965257i \(0.415848\pi\)
\(8\) 1.34218e8 0.353553
\(9\) −8.61300e8 −0.741055
\(10\) 1.00000e9 0.316228
\(11\) −5.35830e9 −0.685167 −0.342584 0.939487i \(-0.611302\pi\)
−0.342584 + 0.939487i \(0.611302\pi\)
\(12\) −4.54774e9 −0.254433
\(13\) 6.91495e10 1.80854 0.904268 0.426965i \(-0.140417\pi\)
0.904268 + 0.426965i \(0.140417\pi\)
\(14\) 2.85677e10 0.369538
\(15\) −3.38833e10 −0.227572
\(16\) 6.87195e10 0.250000
\(17\) 8.22149e11 1.68146 0.840728 0.541457i \(-0.182127\pi\)
0.840728 + 0.541457i \(0.182127\pi\)
\(18\) −4.40986e11 −0.524005
\(19\) −3.19849e11 −0.227397 −0.113699 0.993515i \(-0.536270\pi\)
−0.113699 + 0.993515i \(0.536270\pi\)
\(20\) 5.12000e11 0.223607
\(21\) −9.67968e11 −0.265936
\(22\) −2.74345e12 −0.484486
\(23\) 8.01595e12 0.927984 0.463992 0.885839i \(-0.346417\pi\)
0.463992 + 0.885839i \(0.346417\pi\)
\(24\) −2.32844e12 −0.179911
\(25\) 3.81470e12 0.200000
\(26\) 3.54046e13 1.27883
\(27\) 3.51052e13 0.885964
\(28\) 1.46267e13 0.261303
\(29\) 2.41111e13 0.308629 0.154314 0.988022i \(-0.450683\pi\)
0.154314 + 0.988022i \(0.450683\pi\)
\(30\) −1.73482e13 −0.160918
\(31\) 2.07759e14 1.41132 0.705658 0.708553i \(-0.250652\pi\)
0.705658 + 0.708553i \(0.250652\pi\)
\(32\) 3.51844e13 0.176777
\(33\) 9.29571e13 0.348659
\(34\) 4.20940e14 1.18897
\(35\) 1.08977e14 0.233716
\(36\) −2.25785e14 −0.370528
\(37\) −2.90774e14 −0.367823 −0.183912 0.982943i \(-0.558876\pi\)
−0.183912 + 0.982943i \(0.558876\pi\)
\(38\) −1.63763e14 −0.160794
\(39\) −1.19962e15 −0.920303
\(40\) 2.62144e14 0.158114
\(41\) 9.57878e14 0.456945 0.228472 0.973550i \(-0.426627\pi\)
0.228472 + 0.973550i \(0.426627\pi\)
\(42\) −4.95600e14 −0.188045
\(43\) −5.00622e15 −1.51901 −0.759503 0.650503i \(-0.774558\pi\)
−0.759503 + 0.650503i \(0.774558\pi\)
\(44\) −1.40465e15 −0.342584
\(45\) −1.68223e15 −0.331410
\(46\) 4.10417e15 0.656184
\(47\) −3.13847e14 −0.0409062 −0.0204531 0.999791i \(-0.506511\pi\)
−0.0204531 + 0.999791i \(0.506511\pi\)
\(48\) −1.19216e15 −0.127217
\(49\) −8.28567e15 −0.726883
\(50\) 1.95312e15 0.141421
\(51\) −1.42628e16 −0.855636
\(52\) 1.81271e16 0.904268
\(53\) −1.16717e16 −0.485861 −0.242930 0.970044i \(-0.578109\pi\)
−0.242930 + 0.970044i \(0.578109\pi\)
\(54\) 1.79739e16 0.626471
\(55\) −1.04654e16 −0.306416
\(56\) 7.48886e15 0.184769
\(57\) 5.54882e15 0.115715
\(58\) 1.23449e16 0.218233
\(59\) −2.11995e16 −0.318590 −0.159295 0.987231i \(-0.550922\pi\)
−0.159295 + 0.987231i \(0.550922\pi\)
\(60\) −8.88230e15 −0.113786
\(61\) 4.20425e16 0.460314 0.230157 0.973153i \(-0.426076\pi\)
0.230157 + 0.973153i \(0.426076\pi\)
\(62\) 1.06373e17 0.997951
\(63\) −4.80574e16 −0.387280
\(64\) 1.80144e16 0.125000
\(65\) 1.35058e17 0.808802
\(66\) 4.75940e16 0.246539
\(67\) 2.45879e17 1.10410 0.552052 0.833810i \(-0.313845\pi\)
0.552052 + 0.833810i \(0.313845\pi\)
\(68\) 2.15521e17 0.840728
\(69\) −1.39063e17 −0.472220
\(70\) 5.57963e16 0.165262
\(71\) −3.98392e17 −1.03123 −0.515616 0.856820i \(-0.672437\pi\)
−0.515616 + 0.856820i \(0.672437\pi\)
\(72\) −1.15602e17 −0.262003
\(73\) −5.53484e17 −1.10037 −0.550184 0.835044i \(-0.685442\pi\)
−0.550184 + 0.835044i \(0.685442\pi\)
\(74\) −1.48876e17 −0.260090
\(75\) −6.61783e16 −0.101773
\(76\) −8.38464e16 −0.113699
\(77\) −2.98973e17 −0.358072
\(78\) −6.14207e17 −0.650753
\(79\) −1.76148e18 −1.65356 −0.826780 0.562526i \(-0.809830\pi\)
−0.826780 + 0.562526i \(0.809830\pi\)
\(80\) 1.34218e17 0.111803
\(81\) 3.92041e17 0.290218
\(82\) 4.90433e17 0.323109
\(83\) 1.89559e18 1.11302 0.556511 0.830841i \(-0.312140\pi\)
0.556511 + 0.830841i \(0.312140\pi\)
\(84\) −2.53747e17 −0.132968
\(85\) 1.60576e18 0.751970
\(86\) −2.56318e18 −1.07410
\(87\) −4.18286e17 −0.157051
\(88\) −7.19179e17 −0.242243
\(89\) 4.56180e18 1.38016 0.690082 0.723731i \(-0.257574\pi\)
0.690082 + 0.723731i \(0.257574\pi\)
\(90\) −8.61300e17 −0.234342
\(91\) 3.85829e18 0.945152
\(92\) 2.10133e18 0.463992
\(93\) −3.60426e18 −0.718171
\(94\) −1.60690e17 −0.0289250
\(95\) −6.24705e17 −0.101695
\(96\) −6.10387e17 −0.0899557
\(97\) 1.22283e19 1.63318 0.816592 0.577215i \(-0.195860\pi\)
0.816592 + 0.577215i \(0.195860\pi\)
\(98\) −4.24226e18 −0.513984
\(99\) 4.61510e18 0.507747
\(100\) 1.00000e18 0.100000
\(101\) −1.90414e18 −0.173239 −0.0866194 0.996241i \(-0.527606\pi\)
−0.0866194 + 0.996241i \(0.527606\pi\)
\(102\) −7.30257e18 −0.605026
\(103\) 5.63296e17 0.0425386 0.0212693 0.999774i \(-0.493229\pi\)
0.0212693 + 0.999774i \(0.493229\pi\)
\(104\) 9.28109e18 0.639414
\(105\) −1.89056e18 −0.118930
\(106\) −5.97589e18 −0.343556
\(107\) −3.13426e19 −1.64812 −0.824062 0.566500i \(-0.808297\pi\)
−0.824062 + 0.566500i \(0.808297\pi\)
\(108\) 9.20263e18 0.442982
\(109\) 1.61520e19 0.712319 0.356160 0.934425i \(-0.384086\pi\)
0.356160 + 0.934425i \(0.384086\pi\)
\(110\) −5.35830e18 −0.216669
\(111\) 5.04442e18 0.187173
\(112\) 3.83429e18 0.130651
\(113\) −4.99152e19 −1.56310 −0.781551 0.623841i \(-0.785571\pi\)
−0.781551 + 0.623841i \(0.785571\pi\)
\(114\) 2.84099e18 0.0818228
\(115\) 1.56562e19 0.415007
\(116\) 6.32058e18 0.154314
\(117\) −5.95585e19 −1.34023
\(118\) −1.08542e19 −0.225277
\(119\) 4.58729e19 0.878739
\(120\) −4.54774e18 −0.0804588
\(121\) −3.24477e19 −0.530546
\(122\) 2.15257e19 0.325491
