Properties

Label 10.20.a.d.1.2
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 1.2
Root \(-1710.01\) 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)
 
\(f(q)\) \(=\) \(q+512.000 q^{2} +51072.2 q^{3} +262144. q^{4} +1.95312e6 q^{5} +2.61490e7 q^{6} +2.72650e7 q^{7} +1.34218e8 q^{8} +1.44611e9 q^{9} +1.00000e9 q^{10} +8.90778e9 q^{11} +1.33883e10 q^{12} -3.88993e10 q^{13} +1.39597e10 q^{14} +9.97505e10 q^{15} +6.87195e10 q^{16} -4.92493e11 q^{17} +7.40410e11 q^{18} +7.02926e11 q^{19} +5.12000e11 q^{20} +1.39248e12 q^{21} +4.56078e12 q^{22} -5.93310e12 q^{23} +6.85480e12 q^{24} +3.81470e12 q^{25} -1.99164e13 q^{26} +1.44969e13 q^{27} +7.14735e12 q^{28} +1.42477e14 q^{29} +5.10722e13 q^{30} +2.10392e14 q^{31} +3.51844e13 q^{32} +4.54940e14 q^{33} -2.52156e14 q^{34} +5.32519e13 q^{35} +3.79090e14 q^{36} -1.00145e15 q^{37} +3.59898e14 q^{38} -1.98667e15 q^{39} +2.62144e14 q^{40} -4.08507e15 q^{41} +7.12952e14 q^{42} +3.21023e15 q^{43} +2.33512e15 q^{44} +2.82444e15 q^{45} -3.03774e15 q^{46} -6.44781e15 q^{47} +3.50966e15 q^{48} -1.06555e16 q^{49} +1.95312e15 q^{50} -2.51527e16 q^{51} -1.01972e16 q^{52} +1.88410e15 q^{53} +7.42243e15 q^{54} +1.73980e16 q^{55} +3.65944e15 q^{56} +3.59000e16 q^{57} +7.29480e16 q^{58} -4.28479e15 q^{59} +2.61490e16 q^{60} -1.19883e17 q^{61} +1.07721e17 q^{62} +3.94282e16 q^{63} +1.80144e16 q^{64} -7.59752e16 q^{65} +2.32929e17 q^{66} +1.93776e17 q^{67} -1.29104e17 q^{68} -3.03017e17 q^{69} +2.72650e16 q^{70} +3.68964e17 q^{71} +1.94094e17 q^{72} -9.09825e17 q^{73} -5.12740e17 q^{74} +1.94825e17 q^{75} +1.84268e17 q^{76} +2.42870e17 q^{77} -1.01718e18 q^{78} +5.25508e17 q^{79} +1.34218e17 q^{80} -9.40371e17 q^{81} -2.09156e18 q^{82} -2.77280e18 q^{83} +3.65031e17 q^{84} -9.61900e17 q^{85} +1.64364e18 q^{86} +7.27660e18 q^{87} +1.19558e18 q^{88} -3.88866e17 q^{89} +1.44611e18 q^{90} -1.06059e18 q^{91} -1.55533e18 q^{92} +1.07452e19 q^{93} -3.30128e18 q^{94} +1.37290e18 q^{95} +1.79694e18 q^{96} -9.73741e17 q^{97} -5.45562e18 q^{98} +1.28817e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1024 q^{2} + 33724 q^{3} + 524288 q^{4} + 3906250 q^{5} + 17266688 q^{6} + 83061292 q^{7} + 268435456 q^{8} + 584812954 q^{9} + 2000000000 q^{10} + 3549480144 q^{11} + 8840544256 q^{12} + 30250225564 q^{13}+ \cdots + 17\!\cdots\!88 q^{99}+O(q^{100}) \) 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
\(3\) 51072.2 1.49807 0.749037 0.662529i \(-0.230517\pi\)
0.749037 + 0.662529i \(0.230517\pi\)
\(4\) 262144. 0.500000
\(5\) 1.95312e6 0.447214
\(6\) 2.61490e7 1.05930
\(7\) 2.72650e7 0.255372 0.127686 0.991815i \(-0.459245\pi\)
0.127686 + 0.991815i \(0.459245\pi\)
\(8\) 1.34218e8 0.353553
\(9\) 1.44611e9 1.24422
\(10\) 1.00000e9 0.316228
\(11\) 8.90778e9 1.13904 0.569520 0.821977i \(-0.307129\pi\)
0.569520 + 0.821977i \(0.307129\pi\)
\(12\) 1.33883e10 0.749037
\(13\) −3.88993e10 −1.01737 −0.508686 0.860952i \(-0.669869\pi\)
−0.508686 + 0.860952i \(0.669869\pi\)
\(14\) 1.39597e10 0.180575
\(15\) 9.97505e10 0.669959
\(16\) 6.87195e10 0.250000
\(17\) −4.92493e11 −1.00725 −0.503623 0.863924i \(-0.668000\pi\)
−0.503623 + 0.863924i \(0.668000\pi\)
\(18\) 7.40410e11 0.879799
\(19\) 7.02926e11 0.499747 0.249873 0.968278i \(-0.419611\pi\)
0.249873 + 0.968278i \(0.419611\pi\)
\(20\) 5.12000e11 0.223607
\(21\) 1.39248e12 0.382566
\(22\) 4.56078e12 0.805423
\(23\) −5.93310e12 −0.686858 −0.343429 0.939179i \(-0.611588\pi\)
−0.343429 + 0.939179i \(0.611588\pi\)
\(24\) 6.85480e12 0.529649
\(25\) 3.81470e12 0.200000