\(123\) −1.66175e19 −0.232524
\(124\) 5.44628e19 0.705658
\(125\) 7.45058e18 0.0894427
\(126\) −2.46054e19 −0.273848
\(127\) −5.46401e19 −0.564126 −0.282063 0.959396i \(-0.591019\pi\)
−0.282063 + 0.959396i \(0.591019\pi\)
\(128\) 9.22337e18 0.0883883
\(129\) 8.68491e19 0.772971
\(130\) 6.91495e19 0.571910
\(131\) −1.25613e20 −0.965958 −0.482979 0.875632i \(-0.660445\pi\)
−0.482979 + 0.875632i \(0.660445\pi\)
\(132\) 2.43682e19 0.174329
\(133\) −1.78464e19 −0.118839
\(134\) 1.25890e20 0.780720
\(135\) 6.85649e19 0.396215
\(136\) 1.10347e20 0.594485
\(137\) 2.02938e20 1.01981 0.509904 0.860231i \(-0.329681\pi\)
0.509904 + 0.860231i \(0.329681\pi\)
\(138\) −7.12001e19 −0.333910
\(139\) 3.97467e20 1.74045 0.870223 0.492657i \(-0.163974\pi\)
0.870223 + 0.492657i \(0.163974\pi\)
\(140\) 2.85677e19 0.116858
\(141\) 5.44470e18 0.0208158
\(142\) −2.03977e20 −0.729191
\(143\) −3.70524e20 −1.23915
\(144\) −5.91881e19 −0.185264
\(145\) 4.70920e19 0.138023
\(146\) −2.83384e20 −0.778077
\(147\) 1.43742e20 0.369886
\(148\) −7.62246e19 −0.183912
\(149\) 7.30310e19 0.165287 0.0826433 0.996579i \(-0.473664\pi\)
0.0826433 + 0.996579i \(0.473664\pi\)
\(150\) −3.38833e19 −0.0719646
\(151\) −5.89188e20 −1.17482 −0.587412 0.809288i \(-0.699853\pi\)
−0.587412 + 0.809288i \(0.699853\pi\)
\(152\) −4.29294e19 −0.0803971
\(153\) −7.08116e20 −1.24605
\(154\) −1.53074e20 −0.253195
\(155\) 4.05780e20 0.631159
\(156\) −3.14474e20 −0.460152
\(157\) 4.26722e20 0.587623 0.293811 0.955863i \(-0.405076\pi\)
0.293811 + 0.955863i \(0.405076\pi\)
\(158\) −9.01876e20 −1.16924
\(159\) 2.02483e20 0.247238
\(160\) 6.87195e19 0.0790569
\(161\) 4.47261e20 0.484970
\(162\) 2.00725e20 0.205215
\(163\) −1.38073e21 −1.33146 −0.665729 0.746194i \(-0.731879\pi\)
−0.665729 + 0.746194i \(0.731879\pi\)
\(164\) 2.51102e20 0.228472
\(165\) 1.81557e20 0.155925
\(166\) 9.70544e20 0.787025
\(167\) 1.82808e19 0.0140020 0.00700099 0.999975i \(-0.497771\pi\)
0.00700099 + 0.999975i \(0.497771\pi\)
\(168\) −1.29918e20 −0.0940227
\(169\) 3.31974e21 2.27081
\(170\) 8.22149e20 0.531723
\(171\) 2.75486e20 0.168514
\(172\) −1.31235e21 −0.759503
\(173\) 7.10579e20 0.389201 0.194601 0.980883i \(-0.437659\pi\)
0.194601 + 0.980883i \(0.437659\pi\)
\(174\) −2.14162e20 −0.111052
\(175\) 2.12846e20 0.104521
\(176\) −3.68220e20 −0.171292
\(177\) 3.67775e20 0.162120
\(178\) 2.33564e21 0.975923
\(179\) −1.09003e21 −0.431851 −0.215926 0.976410i \(-0.569277\pi\)
−0.215926 + 0.976410i \(0.569277\pi\)
\(180\) −4.40986e20 −0.165705
\(181\) −2.79419e21 −0.996114 −0.498057 0.867144i \(-0.665953\pi\)
−0.498057 + 0.867144i \(0.665953\pi\)
\(182\) 1.97544e21 0.668323
\(183\) −7.29363e20 −0.234239
\(184\) 1.07588e21 0.328092
\(185\) −5.67918e20 −0.164495
\(186\) −1.84538e21 −0.507823
\(187\) −4.40532e21 −1.15208
\(188\) −8.22732e19 −0.0204531
\(189\) 1.95874e21 0.463010
\(190\) −3.19849e20 −0.0719094
\(191\) 4.63773e21 0.991947 0.495974 0.868338i \(-0.334811\pi\)
0.495974 + 0.868338i \(0.334811\pi\)
\(192\) −3.12518e20 −0.0636083
\(193\) −8.23942e20 −0.159625 −0.0798127 0.996810i \(-0.525432\pi\)
−0.0798127 + 0.996810i \(0.525432\pi\)
\(194\) 6.26090e21 1.15484
\(195\) −2.34301e21 −0.411572
\(196\) −2.17204e21 −0.363442
\(197\) −8.41762e20 −0.134202 −0.0671012 0.997746i \(-0.521375\pi\)
−0.0671012 + 0.997746i \(0.521375\pi\)
\(198\) 2.36293e21 0.359031
\(199\) −1.16067e22 −1.68114 −0.840571 0.541701i \(-0.817780\pi\)
−0.840571 + 0.541701i \(0.817780\pi\)
\(200\) 5.12000e20 0.0707107
\(201\) −4.26556e21 −0.561841
\(202\) −9.74918e20 −0.122498
\(203\) 1.34531e21 0.161291
\(204\) −3.73892e21 −0.427818
\(205\) 1.87086e21 0.204352
\(206\) 2.88407e20 0.0300793
\(207\) −6.90414e21 −0.687688
\(208\) 4.75192e21 0.452134
\(209\) 1.71385e21 0.155805
\(210\) −9.67968e20 −0.0840965
\(211\) 7.16116e20 0.0594703 0.0297352 0.999558i \(-0.490534\pi\)
0.0297352 + 0.999558i \(0.490534\pi\)
\(212\) −3.05966e21 −0.242930
\(213\) 6.91140e21 0.524759
\(214\) −1.60474e22 −1.16540
\(215\) −9.77777e21 −0.679320
\(216\) 4.71175e21 0.313236
\(217\) 1.15922e22 0.737561
\(218\) 8.26983e21 0.503686
\(219\) 9.60197e21 0.559940
\(220\) −2.74345e21 −0.153208
\(221\) 5.68512e22 3.04098
\(222\) 2.58274e21 0.132351
\(223\) −1.19410e22 −0.586335 −0.293168 0.956061i \(-0.594709\pi\)
−0.293168 + 0.956061i \(0.594709\pi\)
\(224\) 1.96316e21 0.0923845
\(225\) −3.28560e21 −0.148211
\(226\) −2.55566e22 −1.10528
\(227\) 2.69537e22 1.11782 0.558911 0.829228i \(-0.311219\pi\)
0.558911 + 0.829228i \(0.311219\pi\)
\(228\) 1.45459e21 0.0578574
\(229\) −4.02051e22 −1.53406 −0.767032 0.641609i \(-0.778267\pi\)
−0.767032 + 0.641609i \(0.778267\pi\)
\(230\) 8.01595e21 0.293454
\(231\) 5.18667e21 0.182211
\(232\) 3.23614e21 0.109117
\(233\) 5.19383e22 1.68115 0.840575 0.541696i \(-0.182217\pi\)
0.840575 + 0.541696i \(0.182217\pi\)
\(234\) −3.04939e22 −0.947683
\(235\) −6.12983e20 −0.0182938
\(236\) −5.55733e21 −0.159295
\(237\) 3.05585e22 0.841440
\(238\) 2.34869e22 0.621362
\(239\) −5.71264e22 −1.45230 −0.726151 0.687535i \(-0.758693\pi\)
−0.726151 + 0.687535i \(0.758693\pi\)
\(240\) −2.32844e21 −0.0568930
\(241\) 2.38712e22 0.560675 0.280338 0.959901i \(-0.409554\pi\)
0.280338 + 0.959901i \(0.409554\pi\)