\(26\) −1.99164e13 −0.719391
\(27\) 1.44969e13 0.365864
\(28\) 7.14735e12 0.127686
\(29\) 1.42477e14 1.82374 0.911869 0.410481i \(-0.134639\pi\)
0.911869 + 0.410481i \(0.134639\pi\)
\(30\) 5.10722e13 0.473732
\(31\) 2.10392e14 1.42920 0.714600 0.699533i \(-0.246609\pi\)
0.714600 + 0.699533i \(0.246609\pi\)
\(32\) 3.51844e13 0.176777
\(33\) 4.54940e14 1.70637
\(34\) −2.52156e14 −0.712230
\(35\) 5.32519e13 0.114206
\(36\) 3.79090e14 0.622112
\(37\) −1.00145e15 −1.26681 −0.633404 0.773821i \(-0.718343\pi\)
−0.633404 + 0.773821i \(0.718343\pi\)
\(38\) 3.59898e14 0.353374
\(39\) −1.98667e15 −1.52410
\(40\) 2.62144e14 0.158114
\(41\) −4.08507e15 −1.94874 −0.974368 0.224961i \(-0.927775\pi\)
−0.974368 + 0.224961i \(0.927775\pi\)
\(42\) 7.12952e14 0.270515
\(43\) 3.21023e15 0.974060 0.487030 0.873385i \(-0.338080\pi\)
0.487030 + 0.873385i \(0.338080\pi\)
\(44\) 2.33512e15 0.569520
\(45\) 2.82444e15 0.556434
\(46\) −3.03774e15 −0.485682
\(47\) −6.44781e15 −0.840393 −0.420196 0.907433i \(-0.638039\pi\)
−0.420196 + 0.907433i \(0.638039\pi\)
\(48\) 3.50966e15 0.374518
\(49\) −1.06555e16 −0.934785
\(50\) 1.95312e15 0.141421
\(51\) −2.51527e16 −1.50893
\(52\) −1.01972e16 −0.508686
\(53\) 1.88410e15 0.0784300 0.0392150 0.999231i \(-0.487514\pi\)
0.0392150 + 0.999231i \(0.487514\pi\)
\(54\) 7.42243e15 0.258705
\(55\) 1.73980e16 0.509394
\(56\) 3.65944e15 0.0902877
\(57\) 3.59000e16 0.748658
\(58\) 7.29480e16 1.28958
\(59\) −4.28479e15 −0.0643924 −0.0321962 0.999482i \(-0.510250\pi\)
−0.0321962 + 0.999482i \(0.510250\pi\)
\(60\) 2.61490e16 0.334979
\(61\) −1.19883e17 −1.31257 −0.656287 0.754511i \(-0.727874\pi\)
−0.656287 + 0.754511i \(0.727874\pi\)
\(62\) 1.07721e17 1.01060
\(63\) 3.94282e16 0.317740
\(64\) 1.80144e16 0.125000
\(65\) −7.59752e16 −0.454983
\(66\) 2.32929e17 1.20658
\(67\) 1.93776e17 0.870141 0.435071 0.900396i \(-0.356723\pi\)
0.435071 + 0.900396i \(0.356723\pi\)
\(68\) −1.29104e17 −0.503623
\(69\) −3.03017e17 −1.02896
\(70\) 2.72650e16 0.0807558
\(71\) 3.68964e17 0.955058 0.477529 0.878616i \(-0.341532\pi\)
0.477529 + 0.878616i \(0.341532\pi\)
\(72\) 1.94094e17 0.439899
\(73\) −9.09825e17 −1.80880 −0.904400 0.426685i \(-0.859681\pi\)
−0.904400 + 0.426685i \(0.859681\pi\)
\(74\) −5.12740e17 −0.895769
\(75\) 1.94825e17 0.299615
\(76\) 1.84268e17 0.249873
\(77\) 2.42870e17 0.290879
\(78\) −1.01718e18 −1.07770
\(79\) 5.25508e17 0.493313 0.246656 0.969103i \(-0.420668\pi\)
0.246656 + 0.969103i \(0.420668\pi\)
\(80\) 1.34218e17 0.111803
\(81\) −9.40371e17 −0.696132
\(82\) −2.09156e18 −1.37796
\(83\) −2.77280e18 −1.62808 −0.814042 0.580807i \(-0.802737\pi\)
−0.814042 + 0.580807i \(0.802737\pi\)
\(84\) 3.65031e17 0.191283
\(85\) −9.61900e17 −0.450454
\(86\) 1.64364e18 0.688764
\(87\) 7.27660e18 2.73209
\(88\) 1.19558e18 0.402712
\(89\) −3.88866e17 −0.117651 −0.0588254 0.998268i \(-0.518736\pi\)
−0.0588254 + 0.998268i \(0.518736\pi\)
\(90\) 1.44611e18 0.393458
\(91\) −1.06059e18 −0.259809
\(92\) −1.55533e18 −0.343429
\(93\) 1.07452e19 2.14105
\(94\) −3.30128e18 −0.594247
\(95\) 1.37290e18 0.223494
\(96\) 1.79694e18 0.264824
\(97\) −9.73741e17 −0.130050 −0.0650252 0.997884i \(-0.520713\pi\)
−0.0650252 + 0.997884i \(0.520713\pi\)
\(98\) −5.45562e18 −0.660993
\(99\) 1.28817e19 1.41722
\(100\) 1.00000e18 0.100000
\(101\) −1.67461e19 −1.52356 −0.761781 0.647834i \(-0.775675\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(102\) −1.28782e19 −1.06697
\(103\) 4.40800e18 0.332880 0.166440 0.986052i \(-0.446773\pi\)