\(242\) −1.66132e22 −0.375153
\(243\) −4.76027e22 −1.03365
\(244\) 1.10212e22 0.230157
\(245\) −1.61829e22 −0.325072
\(246\) −8.50816e21 −0.164419
\(247\) −2.21174e22 −0.411257
\(248\) 2.78850e22 0.498975
\(249\) −3.28852e22 −0.566379
\(250\) 3.81470e21 0.0632456
\(251\) −9.12101e22 −1.45594 −0.727968 0.685611i \(-0.759535\pi\)
−0.727968 + 0.685611i \(0.759535\pi\)
\(252\) −1.25979e22 −0.193640
\(253\) −4.29519e22 −0.635825
\(254\) −2.79757e22 −0.398897
\(255\) −2.78571e22 −0.382652
\(256\) 4.72237e21 0.0625000
\(257\) 9.15373e22 1.16744 0.583718 0.811956i \(-0.301597\pi\)
0.583718 + 0.811956i \(0.301597\pi\)
\(258\) 4.44668e22 0.546573
\(259\) −1.62241e22 −0.192226
\(260\) 3.54046e22 0.404401
\(261\) −2.07669e22 −0.228711
\(262\) −6.43140e22 −0.683036
\(263\) 5.82421e22 0.596565 0.298282 0.954478i \(-0.403586\pi\)
0.298282 + 0.954478i \(0.403586\pi\)
\(264\) 1.24765e22 0.123269
\(265\) −2.27962e22 −0.217284
\(266\) −9.13735e21 −0.0840320
\(267\) −7.91391e22 −0.702319
\(268\) 6.44556e22 0.552052
\(269\) 2.11389e23 1.74758 0.873788 0.486307i \(-0.161656\pi\)
0.873788 + 0.486307i \(0.161656\pi\)
\(270\) 3.51052e22 0.280166
\(271\) 1.73503e23 1.33690 0.668451 0.743757i \(-0.266958\pi\)
0.668451 + 0.743757i \(0.266958\pi\)
\(272\) 5.64976e22 0.420364
\(273\) −6.69345e22 −0.480956
\(274\) 1.03904e23 0.721113
\(275\) −2.04403e22 −0.137033
\(276\) −3.64545e22 −0.236110
\(277\) −8.04324e22 −0.503354 −0.251677 0.967811i \(-0.580982\pi\)
−0.251677 + 0.967811i \(0.580982\pi\)
\(278\) 2.03503e23 1.23068
\(279\) −1.78943e23 −1.04586
\(280\) 1.46267e22 0.0826312
\(281\) 3.28300e23 1.79292 0.896459 0.443127i \(-0.146131\pi\)
0.896459 + 0.443127i \(0.146131\pi\)
\(282\) 2.78769e21 0.0147190
\(283\) −3.19635e23 −1.63186 −0.815930 0.578151i \(-0.803775\pi\)
−0.815930 + 0.578151i \(0.803775\pi\)
\(284\) −1.04436e23 −0.515616
\(285\) 1.08375e22 0.0517493
\(286\) −1.89708e23 −0.876212
\(287\) 5.34461e22 0.238802
\(288\) −3.03043e22 −0.131001
\(289\) 4.36856e23 1.82730
\(290\) 2.41111e22 0.0975970
\(291\) −2.12140e23 −0.831073
\(292\) −1.45092e23 −0.550184
\(293\) −4.15387e23 −1.52479 −0.762396 0.647110i \(-0.775977\pi\)
−0.762396 + 0.647110i \(0.775977\pi\)
\(294\) 7.35958e22 0.261549
\(295\) −4.14053e22 −0.142478
\(296\) −3.90270e22 −0.130045
\(297\) −1.88104e23 −0.607034
\(298\) 3.73919e22 0.116875
\(299\) 5.54299e23 1.67829
\(300\) −1.73482e22 −0.0508866
\(301\) −2.79329e23 −0.793842
\(302\) −3.01664e23 −0.830726
\(303\) 3.30334e22 0.0881553
\(304\) −2.19798e22 −0.0568494
\(305\) 8.21142e22 0.205859
\(306\) −3.62556e23 −0.881092
\(307\) 4.98091e23 1.17353 0.586765 0.809757i \(-0.300401\pi\)
0.586765 + 0.809757i \(0.300401\pi\)
\(308\) −7.83741e22 −0.179036
\(309\) −9.77219e21 −0.0216464
\(310\) 2.07759e23 0.446297
\(311\) 4.96675e23 1.03478 0.517390 0.855750i \(-0.326904\pi\)
0.517390 + 0.855750i \(0.326904\pi\)
\(312\) −1.61011e23 −0.325376
\(313\) −8.89041e23 −1.74281 −0.871406 0.490562i \(-0.836792\pi\)
−0.871406 + 0.490562i \(0.836792\pi\)
\(314\) 2.18482e23 0.415512
\(315\) −9.38620e22 −0.173197
\(316\) −4.61761e23 −0.826780
\(317\) 2.61249e23 0.453934 0.226967 0.973902i \(-0.427119\pi\)
0.226967 + 0.973902i \(0.427119\pi\)
\(318\) 1.03671e23 0.174824
\(319\) −1.29195e23 −0.211462
\(320\) 3.51844e22 0.0559017
\(321\) 5.43740e23 0.838674
\(322\) 2.28997e23 0.342926
\(323\) −2.62963e23 −0.382359
\(324\) 1.02771e23 0.145109
\(325\) 2.63784e23 0.361707
\(326\) −7.06935e23 −0.941483
\(327\) −2.80209e23 −0.362475
\(328\) 1.28564e23 0.161554
\(329\) −1.75115e22 −0.0213778
\(330\) 9.29571e22 0.110256
\(331\) 9.76825e22 0.112577 0.0562887 0.998415i \(-0.482073\pi\)
0.0562887 + 0.998415i \(0.482073\pi\)
\(332\) 4.96918e23 0.556511
\(333\) 2.50443e23 0.272577
\(334\) 9.35979e21 0.00990090
\(335\) 4.80232e23 0.493770
\(336\) −6.65183e22 −0.0664841
\(337\) 3.88676e23 0.377662 0.188831 0.982010i \(-0.439530\pi\)
0.188831 + 0.982010i \(0.439530\pi\)
\(338\) 1.69970e24 1.60570
\(339\) 8.65941e23 0.795410
\(340\) 4.20940e23 0.375985
\(341\) −1.11324e24 −0.966987
\(342\) 1.41049e23 0.119157
\(343\) −1.09833e24 −0.902479
\(344\) −6.71924e23 −0.537050
\(345\) −2.71607e23 −0.211183
\(346\) 3.63816e23 0.275207
\(347\) 6.38185e23 0.469696 0.234848 0.972032i \(-0.424541\pi\)
0.234848 + 0.972032i \(0.424541\pi\)
\(348\) −1.09651e23 −0.0785254
\(349\) −7.25583e23 −0.505645 −0.252822 0.967513i \(-0.581359\pi\)
−0.252822 + 0.967513i \(0.581359\pi\)
\(350\) 1.08977e23 0.0739076
\(351\) 2.42751e24 1.60230
\(352\) −1.88528e23 −0.121122
\(353\) −8.60947e23 −0.538415 −0.269207 0.963082i \(-0.586762\pi\)
−0.269207 + 0.963082i \(0.586762\pi\)
\(354\) 1.88301e23 0.114636
\(355\) −7.78109e23 −0.461181
\(356\) 1.19585e24 0.690082
\(357\) −7.95814e23 −0.447160
\(358\) −5.58095e23 −0.305365
\(359\) −1.95940e24 −1.04406 −0.522030 0.852927i \(-0.674825\pi\)
−0.522030 + 0.852927i \(0.674825\pi\)
\(360\) −2.25785e23 −0.117171
\(361\) −1.87612e24 −0.948290
\(362\) −1.43062e24 −0.704359
\(363\) 5.62911e23 0.269977
\(364\) 1.01143e24 0.472576
\(365\) −1.08102e24 −0.492099
\(366\) −3.73434e23 −0.165632
\(367\) −1.70386e24 −0.736387 −0.368193 0.929749i \(-0.620024\pi\)
−0.368193 + 0.929749i \(0.620024\pi\)
\(368\) 5.50852e23 0.231996