0.166440 + 0.986052i \(0.446773\pi\)
\(104\) −5.22098e18 −0.359695
\(105\) 2.71969e18 0.171089
\(106\) 9.64657e17 0.0554584
\(107\) −8.71914e18 −0.458488 −0.229244 0.973369i \(-0.573625\pi\)
−0.229244 + 0.973369i \(0.573625\pi\)
\(108\) 3.80028e18 0.182932
\(109\) 4.17114e19 1.83951 0.919757 0.392487i \(-0.128385\pi\)
0.919757 + 0.392487i \(0.128385\pi\)
\(110\) 8.90778e18 0.360196
\(111\) −5.11461e19 −1.89777
\(112\) 1.87363e18 0.0638431
\(113\) −8.70592e18 −0.272627 −0.136314 0.990666i \(-0.543525\pi\)
−0.136314 + 0.990666i \(0.543525\pi\)
\(114\) 1.83808e19 0.529381
\(115\) −1.15881e19 −0.307172
\(116\) 3.73494e19 0.911869
\(117\) −5.62528e19 −1.26584
\(118\) −2.19381e18 −0.0455323
\(119\) −1.34278e19 −0.257222
\(120\) 1.33883e19 0.236866
\(121\) 1.81895e19 0.297413
\(122\) −6.13801e19 −0.928130
\(123\) −2.08634e20 −2.91935
\(124\) 5.51530e19 0.714600
\(125\) 7.45058e18 0.0894427
\(126\) 2.01873e19 0.224676
\(127\) 6.07257e19 0.626956 0.313478 0.949595i \(-0.398506\pi\)
0.313478 + 0.949595i \(0.398506\pi\)
\(128\) 9.22337e18 0.0883883
\(129\) 1.63954e20 1.45921
\(130\) −3.88993e19 −0.321721
\(131\) 1.94831e20 1.49824 0.749120 0.662434i \(-0.230477\pi\)
0.749120 + 0.662434i \(0.230477\pi\)
\(132\) 1.19260e20 0.853183
\(133\) 1.91652e19 0.127622
\(134\) 9.92134e19 0.615283
\(135\) 2.83143e19 0.163620
\(136\) −6.61013e19 −0.356115
\(137\) 1.21216e20 0.609135 0.304567 0.952491i \(-0.401488\pi\)
0.304567 + 0.952491i \(0.401488\pi\)
\(138\) −1.55144e20 −0.727587
\(139\) 2.22493e20 0.974263 0.487131 0.873329i \(-0.338043\pi\)
0.487131 + 0.873329i \(0.338043\pi\)
\(140\) 1.39597e19 0.0571030
\(141\) −3.29304e20 −1.25897
\(142\) 1.88910e20 0.675328
\(143\) −3.46506e20 −1.15883
\(144\) 9.93761e19 0.311056
\(145\) 2.78275e20 0.815600
\(146\) −4.65830e20 −1.27902
\(147\) −5.44201e20 −1.40038
\(148\) −2.62523e20 −0.633404
\(149\) 3.46894e20 0.785104 0.392552 0.919730i \(-0.371592\pi\)
0.392552 + 0.919730i \(0.371592\pi\)
\(150\) 9.97505e19 0.211860
\(151\) −1.05205e19 −0.0209776 −0.0104888 0.999945i \(-0.503339\pi\)
−0.0104888 + 0.999945i \(0.503339\pi\)
\(152\) 9.43451e19 0.176687
\(153\) −7.12200e20 −1.25324
\(154\) 1.24350e20 0.205683
\(155\) 4.10922e20 0.639158
\(156\) −5.20795e20 −0.762049
\(157\) −3.07714e19 −0.0423741 −0.0211871 0.999776i \(-0.506745\pi\)
−0.0211871 + 0.999776i \(0.506745\pi\)
\(158\) 2.69060e20 0.348825
\(159\) 9.62250e19 0.117494
\(160\) 6.87195e19 0.0790569
\(161\) −1.61766e20 −0.175404
\(162\) −4.81470e20 −0.492239
\(163\) 5.76318e20 0.555751 0.277875 0.960617i \(-0.410370\pi\)
0.277875 + 0.960617i \(0.410370\pi\)
\(164\) −1.07088e21 −0.974368
\(165\) 8.88555e20 0.763110
\(166\) −1.41967e21 −1.15123
\(167\) 6.89550e20 0.528153 0.264076 0.964502i \(-0.414933\pi\)
0.264076 + 0.964502i \(0.414933\pi\)
\(168\) 1.86896e20 0.135258
\(169\) 5.12350e19 0.0350464
\(170\) −4.92493e20 −0.318519
\(171\) 1.01651e21 0.621797
\(172\) 8.41542e20 0.487030
\(173\) −4.14244e19 −0.0226892 −0.0113446 0.999936i \(-0.503611\pi\)
−0.0113446 + 0.999936i \(0.503611\pi\)
\(174\) 3.72562e21 1.93188
\(175\) 1.04008e20 0.0510744
\(176\) 6.12138e20 0.284760
\(177\) −2.18834e20 −0.0964646
\(178\) −1.99099e20 −0.0831917
\(179\) 3.65179e21 1.44678 0.723388 0.690441i \(-0.242584\pi\)
0.723388 + 0.690441i \(0.242584\pi\)
\(180\) 7.40410e20 0.278217
\(181\) 1.05232e21 0.375145 0.187573 0.982251i \(-0.439938\pi\)
0.187573 + 0.982251i \(0.439938\pi\)
\(182\) −5.43021e20 −0.183712
\(183\) −6.12269e21 −1.96633
\(184\) −7.96327e20 −0.242841