\(369\) −8.25020e23 −0.338621
\(370\) −2.90774e23 −0.116316
\(371\) −6.51236e23 −0.253914
\(372\) −9.44834e23 −0.359085
\(373\) −5.66530e22 −0.0209889 −0.0104944 0.999945i \(-0.503341\pi\)
−0.0104944 + 0.999945i \(0.503341\pi\)
\(374\) −2.25552e24 −0.814643
\(375\) −1.29255e23 −0.0455144
\(376\) −4.21239e22 −0.0144625
\(377\) 1.66727e24 0.558166
\(378\) 1.00288e24 0.327397
\(379\) 2.57014e23 0.0818247 0.0409123 0.999163i \(-0.486974\pi\)
0.0409123 + 0.999163i \(0.486974\pi\)
\(380\) −1.63763e23 −0.0508476
\(381\) 9.47909e23 0.287065
\(382\) 2.37452e24 0.701413
\(383\) 4.38632e24 1.26390 0.631949 0.775010i \(-0.282255\pi\)
0.631949 + 0.775010i \(0.282255\pi\)
\(384\) −1.60009e23 −0.0449778
\(385\) −5.83933e23 −0.160135
\(386\) −4.21858e23 −0.112872
\(387\) 4.31186e24 1.12567
\(388\) 3.20558e24 0.816592
\(389\) −1.37553e24 −0.341939 −0.170970 0.985276i \(-0.554690\pi\)
−0.170970 + 0.985276i \(0.554690\pi\)
\(390\) −1.19962e24 −0.291025
\(391\) 6.59030e24 1.56037
\(392\) −1.11208e24 −0.256992
\(393\) 2.17917e24 0.491544
\(394\) −4.30982e23 −0.0948954
\(395\) −3.44039e24 −0.739494
\(396\) 1.20982e24 0.253873
\(397\) −4.17970e24 −0.856319 −0.428160 0.903703i \(-0.640838\pi\)
−0.428160 + 0.903703i \(0.640838\pi\)
\(398\) −5.94263e24 −1.18875
\(399\) 3.09603e23 0.0604733
\(400\) 2.62144e23 0.0500000
\(401\) 3.45018e23 0.0642644 0.0321322 0.999484i \(-0.489770\pi\)
0.0321322 + 0.999484i \(0.489770\pi\)
\(402\) −2.18397e24 −0.397282
\(403\) 1.43664e25 2.55242
\(404\) −4.99158e23 −0.0866194
\(405\) 7.65706e23 0.129789
\(406\) 6.88799e23 0.114050
\(407\) 1.55805e24 0.252020
\(408\) −1.91433e24 −0.302513
\(409\) 1.82296e24 0.281452 0.140726 0.990049i \(-0.455056\pi\)
0.140726 + 0.990049i \(0.455056\pi\)
\(410\) 9.57878e23 0.144499
\(411\) −3.52062e24 −0.518946
\(412\) 1.47665e23 0.0212693
\(413\) −1.18286e24 −0.166497
\(414\) −3.53492e24 −0.486269
\(415\) 3.70233e24 0.497758
\(416\) 2.43298e24 0.319707
\(417\) −6.89536e24 −0.885655
\(418\) 8.77489e23 0.110171
\(419\) −1.44355e25 −1.77174 −0.885869 0.463935i \(-0.846437\pi\)
−0.885869 + 0.463935i \(0.846437\pi\)
\(420\) −4.95600e23 −0.0594652
\(421\) −4.71956e24 −0.553632 −0.276816 0.960923i \(-0.589279\pi\)
−0.276816 + 0.960923i \(0.589279\pi\)
\(422\) 3.66652e23 0.0420519
\(423\) 2.70317e23 0.0303137
\(424\) −1.56654e24 −0.171778
\(425\) 3.13625e24 0.336291
\(426\) 3.53864e24 0.371061
\(427\) 2.34581e24 0.240563
\(428\) −8.21629e24 −0.824062
\(429\) 6.42794e24 0.630562
\(430\) −5.00622e24 −0.480352
\(431\) −1.44126e25 −1.35272 −0.676361 0.736570i \(-0.736444\pi\)
−0.676361 + 0.736570i \(0.736444\pi\)
\(432\) 2.41241e24 0.221491
\(433\) −1.38601e25 −1.24489 −0.622447 0.782662i \(-0.713862\pi\)
−0.622447 + 0.782662i \(0.713862\pi\)
\(434\) 5.93520e24 0.521535
\(435\) −8.16964e23 −0.0702352
\(436\) 4.23415e24 0.356160
\(437\) −2.56389e24 −0.211021
\(438\) 4.91621e24 0.395937
\(439\) −1.25797e25 −0.991418 −0.495709 0.868489i \(-0.665092\pi\)
−0.495709 + 0.868489i \(0.665092\pi\)
\(440\) −1.40465e24 −0.108334
\(441\) 7.13644e24 0.538661
\(442\) 2.91078e25 2.15029
\(443\) 2.28568e24 0.165264 0.0826322 0.996580i \(-0.473667\pi\)
0.0826322 + 0.996580i \(0.473667\pi\)
\(444\) 1.32236e24 0.0935864
\(445\) 8.90976e24 0.617228
\(446\) −6.11381e24 −0.414601
\(447\) −1.26696e24 −0.0841088
\(448\) 1.00514e24 0.0653257
\(449\) −4.27635e24 −0.272103 −0.136051 0.990702i \(-0.543441\pi\)
−0.136051 + 0.990702i \(0.543441\pi\)
\(450\) −1.68223e24 −0.104801
\(451\) −5.13260e24 −0.313083
\(452\) −1.30850e25 −0.781551
\(453\) 1.02214e25 0.597828
\(454\) 1.38003e25 0.790420
\(455\) 7.53572e24 0.422685
\(456\) 7.44749e23 0.0409114
\(457\) 1.82618e25 0.982515 0.491257 0.871014i \(-0.336537\pi\)
0.491257 + 0.871014i \(0.336537\pi\)
\(458\) −2.05850e25 −1.08475
\(459\) 2.88617e25 1.48971
\(460\) 4.10417e24 0.207504
\(461\) −7.90654e24 −0.391587 −0.195793 0.980645i \(-0.562728\pi\)
−0.195793 + 0.980645i \(0.562728\pi\)
\(462\) 2.65557e24 0.128843
\(463\) −2.91664e25 −1.38632 −0.693160 0.720784i \(-0.743782\pi\)
−0.693160 + 0.720784i \(0.743782\pi\)
\(464\) 1.65690e24 0.0771572
\(465\) −7.03956e24 −0.321176
\(466\) 2.65924e25 1.18875
\(467\) −2.90132e25 −1.27082 −0.635411 0.772174i \(-0.719169\pi\)
−0.635411 + 0.772174i \(0.719169\pi\)
\(468\) −1.56129e25 −0.670113
\(469\) 1.37191e25 0.577011
\(470\) −3.13847e23 −0.0129357
\(471\) −7.40288e24 −0.299021
\(472\) −2.84535e24 −0.112639
\(473\) 2.68248e25 1.04077
\(474\) 1.56460e25 0.594988
\(475\) −1.22013e24 −0.0454795
\(476\) 1.20253e25 0.439369
\(477\) 1.00528e25 0.360050
\(478\) −2.92487e25 −1.02693
\(479\) 4.33547e25 1.49228 0.746138 0.665792i \(-0.231906\pi\)
0.746138 + 0.665792i \(0.231906\pi\)
\(480\) −1.19216e24 −0.0402294
\(481\) −2.01069e25 −0.665222
\(482\) 1.22220e25 0.396457
\(483\) −7.75919e24 −0.246785
\(484\) −8.50597e24 −0.265273
\(485\) 2.38834e25 0.730382
\(486\) −2.43726e25 −0.730898
\(487\) −2.20894e25 −0.649619 −0.324809 0.945780i \(-0.605300\pi\)
−0.324809 + 0.945780i \(0.605300\pi\)
\(488\) 5.64284e24 0.162746
\(489\) 2.39533e25 0.677534
\(490\) −8.28567e24 −0.229861
\(491\) −1.76617e25 −0.480573 −0.240286 0.970702i \(-0.577241\pi\)
−0.240286 + 0.970702i \(0.577241\pi\)
\(492\) −4.35618e24 −0.116262
\(493\) 1.98229e25 0.518946
\(494\) −1.13241e25 −0.290802
\(495\) 9.01388e24 0.227071
\(496\) 1.42771e25 0.352829
\(497\) −2.22288e25 −0.538928
\(498\) −1.68372e25 −0.400490
\(499\) −4.52317e25 −1.05557 −0.527786 0.849377i \(-0.676978\pi\)
−0.527786 + 0.849377i \(0.676978\pi\)
\(500\) 1.95313e24 0.0447214
\(501\) −3.17140e23 −0.00712514
\(502\) −4.66996e25 −1.02950
\(503\) 1.28554e25 0.278092 0.139046 0.990286i \(-0.455596\pi\)
0.139046 + 0.990286i \(0.455596\pi\)
\(504\) −6.45015e24 −0.136924
\(505\) −3.71902e24 −0.0774747
\(506\) −2.19914e25 −0.449596
\(507\) −5.75916e25 −1.15554
\(508\) −1.43236e25 −0.282063
\(509\) 4.19437e25 0.810677 0.405338 0.914167i \(-0.367154\pi\)
0.405338 + 0.914167i \(0.367154\pi\)
\(510\) −1.42628e25 −0.270576
\(511\) −3.08824e25 −0.575058
\(512\) 2.41785e24 0.0441942
\(513\) −1.12284e25 −0.201466
\(514\) 4.68671e25 0.825502
\(515\) 1.10019e24 0.0190238
\(516\) 2.27670e25 0.386486
\(517\) 1.68169e24 0.0280276
\(518\) −8.30674e24 −0.135925
\(519\) −1.23273e25 −0.198051
\(520\) 1.81271e25 0.285955
\(521\) 3.05094e25 0.472581 0.236290 0.971682i \(-0.424068\pi\)
0.236290 + 0.971682i \(0.424068\pi\)
\(522\) −1.06327e25 −0.161723
\(523\) 6.76475e23 0.0101038 0.00505191 0.999987i \(-0.498392\pi\)
0.00505191 + 0.999987i \(0.498392\pi\)
\(524\) −3.29288e25 −0.482979
\(525\) −3.69251e24 −0.0531873
\(526\) 2.98200e25 0.421835
\(527\) 1.70809e26 2.37306
\(528\) 6.38797e24 0.0871646
\(529\) −1.03600e25 −0.138845
\(530\) −1.16717e25 −0.153643
\(531\) 1.82592e25 0.236093
\(532\) −4.67832e24 −0.0594196
\(533\) 6.62368e25 0.826401
\(534\) −4.05192e25 −0.496614
\(535\) −6.12161e25 −0.737063
\(536\) 3.30013e25 0.390360
\(537\) 1.89101e25 0.219755
\(538\) 1.08231e26 1.23572
\(539\) 4.43971e25 0.498037
\(540\) 1.79739e25 0.198108
\(541\) −5.91055e25 −0.640109 −0.320055 0.947399i \(-0.603701\pi\)
−0.320055 + 0.947399i \(0.603701\pi\)
\(542\) 8.88336e25 0.945332
\(543\) 4.84742e25 0.506889
\(544\) 2.89268e25 0.297242
\(545\) 3.15469e25 0.318559
\(546\) −3.42705e25 −0.340087
\(547\) −8.46391e25 −0.825452 −0.412726 0.910855i \(-0.635423\pi\)
−0.412726 + 0.910855i \(0.635423\pi\)
\(548\) 5.31990e25 0.509904
\(549\) −3.62112e25 −0.341118
\(550\) −1.04654e25 −0.0968973
\(551\) −7.71191e24 −0.0701814
\(552\) −1.86647e25 −0.166955
\(553\) −9.82840e25 −0.864159
\(554\) −4.11814e25 −0.355925
\(555\) 9.85238e24 0.0837062
\(556\) 1.04194e26 0.870223
\(557\) 4.71567e25 0.387185 0.193593 0.981082i \(-0.437986\pi\)
0.193593 + 0.981082i \(0.437986\pi\)
\(558\) −9.16188e25 −0.739536
\(559\) −3.46178e26 −2.74718
\(560\) 7.48886e24 0.0584291
\(561\) 7.64246e25 0.586254
\(562\) 1.68089e26 1.26778
\(563\) −5.00520e25 −0.371186 −0.185593 0.982627i \(-0.559421\pi\)
−0.185593 + 0.982627i \(0.559421\pi\)
\(564\) 1.42730e24 0.0104079
\(565\) −9.74906e25 −0.699041
\(566\) −1.63653e26 −1.15390
\(567\) 2.18745e25 0.151670
\(568\) −5.34713e25 −0.364595
\(569\) −2.44961e26 −1.64259 −0.821296 0.570502i \(-0.806749\pi\)
−0.821296 + 0.570502i \(0.806749\pi\)
\(570\) 5.54882e24 0.0365923
\(571\) 2.66645e26 1.72938 0.864690 0.502306i \(-0.167515\pi\)
0.864690 + 0.502306i \(0.167515\pi\)
\(572\) −9.71306e25 −0.619575
\(573\) −8.04565e25 −0.504768
\(574\) 2.73644e25 0.168858
\(575\) 3.05784e25 0.185597
\(576\) −1.55158e25 −0.0926319
\(577\) −1.31556e25 −0.0772576 −0.0386288 0.999254i \(-0.512299\pi\)
−0.0386288 + 0.999254i \(0.512299\pi\)
\(578\) 2.23670e26 1.29209
\(579\) 1.42939e25 0.0812280
\(580\) 1.23449e25 0.0690115
\(581\) 1.05767e26 0.581671
\(582\) −1.08616e26 −0.587657
\(583\) 6.25403e25 0.332896
\(584\) −7.42873e25 −0.389039
\(585\) −1.16325e26 −0.599367
\(586\) −2.12678e26 −1.07819
\(587\) −1.54368e26 −0.770009 −0.385005 0.922915i \(-0.625800\pi\)
−0.385005 + 0.922915i \(0.625800\pi\)
\(588\) 3.76810e25 0.184943
\(589\) −6.64515e25 −0.320929
\(590\) −2.11995e25 −0.100747
\(591\) 1.46031e25 0.0682911
\(592\) −1.99818e25 −0.0919558
\(593\) 4.05526e26 1.83654 0.918268 0.395959i \(-0.129588\pi\)
0.918268 + 0.395959i \(0.129588\pi\)
\(594\) −9.63095e25 −0.429238
\(595\) 8.95954e25 0.392984
\(596\) 1.91446e25 0.0826433
\(597\) 2.01356e26 0.855476
\(598\) 2.83801e26 1.18673
\(599\) 4.12101e26 1.69609 0.848045 0.529925i \(-0.177780\pi\)
0.848045 + 0.529925i \(0.177780\pi\)
\(600\) −8.88230e24 −0.0359823
\(601\) −2.12493e25 −0.0847300 −0.0423650 0.999102i \(-0.513489\pi\)
−0.0423650 + 0.999102i \(0.513489\pi\)
\(602\) −1.43016e26 −0.561331
\(603\) −2.11775e26 −0.818202
\(604\) −1.54452e26 −0.587412
\(605\) −6.33744e25 −0.237267
\(606\) 1.69131e25 0.0623352
\(607\) 2.90231e26 1.05305 0.526527 0.850158i \(-0.323494\pi\)
0.526527 + 0.850158i \(0.323494\pi\)
\(608\) −1.12537e25 −0.0401986
\(609\) −2.33388e25 −0.0820756
\(610\) 4.20425e25 0.145564
\(611\) −2.17024e25 −0.0739803
\(612\) −1.85628e26 −0.623026
\(613\) −4.17018e26 −1.37810 −0.689049 0.724715i \(-0.741972\pi\)
−0.689049 + 0.724715i \(0.741972\pi\)
\(614\) 2.55023e26 0.829811
\(615\) −3.24561e25 −0.103988
\(616\) −4.01275e25 −0.126598
\(617\) −1.23091e26 −0.382400 −0.191200 0.981551i \(-0.561238\pi\)
−0.191200 + 0.981551i \(0.561238\pi\)
\(618\) −5.00336e24 −0.0153063
\(619\) 5.61407e26 1.69128 0.845642 0.533750i \(-0.179218\pi\)