\(185\) −1.95595e21 −0.566534
\(186\) 5.50154e21 1.51395
\(187\) −4.38702e21 −1.14729
\(188\) −1.69025e21 −0.420196
\(189\) 3.95258e20 0.0934316
\(190\) 7.02926e20 0.158034
\(191\) −6.63681e21 −1.41952 −0.709761 0.704442i \(-0.751197\pi\)
−0.709761 + 0.704442i \(0.751197\pi\)
\(192\) 9.20036e20 0.187259
\(193\) −2.65526e21 −0.514414 −0.257207 0.966356i \(-0.582802\pi\)
−0.257207 + 0.966356i \(0.582802\pi\)
\(194\) −4.98555e20 −0.0919596
\(195\) −3.88022e21 −0.681597
\(196\) −2.79328e21 −0.467393
\(197\) 1.02356e22 1.63186 0.815929 0.578153i \(-0.196226\pi\)
0.815929 + 0.578153i \(0.196226\pi\)
\(198\) 6.59541e21 1.00213
\(199\) −2.53082e21 −0.366570 −0.183285 0.983060i \(-0.558673\pi\)
−0.183285 + 0.983060i \(0.558673\pi\)
\(200\) 5.12000e20 0.0707107
\(201\) 9.89659e21 1.30354
\(202\) −8.57400e21 −1.07732
\(203\) 3.88462e21 0.465732
\(204\) −6.59363e21 −0.754463
\(205\) −7.97865e21 −0.871501
\(206\) 2.25689e21 0.235382
\(207\) −8.57993e21 −0.854605
\(208\) −2.67314e21 −0.254343
\(209\) 6.26151e21 0.569232
\(210\) 1.39248e21 0.120978
\(211\) −1.43232e22 −1.18948 −0.594740 0.803918i \(-0.702745\pi\)
−0.594740 + 0.803918i \(0.702745\pi\)
\(212\) 4.93904e20 0.0392150
\(213\) 1.88438e22 1.43075
\(214\) −4.46420e21 −0.324200
\(215\) 6.26998e21 0.435613
\(216\) 1.94574e21 0.129353
\(217\) 5.73633e21 0.364978
\(218\) 2.13562e22 1.30073
\(219\) −4.64668e22 −2.70972
\(220\) 4.56078e21 0.254697
\(221\) 1.91576e22 1.02474
\(222\) −2.61868e22 −1.34193
\(223\) 1.40599e22 0.690375 0.345187 0.938534i \(-0.387815\pi\)
0.345187 + 0.938534i \(0.387815\pi\)
\(224\) 9.59301e20 0.0451439
\(225\) 5.51648e21 0.248845
\(226\) −4.45743e21 −0.192777
\(227\) 3.41441e22 1.41602 0.708011 0.706201i \(-0.249592\pi\)
0.708011 + 0.706201i \(0.249592\pi\)
\(228\) 9.41097e21 0.374329
\(229\) 2.26978e22 0.866057 0.433028 0.901380i \(-0.357445\pi\)
0.433028 + 0.901380i \(0.357445\pi\)
\(230\) −5.93310e21 −0.217204
\(231\) 1.24039e22 0.435758
\(232\) 1.91229e22 0.644789
\(233\) 3.14636e22 1.01842 0.509211 0.860642i \(-0.329937\pi\)
0.509211 + 0.860642i \(0.329937\pi\)
\(234\) −2.88014e22 −0.895083
\(235\) −1.25934e22 −0.375835
\(236\) −1.12323e21 −0.0321962
\(237\) 2.68389e22 0.739018
\(238\) −6.87504e21 −0.181884
\(239\) −1.24777e22 −0.317216 −0.158608 0.987342i \(-0.550701\pi\)
−0.158608 + 0.987342i \(0.550701\pi\)
\(240\) 6.85480e21 0.167490
\(241\) −1.16264e21 −0.0273076 −0.0136538 0.999907i \(-0.504346\pi\)
−0.0136538 + 0.999907i \(0.504346\pi\)
\(242\) 9.31301e21 0.210302
\(243\) −6.48761e22 −1.40872
\(244\) −3.14266e22 −0.656287
\(245\) −2.08116e22 −0.418049
\(246\) −1.06820e23 −2.06429
\(247\) −2.73433e22 −0.508429
\(248\) 2.82383e22 0.505299
\(249\) −1.41613e23 −2.43899
\(250\) 3.81470e21 0.0632456
\(251\) 1.09655e23 1.75036 0.875182 0.483793i \(-0.160741\pi\)
0.875182 + 0.483793i \(0.160741\pi\)
\(252\) 1.03359e22 0.158870
\(253\) −5.28507e22 −0.782359
\(254\) 3.10916e22 0.443325
\(255\) −4.91264e22 −0.674813
\(256\) 4.72237e21 0.0625000
\(257\) −2.11998e22 −0.270375 −0.135187 0.990820i \(-0.543164\pi\)
−0.135187 + 0.990820i \(0.543164\pi\)
\(258\) 8.39442e22 1.03182
\(259\) −2.73044e22 −0.323508
\(260\) −1.99164e22 −0.227491
\(261\) 2.06037e23 2.26914
\(262\) 9.97536e22 1.05942
\(263\) −1.06654e23 −1.09244 −0.546222 0.837641i \(-0.683934\pi\)
−0.546222 + 0.837641i \(0.683934\pi\)
\(264\) 6.10611e22 0.603291
\(265\) 3.67987e21 0.0350750
\(266\) 9.81261e21 0.0902420
\(267\) −1.98603e22 −0.176250
\(268\) 5.07973e22 0.435071
\(269\) 9.27742e22 0.766974 0.383487 0.923546i \(-0.374723\pi\)
0.383487 + 0.923546i \(0.374723\pi\)
\(270\) 1.44969e22 0.115696
\(271\) −9.46299e22 −0.729156 −0.364578 0.931173i \(-0.618787\pi\)
−0.364578 + 0.931173i \(0.618787\pi\)
\(272\) −3.38438e22 −0.251811
\(273\) −5.41666e22 −0.389212
\(274\) 6.20624e22 0.430723
\(275\) 3.39805e22 0.227808
\(276\) −7.94340e22 −0.514482
\(277\) −7.04507e21 −0.0440887 −0.0220444 0.999757i \(-0.507018\pi\)
−0.0220444 + 0.999757i \(0.507018\pi\)
\(278\) 1.13916e23 0.688908
\(279\) 3.04251e23 1.77824
\(280\) 7.14735e21 0.0403779
\(281\) −1.00259e22 −0.0547539 −0.0273770 0.999625i \(-0.508715\pi\)
−0.0273770 + 0.999625i \(0.508715\pi\)
\(282\) −1.68604e23 −0.890226
\(283\) −2.12867e23 −1.08677 −0.543383 0.839485i \(-0.682857\pi\)
−0.543383 + 0.839485i \(0.682857\pi\)
\(284\) 9.67218e22 0.477529
\(285\) 7.01172e22 0.334810
\(286\) −1.77411e23 −0.819415
\(287\) −1.11379e23 −0.497653
\(288\) 5.08806e22 0.219950
\(289\) 3.47675e21 0.0145427
\(290\) 1.42477e23 0.576717
\(291\) −4.97311e22 −0.194825
\(292\) −2.38505e23 −0.904400
\(293\) 5.10289e23 1.87315 0.936577 0.350462i \(-0.113975\pi\)
0.936577 + 0.350462i \(0.113975\pi\)
\(294\) −2.78631e23 −0.990216
\(295\) −8.36872e21 −0.0287972
\(296\) −1.34412e23 −0.447885
\(297\) 1.29135e23 0.416734
\(298\) 1.77610e23 0.555152
\(299\) 2.30793e23 0.698790
\(300\) 5.10722e22 0.149807
\(301\) 8.75268e22 0.248748
\(302\) −5.38650e21 −0.0148334
\(303\) −8.55260e23 −2.28241
\(304\) 4.83047e22 0.124937
\(305\) −2.34146e23 −0.587001
\(306\) −3.64646e23 −0.886173
\(307\) −4.42100e23 −1.04161 −0.520806 0.853675i \(-0.674369\pi\)
−0.520806 + 0.853675i \(0.674369\pi\)
\(308\) 6.36670e22 0.145440
\(309\) 2.25126e23 0.498678
\(310\) 2.10392e23 0.451953
\(311\) −3.04953e23 −0.635344 −0.317672 0.948201i \(-0.602901\pi\)
−0.317672 + 0.948201i \(0.602901\pi\)
\(312\) −2.66647e23 −0.538850
\(313\) −6.44559e23 −1.26355 −0.631774 0.775153i \(-0.717673\pi\)
−0.631774 + 0.775153i \(0.717673\pi\)
\(314\) −1.57550e22 −0.0299630
\(315\) 7.70083e22 0.142098
\(316\) 1.37759e23 0.246656
\(317\) 1.02667e24 1.78389 0.891947 0.452140i \(-0.149339\pi\)
0.891947 + 0.452140i \(0.149339\pi\)
\(318\) 4.92672e22 0.0830807
\(319\) 1.26915e24 2.07731
\(320\) 3.51844e22 0.0559017
\(321\) −4.45306e23 −0.686849
\(322\) −8.28240e22 −0.124030
\(323\) −3.46186e23 −0.503368
\(324\) −2.46513e23 −0.348066
\(325\) −1.48389e23 −0.203474
\(326\) 2.95075e23 0.392975
\(327\) 2.13030e24 2.75573
\(328\) −5.48289e23 −0.688982
\(329\) −1.75799e23 −0.214613
\(330\) 4.54940e23 0.539600
\(331\) 8.39874e23 0.967939 0.483970 0.875085i \(-0.339194\pi\)
0.483970 + 0.875085i \(0.339194\pi\)
\(332\) −7.26873e23 −0.814042
\(333\) −1.44820e24 −1.57619
\(334\) 3.53050e23 0.373460
\(335\) 3.78469e23 0.389139
\(336\) 9.56907e22 0.0956416
\(337\) −7.88161e23 −0.765827 −0.382913 0.923784i \(-0.625079\pi\)
−0.382913 + 0.923784i \(0.625079\pi\)
\(338\) 2.62323e22 0.0247815
\(339\) −4.44631e23 −0.408416
\(340\) −2.52156e23 −0.225227
\(341\) 1.87413e24 1.62792
\(342\) 5.20453e23 0.439677
\(343\) −6.01313e23 −0.494090
\(344\) 4.30870e23 0.344382
\(345\) −5.91829e23 −0.460166
\(346\) −2.12093e22 −0.0160437
\(347\) −3.27984e23 −0.241392 −0.120696 0.992690i \(-0.538513\pi\)
−0.120696 + 0.992690i \(0.538513\pi\)
\(348\) 1.90752e24 1.36605
\(349\) −1.57618e24 −1.09841 −0.549204 0.835689i \(-0.685069\pi\)
−0.549204 + 0.835689i \(0.685069\pi\)
\(350\) 5.32519e22 0.0361151
\(351\) −5.63920e23 −0.372220
\(352\) 3.13415e23 0.201356
\(353\) −1.44980e23 −0.0906667 −0.0453333 0.998972i \(-0.514435\pi\)