0.845642 + 0.533750i \(0.179218\pi\)
\(620\) 1.06373e26 0.315580
\(621\) 2.81402e26 0.822161
\(622\) 2.54297e26 0.731700
\(623\) 2.54531e26 0.721282
\(624\) −8.24375e25 −0.230076
\(625\) 1.45519e25 0.0400000
\(626\) −4.55189e26 −1.23235
\(627\) −2.97322e25 −0.0792840
\(628\) 1.11863e26 0.293811
\(629\) −2.39059e26 −0.618478
\(630\) −4.80574e25 −0.122469
\(631\) 2.24570e25 0.0563732 0.0281866 0.999603i \(-0.491027\pi\)
0.0281866 + 0.999603i \(0.491027\pi\)
\(632\) −2.36421e26 −0.584621
\(633\) −1.24234e25 −0.0302624
\(634\) 1.33760e26 0.320980
\(635\) −1.06719e26 −0.252285
\(636\) 5.30797e25 0.123619
\(637\) −5.72950e26 −1.31460
\(638\) −6.61476e25 −0.149526
\(639\) 3.43135e26 0.764199
\(640\) 1.80144e25 0.0395285
\(641\) 3.71414e25 0.0802985 0.0401492 0.999194i \(-0.487217\pi\)
0.0401492 + 0.999194i \(0.487217\pi\)
\(642\) 2.78395e26 0.593032
\(643\) −3.15707e26 −0.662643 −0.331321 0.943518i \(-0.607494\pi\)
−0.331321 + 0.943518i \(0.607494\pi\)
\(644\) 1.17247e26 0.242485
\(645\) 1.69627e26 0.345683
\(646\) −1.34637e26 −0.270368
\(647\) −1.43071e25 −0.0283114 −0.0141557 0.999900i \(-0.504506\pi\)
−0.0141557 + 0.999900i \(0.504506\pi\)
\(648\) 5.26189e25 0.102608
\(649\) 1.13593e26 0.218287
\(650\) 1.35058e26 0.255766
\(651\) −2.01104e26 −0.375320
\(652\) −3.61951e26 −0.665729
\(653\) −8.73760e26 −1.58386 −0.791930 0.610612i \(-0.790924\pi\)
−0.791930 + 0.610612i \(0.790924\pi\)
\(654\) −1.43467e26 −0.256309
\(655\) −2.45339e26 −0.431990
\(656\) 6.58249e25 0.114236
\(657\) 4.76715e26 0.815433
\(658\) −8.96591e24 −0.0151164
\(659\) 5.24673e25 0.0871922 0.0435961 0.999049i \(-0.486119\pi\)
0.0435961 + 0.999049i \(0.486119\pi\)
\(660\) 4.75940e25 0.0779624
\(661\) −2.51792e26 −0.406563 −0.203281 0.979120i \(-0.565161\pi\)
−0.203281 + 0.979120i \(0.565161\pi\)
\(662\) 5.00134e25 0.0796042
\(663\) −9.86268e26 −1.54745
\(664\) 2.54422e26 0.393512
\(665\) −3.48562e25 −0.0531465
\(666\) 1.28227e26 0.192741
\(667\) 1.93274e26 0.286403
\(668\) 4.79221e24 0.00700099
\(669\) 2.07156e26 0.298366
\(670\) 2.45879e26 0.349148
\(671\) −2.25276e26 −0.315392
\(672\) −3.40574e25 −0.0470114
\(673\) −7.15963e26 −0.974423 −0.487212 0.873284i \(-0.661986\pi\)
−0.487212 + 0.873284i \(0.661986\pi\)
\(674\) 1.99002e26 0.267047
\(675\) 1.33916e26 0.177193
\(676\) 8.70249e26 1.13540
\(677\) −4.37826e26 −0.563260 −0.281630 0.959523i \(-0.590875\pi\)
−0.281630 + 0.959523i \(0.590875\pi\)
\(678\) 4.43362e26 0.562440
\(679\) 6.82295e26 0.853512
\(680\) 2.15521e26 0.265862
\(681\) −4.67599e26 −0.568822
\(682\) −5.69977e26 −0.683763
\(683\) 4.02600e26 0.476296 0.238148 0.971229i \(-0.423460\pi\)
0.238148 + 0.971229i \(0.423460\pi\)
\(684\) 7.22169e25 0.0842570
\(685\) 3.96364e26 0.456072
\(686\) −5.62343e26 −0.638149
\(687\) 6.97487e26 0.780633
\(688\) −3.44025e26 −0.379752
\(689\) −8.07090e26 −0.878697
\(690\) −1.39063e26 −0.149329
\(691\) 5.80847e26 0.615206 0.307603 0.951515i \(-0.400473\pi\)
0.307603 + 0.951515i \(0.400473\pi\)
\(692\) 1.86274e26 0.194601
\(693\) 2.57506e26 0.265351
\(694\) 3.26751e26 0.332125
\(695\) 7.76303e26 0.778351
\(696\) −5.61413e25 −0.0555258
\(697\) 7.87518e26 0.768332
\(698\) −3.71498e26 −0.357545
\(699\) −9.01038e26 −0.855480
\(700\) 5.57963e25 0.0522606
\(701\) −6.65373e26 −0.614814 −0.307407 0.951578i \(-0.599461\pi\)
−0.307407 + 0.951578i \(0.599461\pi\)
\(702\) 1.24289e27 1.13300
\(703\) 9.30037e25 0.0836420
\(704\) −9.65266e25 −0.0856459
\(705\) 1.06342e25 0.00930910
\(706\) −4.40805e26 −0.380717
\(707\) −1.06244e26 −0.0905356
\(708\) 9.64099e25 0.0810599
\(709\) 2.09393e27 1.73709 0.868546 0.495608i \(-0.165055\pi\)
0.868546 + 0.495608i \(0.165055\pi\)
\(710\) −3.98392e26 −0.326104
\(711\) 1.51716e27 1.22538
\(712\) 6.12274e26 0.487962
\(713\) 1.66539e27 1.30968
\(714\) −4.07457e26 −0.316190
\(715\) −7.23680e26 −0.554165
\(716\) −2.85744e26 −0.215926
\(717\) 9.91043e26 0.739028
\(718\) −1.00321e27 −0.738262
\(719\) 9.61307e25 0.0698132 0.0349066 0.999391i \(-0.488887\pi\)
0.0349066 + 0.999391i \(0.488887\pi\)
\(720\) −1.15602e26 −0.0828525
\(721\) 3.14298e25 0.0222309
\(722\) −9.60572e26 −0.670543
\(723\) −4.14123e26 −0.285309
\(724\) −7.32479e26 −0.498057
\(725\) 9.19766e25 0.0617257
\(726\) 2.88210e26 0.190902
\(727\) −2.28716e27 −1.49527 −0.747637 0.664108i \(-0.768811\pi\)
−0.747637 + 0.664108i \(0.768811\pi\)
\(728\) 5.17851e26 0.334162
\(729\) 3.70169e26 0.235770
\(730\) −5.53484e26 −0.347967
\(731\) −4.11586e27 −2.55414
\(732\) −1.91198e26 −0.117119
\(733\) 9.50008e26 0.574433 0.287217 0.957866i \(-0.407270\pi\)
0.287217 + 0.957866i \(0.407270\pi\)
\(734\) −8.72376e26 −0.520704
\(735\) 2.80746e26 0.165418
\(736\) 2.82036e26 0.164046
\(737\) −1.31749e27 −0.756496
\(738\) −4.22410e26 −0.239441
\(739\) −2.92022e27 −1.63415 −0.817077 0.576529i \(-0.804407\pi\)
−0.817077 + 0.576529i \(0.804407\pi\)
\(740\) −1.48876e26 −0.0822477
\(741\) 3.83698e26 0.209275
\(742\) −3.33433e26 −0.179544
\(743\) 2.06362e27 1.09707 0.548536 0.836127i \(-0.315185\pi\)
0.548536 + 0.836127i \(0.315185\pi\)
\(744\) −4.83755e26 −0.253912
\(745\) 1.42639e26 0.0739185
\(746\) −2.90063e25 −0.0148414
\(747\) −1.63267e27 −0.824810
\(748\) −1.15483e27 −0.576039
\(749\) −1.74880e27 −0.861319