−0.0453333 + 0.998972i \(0.514435\pi\)
\(354\) −1.12043e23 −0.0682108
\(355\) 7.20633e23 0.427115
\(356\) −1.01939e23 −0.0588254
\(357\) −6.85788e23 −0.385338
\(358\) 1.86971e24 1.02303
\(359\) 7.64624e23 0.407428 0.203714 0.979030i \(-0.434699\pi\)
0.203714 + 0.979030i \(0.434699\pi\)
\(360\) 3.79090e23 0.196729
\(361\) −1.48432e24 −0.750253
\(362\) 5.38786e23 0.265268
\(363\) 9.28978e23 0.445546
\(364\) −2.78027e23 −0.129904
\(365\) −1.77700e24 −0.808920
\(366\) −3.13482e24 −1.39041
\(367\) −2.08854e24 −0.902639 −0.451320 0.892362i \(-0.649047\pi\)
−0.451320 + 0.892362i \(0.649047\pi\)
\(368\) −4.07719e23 −0.171714
\(369\) −5.90747e24 −2.42466
\(370\) −1.00145e24 −0.400600
\(371\) 5.13698e22 0.0200288
\(372\) 2.81679e24 1.07052
\(373\) 2.90964e24 1.07797 0.538984 0.842316i \(-0.318808\pi\)
0.538984 + 0.842316i \(0.318808\pi\)
\(374\) −2.24615e24 −0.811258
\(375\) 3.80518e23 0.133992
\(376\) −8.65410e23 −0.297124
\(377\) −5.54224e24 −1.85542
\(378\) 2.02372e23 0.0660661
\(379\) 4.57048e24 1.45509 0.727544 0.686061i \(-0.240662\pi\)
0.727544 + 0.686061i \(0.240662\pi\)
\(380\) 3.59898e23 0.111747
\(381\) 3.10140e24 0.939226
\(382\) −3.39805e24 −1.00375
\(383\) 4.74338e24 1.36678 0.683392 0.730052i \(-0.260504\pi\)
0.683392 + 0.730052i \(0.260504\pi\)
\(384\) 4.71058e23 0.132412
\(385\) 4.74356e23 0.130085
\(386\) −1.35949e24 −0.363745
\(387\) 4.64235e24 1.21195
\(388\) −2.55260e23 −0.0650252
\(389\) −1.60295e24 −0.398472 −0.199236 0.979952i \(-0.563846\pi\)
−0.199236 + 0.979952i \(0.563846\pi\)
\(390\) −1.98667e24 −0.481962
\(391\) 2.92201e24 0.691834
\(392\) −1.43016e24 −0.330496
\(393\) 9.95047e24 2.24447
\(394\) 5.24060e24 1.15390
\(395\) 1.02638e24 0.220616
\(396\) 3.37685e24 0.708610
\(397\) 5.17477e24 1.06018 0.530092 0.847940i \(-0.322157\pi\)
0.530092 + 0.847940i \(0.322157\pi\)
\(398\) −1.29578e24 −0.259204
\(399\) 9.78812e23 0.191186
\(400\) 2.62144e23 0.0500000
\(401\) 1.04892e24 0.195377 0.0976883 0.995217i \(-0.468855\pi\)
0.0976883 + 0.995217i \(0.468855\pi\)
\(402\) 5.06705e24 0.921739
\(403\) −8.18410e24 −1.45403
\(404\) −4.38989e24 −0.761781
\(405\) −1.83666e24 −0.311320
\(406\) 1.98893e24 0.329322
\(407\) −8.92066e24 −1.44295
\(408\) −3.37594e24 −0.533486
\(409\) −9.84304e24 −1.51970 −0.759850 0.650098i \(-0.774728\pi\)
−0.759850 + 0.650098i \(0.774728\pi\)
\(410\) −4.08507e24 −0.616244
\(411\) 6.19076e24 0.912529
\(412\) 1.15553e24 0.166440
\(413\) −1.16825e23 −0.0164440
\(414\) −4.39292e24 −0.604297
\(415\) −5.41562e24 −0.728101
\(416\) −1.36865e24 −0.179848
\(417\) 1.13632e25 1.45952
\(418\) 3.20589e24 0.402508
\(419\) 1.39759e24 0.171533 0.0857664 0.996315i \(-0.472666\pi\)
0.0857664 + 0.996315i \(0.472666\pi\)
\(420\) 7.12952e23 0.0855444
\(421\) 2.96558e24 0.347880 0.173940 0.984756i \(-0.444350\pi\)
0.173940 + 0.984756i \(0.444350\pi\)
\(422\) −7.33348e24 −0.841089
\(423\) −9.32425e24 −1.04564
\(424\) 2.52879e23 0.0277292
\(425\) −1.87871e24 −0.201449
\(426\) 9.64804e24 1.01169
\(427\) −3.26861e24 −0.335195
\(428\) −2.28567e24 −0.229244
\(429\) −1.76969e25 −1.73601
\(430\) 3.21023e24 0.308025
\(431\) −1.73742e25 −1.63069 −0.815343 0.578978i \(-0.803452\pi\)
−0.815343 + 0.578978i \(0.803452\pi\)
\(432\) 9.96221e23 0.0914661
\(433\) 1.97317e25 1.77227 0.886133 0.463432i \(-0.153382\pi\)
0.886133 + 0.463432i \(0.153382\pi\)
\(434\) 2.93700e24 0.258078
\(435\) 1.42121e25 1.22183
\(436\) 1.09344e25 0.919757
\(437\) −4.17052e24 −0.343255
\(438\) −2.37910e25 −1.91606
\(439\) 2.37490e25 1.87168 0.935840 0.352425i \(-0.114643\pi\)
0.935840 + 0.352425i \(0.114643\pi\)
\(440\) 2.33512e24 0.180098
\(441\) −1.54091e25 −1.16308
\(442\) 9.80870e24 0.724603
\(443\) −1.08663e25 −0.785679 −0.392840 0.919607i \(-0.628507\pi\)
−0.392840 + 0.919607i \(0.628507\pi\)
\(444\) −1.34076e25 −0.948886
\(445\) −7.59504e23 −0.0526150
\(446\) 7.19865e24 0.488169
\(447\) 1.77166e25 1.17614
\(448\) 4.91162e23 0.0319215
\(449\) 1.64022e25 1.04367 0.521834 0.853047i \(-0.325248\pi\)
0.521834 + 0.853047i \(0.325248\pi\)
\(450\) 2.82444e24 0.175960
\(451\) −3.63889e25 −2.21969
\(452\) −2.28220e24 −0.136314
\(453\) −5.37306e23 −0.0314260
\(454\) 1.74818e25 1.00128
\(455\) −2.07146e24 −0.116190
\(456\) 4.81841e24 0.264690
\(457\) −1.90127e25 −1.02291 −0.511457 0.859309i \(-0.670894\pi\)
−0.511457 + 0.859309i \(0.670894\pi\)
\(458\) 1.16213e25 0.612395
\(459\) −7.13963e24 −0.368515
\(460\) −3.03774e24 −0.153586
\(461\) −2.73913e25 −1.35660 −0.678302 0.734783i \(-0.737284\pi\)
−0.678302 + 0.734783i \(0.737284\pi\)
\(462\) 6.35082e24 0.308128
\(463\) 2.15999e25 1.02667 0.513337 0.858187i \(-0.328409\pi\)
0.513337 + 0.858187i \(0.328409\pi\)
\(464\) 9.79092e24 0.455935
\(465\) 2.09867e25 0.957505
\(466\) 1.61094e25 0.720133
\(467\) 6.57907e24 0.288174 0.144087 0.989565i \(-0.453976\pi\)
0.144087 + 0.989565i \(0.453976\pi\)
\(468\) −1.47463e25 −0.632919
\(469\) 5.28330e24 0.222210
\(470\) −6.44781e24 −0.265756
\(471\) −1.57157e24 −0.0634796
\(472\) −5.75094e23 −0.0227662
\(473\) 2.85960e25 1.10949
\(474\) 1.37415e25 0.522565
\(475\) 2.68145e24 0.0999494
\(476\) −3.52002e24 −0.128611
\(477\) 2.72461e24 0.0975844
\(478\) −6.38858e24 −0.224305
\(479\) 1.84209e25 0.634049 0.317024 0.948417i \(-0.397316\pi\)
0.317024 + 0.948417i \(0.397316\pi\)
\(480\) 3.50966e24 0.118433
\(481\) 3.89555e25 1.28882
\(482\) −5.95271e23 −0.0193094
\(483\) −8.26174e24 −0.262769
\(484\) 4.76826e24 0.148706
\(485\) −1.90184e24 −0.0581603
\(486\) −3.32165e25 −0.996116
\(487\) −5.85949e25 −1.72320 −0.861598 0.507591i \(-0.830536\pi\)
−0.861598 + 0.507591i \(0.830536\pi\)
\(488\) −1.60904e25 −0.464065
\(489\) 2.94339e25 0.832555
\(490\) −1.06555e25 −0.295605
\(491\) 1.17567e24 0.0319896 0.0159948 0.999872i \(-0.494908\pi\)
0.0159948 + 0.999872i \(0.494908\pi\)
\(492\) −5.46921e25 −1.45967
\(493\) −7.01687e25 −1.83695
\(494\) −1.39998e25 −0.359513
\(495\) 2.51595e25 0.633800
\(496\) 1.44580e25 0.357300
\(497\) 1.00598e25 0.243895
\(498\) −7.25059e25 −1.72462
\(499\) −1.57255e25 −0.366985 −0.183492 0.983021i \(-0.558740\pi\)
−0.183492 + 0.983021i \(0.558740\pi\)
\(500\) 1.95313e24 0.0447214
\(501\) 3.52169e25 0.791211
\(502\) 5.61434e25 1.23769
\(503\) −2.64249e24 −0.0571632 −0.0285816 0.999591i \(-0.509099\pi\)
−0.0285816 + 0.999591i \(0.509099\pi\)
\(504\) 5.29197e24 0.112338
\(505\) −3.27072e25 −0.681358
\(506\) −2.70596e25 −0.553211
\(507\) 2.61669e24 0.0525020
\(508\) 1.59189e25 0.313478
\(509\) 7.94197e25 1.53500 0.767501 0.641048i \(-0.221500\pi\)
0.767501 + 0.641048i \(0.221500\pi\)
\(510\) −2.51527e25 −0.477165
\(511\) −2.48064e25 −0.461918
\(512\) 2.41785e24 0.0441942
\(513\) 1.01903e25 0.182840
\(514\) −1.08543e25 −0.191184
\(515\) 8.60937e24 0.148868
\(516\) 4.29794e25 0.729606
\(517\) −5.74356e25 −0.957241
\(518\) −1.39798e25 −0.228755
\(519\) −2.11564e24 −0.0339900
\(520\) −1.01972e25 −0.160861
\(521\) 4.66972e25 0.723324 0.361662 0.932309i \(-0.382209\pi\)
0.361662 + 0.932309i \(0.382209\pi\)
\(522\) 1.05491e26 1.60452