\(750\) −6.61783e25 −0.0321835
\(751\) 2.97915e27 1.43058 0.715291 0.698827i \(-0.246294\pi\)
0.715291 + 0.698827i \(0.246294\pi\)
\(752\) −2.15674e25 −0.0102265
\(753\) 1.58233e27 0.740877
\(754\) 8.53643e26 0.394683
\(755\) −1.15076e27 −0.525397
\(756\) 5.13473e26 0.231505
\(757\) 2.29450e27 1.02159 0.510796 0.859702i \(-0.329351\pi\)
0.510796 + 0.859702i \(0.329351\pi\)
\(758\) 1.31591e26 0.0578588
\(759\) 7.45140e26 0.323550
\(760\) −8.38464e25 −0.0359547
\(761\) 2.78541e27 1.17960 0.589801 0.807548i \(-0.299206\pi\)
0.589801 + 0.807548i \(0.299206\pi\)
\(762\) 4.85330e26 0.202985
\(763\) 9.01223e26 0.372262
\(764\) 1.21575e27 0.495974
\(765\) −1.38304e27 −0.557251
\(766\) 2.24580e27 0.893710
\(767\) −1.46594e27 −0.576182
\(768\) −8.19248e25 −0.0318041
\(769\) −2.23371e27 −0.856498 −0.428249 0.903661i \(-0.640869\pi\)
−0.428249 + 0.903661i \(0.640869\pi\)
\(770\) −2.98973e26 −0.113232
\(771\) −1.58801e27 −0.594069
\(772\) −2.15991e26 −0.0798127
\(773\) 9.13265e26 0.333343 0.166672 0.986012i \(-0.446698\pi\)
0.166672 + 0.986012i \(0.446698\pi\)
\(774\) 2.20767e27 0.795967
\(775\) 7.92538e26 0.282263
\(776\) 1.64126e27 0.577418
\(777\) 2.81460e26 0.0978176
\(778\) −7.04272e26 −0.241788
\(779\) −3.06376e26 −0.103908
\(780\) −6.14207e26 −0.205786
\(781\) 2.13470e27 0.706566
\(782\) 3.37424e27 1.10334
\(783\) 8.46426e26 0.273434
\(784\) −5.69387e26 −0.181721
\(785\) 8.33442e26 0.262793
\(786\) 1.11574e27 0.347574
\(787\) 5.64471e26 0.173733 0.0868664 0.996220i \(-0.472315\pi\)
0.0868664 + 0.996220i \(0.472315\pi\)
\(788\) −2.20663e26 −0.0671012
\(789\) −1.01040e27 −0.303572
\(790\) −1.76148e27 −0.522901
\(791\) −2.78508e27 −0.816887
\(792\) 6.19429e26 0.179516
\(793\) 2.90722e27 0.832496
\(794\) −2.14001e27 −0.605509
\(795\) 3.95474e26 0.110568
\(796\) −3.04263e27 −0.840571
\(797\) −1.25075e27 −0.341441 −0.170721 0.985319i \(-0.554610\pi\)
−0.170721 + 0.985319i \(0.554610\pi\)
\(798\) 1.58517e26 0.0427610
\(799\) −2.58029e26 −0.0687820
\(800\) 1.34218e26 0.0353553
\(801\) −3.92907e27 −1.02278
\(802\) 1.76649e26 0.0454418
\(803\) 2.96573e27 0.753936
\(804\) −1.11819e27 −0.280921
\(805\) 8.73556e26 0.216885
\(806\) 7.35562e27 1.80483
\(807\) −3.66723e27 −0.889283
\(808\) −2.55569e26 −0.0612491
\(809\) −2.25667e26 −0.0534510 −0.0267255 0.999643i \(-0.508508\pi\)
−0.0267255 + 0.999643i \(0.508508\pi\)
\(810\) 3.92041e26 0.0917750
\(811\) −1.12861e27 −0.261123 −0.130561 0.991440i \(-0.541678\pi\)
−0.130561 + 0.991440i \(0.541678\pi\)
\(812\) 3.52665e26 0.0806456
\(813\) −3.00998e27 −0.680304
\(814\) 7.97724e26 0.178205
\(815\) −2.69674e27 −0.595446
\(816\) −9.80135e26 −0.213909
\(817\) 1.60123e27 0.345418
\(818\) 9.33354e26 0.199017
\(819\) −3.32314e27 −0.700409
\(820\) 4.90433e26 0.102176
\(821\) −4.62062e27 −0.951569 −0.475784 0.879562i \(-0.657836\pi\)
−0.475784 + 0.879562i \(0.657836\pi\)
\(822\) −1.80256e27 −0.366950
\(823\) 5.30846e27 1.06824 0.534121 0.845408i \(-0.320643\pi\)
0.534121 + 0.845408i \(0.320643\pi\)
\(824\) 7.56043e25 0.0150397
\(825\) 3.54603e26 0.0697317
\(826\) −6.05622e26 −0.117731
\(827\) 2.89805e27 0.556933 0.278467 0.960446i \(-0.410174\pi\)
0.278467 + 0.960446i \(0.410174\pi\)
\(828\) −1.80988e27 −0.343844
\(829\) −1.57547e27 −0.295898 −0.147949 0.988995i \(-0.547267\pi\)
−0.147949 + 0.988995i \(0.547267\pi\)
\(830\) 1.89559e27 0.351968
\(831\) 1.39536e27 0.256140
\(832\) 1.24569e27 0.226067
\(833\) −6.81205e27 −1.22222
\(834\) −3.53042e27 −0.626252
\(835\) 3.57048e25 0.00626188
\(836\) 4.49275e26 0.0779026
\(837\) 7.29343e27 1.25037
\(838\) −7.39099e27 −1.25281
\(839\) 4.31895e27 0.723836 0.361918 0.932210i \(-0.382122\pi\)
0.361918 + 0.932210i \(0.382122\pi\)
\(840\) −2.53747e26 −0.0420482
\(841\) −5.52192e27 −0.904748
\(842\) −2.41641e27 −0.391477
\(843\) −5.69542e27 −0.912355
\(844\) 1.87726e26 0.0297352
\(845\) 6.48386e27 1.01553
\(846\) 1.38402e26 0.0214351
\(847\) −1.81046e27 −0.277266
\(848\) −8.02071e26 −0.121465
\(849\) 5.54510e27 0.830398
\(850\) 1.60576e27 0.237794
\(851\) −2.33083e27 −0.341334
\(852\) 1.81178e27 0.262379
\(853\) 4.26621e26 0.0610979 0.0305490 0.999533i \(-0.490274\pi\)
0.0305490 + 0.999533i \(0.490274\pi\)
\(854\) 1.20106e27 0.170104
\(855\) 5.38058e26 0.0753618
\(856\) −4.20674e27 −0.582700
\(857\) 4.56728e27 0.625662 0.312831 0.949809i \(-0.398723\pi\)
0.312831 + 0.949809i \(0.398723\pi\)
\(858\) 3.29111e27 0.445874
\(859\) 1.40555e27 0.188326 0.0941631 0.995557i \(-0.469982\pi\)
0.0941631 + 0.995557i \(0.469982\pi\)
\(860\) −2.56318e27 −0.339660
\(861\) −9.27195e26 −0.121518
\(862\) −7.37926e27 −0.956519
\(863\) 8.51803e27 1.09204 0.546018 0.837774i \(-0.316143\pi\)
0.546018 + 0.837774i \(0.316143\pi\)
\(864\) 1.23516e27 0.156618
\(865\) 1.38785e27 0.174056
\(866\) −7.09639e27 −0.880274
\(867\) −7.57868e27 −0.929849
\(868\) 3.03882e27 0.368781
\(869\) 9.43853e27 1.13296
\(870\) −4.18286e26 −0.0496638
\(871\) 1.70024e28 1.99681
\(872\) 2.16789e27 0.251843
\(873\) −1.05322e28 −1.21028
\(874\) −1.31271e27 −0.149215
\(875\) 4.15715e26 0.0467433
\(876\) 2.51710e27 0.279970
\(877\) −7.77367e27 −0.855322 −0.427661 0.903939i \(-0.640662\pi\)
−0.427661 + 0.903939i \(0.640662\pi\)
\(878\) −6.44080e27 −0.701038
\(879\) 7.20624e27 0.775916
\(880\) −7.19179e26 −0.0766040
\(881\) 9.90182e27 1.04338 0.521692 0.853134i \(-0.325301\pi\)
0.521692 + 0.853134i \(0.325301\pi\)
\(882\) 3.65386e27 0.380891
\(883\) −1.01292e28 −1.04460 −0.522298 0.852763i \(-0.674925\pi\)
−0.522298 + 0.852763i \(0.674925\pi\)
\(884\) 1.49032e28 1.52049
\(885\) 7.18310e26 0.0725021
\(886\) 1.17027e27 0.116860
\(887\) 1.94019e28 1.91677 0.958384 0.285483i \(-0.0921539\pi\)
0.958384 + 0.285483i \(0.0921539\pi\)
\(888\) 6.77050e26 0.0661756
\(889\) −3.04871e27 −0.294815
\(890\) 4.56180e27 0.436446
\(891\) −2.10068e27 −0.198848
\(892\) −3.13027e27 −0.293168
\(893\) 1.00384e26 0.00930196
\(894\) −6.48684e26 −0.0594739
\(895\) −2.12896e27 −0.193130
\(896\) 5.14630e26 0.0461923
\(897\) −9.61612e27 −0.854027
\(898\) −2.18949e27 −0.192406
\(899\) 5.00930e27 0.435572
\(900\) −8.61300e26 −0.0741055
\(901\) −9.59584e27 −0.816954
\(902\) −2.62789e27 −0.221383
\(903\) 4.84586e27 0.403959
\(904\) −6.69950e27 −0.552640
\(905\) −5.45740e27 −0.445476
\(906\) 5.23334e27 0.422728
\(907\) −1.39346e28 −1.11385 −0.556924 0.830563i \(-0.688019\pi\)
−0.556924 + 0.830563i \(0.688019\pi\)
\(908\) 7.06575e27 0.558911
\(909\) 1.64003e27 0.128379
\(910\) 3.85829e27 0.298883
\(911\) −6.06821e27 −0.465196 −0.232598 0.972573i \(-0.574723\pi\)
−0.232598 + 0.972573i \(0.574723\pi\)
\(912\) 3.81312e26 0.0289287
\(913\) −1.01572e28 −0.762606
\(914\) 9.35003e27 0.694743
\(915\) −1.42454e27 −0.104755
\(916\) −1.05395e28 −0.767032
\(917\) −7.00876e27 −0.504815
\(918\) 1.47772e28 1.05338
\(919\) 2.20624e28 1.55652 0.778260 0.627942i \(-0.216102\pi\)
0.778260 + 0.627942i \(0.216102\pi\)
\(920\) 2.10133e27 0.146727
\(921\) −8.64101e27 −0.597170
\(922\) −4.04815e27 −0.276893
\(923\) −2.75486e28 −1.86502
\(924\) 1.35965e27 0.0911055
\(925\) −1.10921e27 −0.0735646
\(926\) −1.49332e28 −0.980276
\(927\) −4.85167e26 −0.0315234
\(928\) 8.48334e26 0.0545584
\(929\) −2.55096e28 −1.62388 −0.811940 0.583740i \(-0.801589\pi\)
−0.811940 + 0.583740i \(0.801589\pi\)
\(930\) −3.60426e27 −0.227106
\(931\) 2.65016e27 0.165291
\(932\) 1.36153e28 0.840575
\(933\) −8.61643e27 −0.526565
\(934\) −1.48547e28 −0.898607
\(935\) −8.60414e27 −0.515225
\(936\) −7.99380e27 −0.473841
\(937\) 1.82517e28 1.07097 0.535485 0.844545i \(-0.320129\pi\)
0.535485 + 0.844545i \(0.320129\pi\)
\(938\) 7.02419e27 0.408009
\(939\) 1.54233e28 0.886859
\(940\) −1.60690e26 −0.00914690
\(941\) 2.32888e28 1.31234 0.656169 0.754614i \(-0.272176\pi\)
0.656169 + 0.754614i \(0.272176\pi\)
\(942\) −3.79028e27 −0.211440
\(943\) 7.67830e27 0.424037
\(944\) −1.45682e27 −0.0796475
\(945\) 3.82567e27 0.207064
\(946\) 1.37343e28 0.735938
\(947\) −2.16819e28 −1.15020 −0.575098 0.818084i \(-0.695036\pi\)
−0.575098 + 0.818084i \(0.695036\pi\)
\(948\) 8.01074e27 0.420720
\(949\) −3.82731e28 −1.99005
\(950\) −6.24705e26 −0.0321589
\(951\) −4.53222e27 −0.230992
\(952\) 6.15695e27 0.310681
\(953\) −4.92023e27 −0.245812 −0.122906 0.992418i \(-0.539221\pi\)
−0.122906 + 0.992418i \(0.539221\pi\)
\(954\) 5.14703e27 0.254594
\(955\) 9.05807e27 0.443612
\(956\) −1.49754e28 −0.726151
\(957\) 2.24130e27 0.107606
\(958\) 2.21976e28 1.05520
\(959\) 1.13232e28 0.532957
\(960\) −6.10387e26 −0.0284465
\(961\) 2.14932e28 0.991811
\(962\) −1.02947e28 −0.470383
\(963\) 2.69954e28 1.22135
\(964\) 6.25768e27 0.280338
\(965\) −1.60926e27 −0.0713866
\(966\) −3.97270e27 −0.174503
\(967\) 1.22764e28 0.533972 0.266986 0.963700i \(-0.413972\pi\)
0.266986 + 0.963700i \(0.413972\pi\)
\(968\) −4.35506e27 −0.187576
\(969\) 4.56195e27 0.194569
\(970\) 1.22283e28 0.516458
\(971\) 2.18467e28 0.913698 0.456849 0.889544i \(-0.348978\pi\)
0.456849 + 0.889544i \(0.348978\pi\)
\(972\) −1.24788e28 −0.516823
\(973\) 2.21772e28 0.909567
\(974\) −1.13098e28 −0.459350
\(975\) −4.57620e27 −0.184061
\(976\) 2.88914e27 0.115079
\(977\) 3.53418e28 1.39409 0.697043 0.717029i \(-0.254499\pi\)
0.697043 + 0.717029i \(0.254499\pi\)
\(978\) 1.22641e28 0.479089
\(979\) −2.44435e28 −0.945643
\(980\) −4.24226e27 −0.162536
\(981\) −1.39117e28 −0.527868
\(982\) −9.04281e27 −0.339816
\(983\) 2.24443e28 0.835309 0.417655 0.908606i \(-0.362852\pi\)
0.417655 + 0.908606i \(0.362852\pi\)
\(984\) −2.23036e27 −0.0822095
\(985\) −1.64407e27 −0.0600171
\(986\) 1.01493e28 0.366950
\(987\) 3.03794e26 0.0108784
\(988\) −5.79794e27 −0.205628
\(989\) −4.01296e28 −1.40961
\(990\) 4.61510e27 0.160564
\(991\) 3.80736e28 1.31197 0.655986 0.754773i \(-0.272253\pi\)
0.655986 + 0.754773i \(0.272253\pi\)
\(992\) 7.30987e27 0.249488
\(993\) −1.69462e27 −0.0572868
\(994\) −1.13811e28 −0.381079
\(995\) −2.26693e28 −0.751830
\(996\) −8.62066e27 −0.283189
\(997\) 4.84284e28 1.57578 0.787892 0.615814i \(-0.211173\pi\)
0.787892 + 0.615814i \(0.211173\pi\)
\(998\) −2.31586e28 −0.746402
\(999\) −1.02077e28 −0.325878
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 10.20.a.d.1.1 2
3.2 odd 2 90.20.a.g.1.2 2
4.3 odd 2 80.20.a.d.1.2 2
5.2 odd 4 50.20.b.f.49.4 4
5.3 odd 4 50.20.b.f.49.1 4
5.4 even 2 50.20.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.20.a.d.1.1 2 1.1 even 1 trivial
50.20.a.f.1.2 2 5.4 even 2
50.20.b.f.49.1 4 5.3 odd 4
50.20.b.f.49.4 4 5.2 odd 4
80.20.a.d.1.2 2 4.3 odd 2
90.20.a.g.1.2 2 3.2 odd 2