\(523\) −6.46849e25 −0.966132 −0.483066 0.875584i \(-0.660477\pi\)
−0.483066 + 0.875584i \(0.660477\pi\)
\(524\) 5.10739e25 0.749120
\(525\) 5.31190e24 0.0765133
\(526\) −5.46070e25 −0.772474
\(527\) −1.03617e26 −1.43955
\(528\) 3.12633e25 0.426591
\(529\) −3.94138e25 −0.528226
\(530\) 1.88410e24 0.0248017
\(531\) −6.19628e24 −0.0801186
\(532\) 5.02405e24 0.0638108
\(533\) 1.58906e26 1.98259
\(534\) −1.01685e25 −0.124627
\(535\) −1.70296e25 −0.205042
\(536\) 2.60082e25 0.307641
\(537\) 1.86505e26 2.16738
\(538\) 4.75004e25 0.542333
\(539\) −9.49170e25 −1.06476
\(540\) 7.42243e24 0.0818098
\(541\) 1.28312e25 0.138961 0.0694805 0.997583i \(-0.477866\pi\)
0.0694805 + 0.997583i \(0.477866\pi\)
\(542\) −4.84505e25 −0.515591
\(543\) 5.37441e25 0.561995
\(544\) −1.73281e25 −0.178057
\(545\) 8.14676e25 0.822656
\(546\) −2.77333e25 −0.275215
\(547\) 5.22682e25 0.509750 0.254875 0.966974i \(-0.417966\pi\)
0.254875 + 0.966974i \(0.417966\pi\)
\(548\) 3.17760e25 0.304567
\(549\) −1.73364e26 −1.63314
\(550\) 1.73980e25 0.161085
\(551\) 1.00150e26 0.911408
\(552\) −4.06702e25 −0.363793
\(553\) 1.43280e25 0.125978
\(554\) −3.60708e24 −0.0311754
\(555\) −9.98947e25 −0.848710
\(556\) 5.83252e25 0.487131
\(557\) 5.10671e25 0.419292 0.209646 0.977777i \(-0.432769\pi\)
0.209646 + 0.977777i \(0.432769\pi\)
\(558\) 1.55776e26 1.25741
\(559\) −1.24876e26 −0.990981
\(560\) 3.65944e24 0.0285515
\(561\) −2.24055e26 −1.71873
\(562\) −5.13328e24 −0.0387169
\(563\) 1.95962e26 1.45325 0.726627 0.687032i \(-0.241087\pi\)
0.726627 + 0.687032i \(0.241087\pi\)
\(564\) −8.63250e25 −0.629485
\(565\) −1.70037e25 −0.121923
\(566\) −1.08988e26 −0.768460
\(567\) −2.56392e25 −0.177773
\(568\) 4.95215e25 0.337664
\(569\) −2.04702e25 −0.137264 −0.0686320 0.997642i \(-0.521863\pi\)
−0.0686320 + 0.997642i \(0.521863\pi\)
\(570\) 3.59000e25 0.236746
\(571\) −1.26019e26 −0.817322 −0.408661 0.912686i \(-0.634004\pi\)
−0.408661 + 0.912686i \(0.634004\pi\)
\(572\) −9.08346e25 −0.579414
\(573\) −3.38957e26 −2.12655
\(574\) −5.70262e25 −0.351894
\(575\) −2.26330e25 −0.137372
\(576\) 2.60509e25 0.155528
\(577\) −1.24271e26 −0.729792 −0.364896 0.931048i \(-0.618895\pi\)
−0.364896 + 0.931048i \(0.618895\pi\)
\(578\) 1.78010e24 0.0102832
\(579\) −1.35610e26 −0.770629
\(580\) 7.29480e25 0.407800
\(581\) −7.56003e25 −0.415767
\(582\) −2.54623e25 −0.137762
\(583\) 1.67831e25 0.0893349
\(584\) −1.22115e26 −0.639508
\(585\) −1.09869e26 −0.566100
\(586\) 2.61268e26 1.32452
\(587\) 3.67268e26 1.83198 0.915991 0.401199i \(-0.131407\pi\)
0.915991 + 0.401199i \(0.131407\pi\)
\(588\) −1.42659e26 −0.700188
\(589\) 1.47890e26 0.714238
\(590\) −4.28479e24 −0.0203627
\(591\) 5.22753e26 2.44464
\(592\) −6.88188e25 −0.316702
\(593\) −1.40519e26 −0.636378 −0.318189 0.948027i \(-0.603075\pi\)
−0.318189 + 0.948027i \(0.603075\pi\)
\(594\) 6.61173e25 0.294676
\(595\) −2.62262e25 −0.115033
\(596\) 9.09361e25 0.392552
\(597\) −1.29255e26 −0.549149
\(598\) 1.18166e26 0.494119
\(599\) 2.05613e26 0.846244 0.423122 0.906073i \(-0.360934\pi\)
0.423122 + 0.906073i \(0.360934\pi\)
\(600\) 2.61490e25 0.105930
\(601\) 2.32345e26 0.926457 0.463228 0.886239i \(-0.346691\pi\)
0.463228 + 0.886239i \(0.346691\pi\)
\(602\) 4.48137e25 0.175891
\(603\) 2.80222e26 1.08265
\(604\) −2.75789e24 −0.0104888
\(605\) 3.55263e25 0.133007
\(606\) −4.37893e26 −1.61391
\(607\) 2.30237e26 0.835376 0.417688 0.908591i \(-0.362841\pi\)
0.417688 + 0.908591i \(0.362841\pi\)
\(608\) 2.47320e25 0.0883436
\(609\) 1.98396e26 0.697701
\(610\) −1.19883e26 −0.415073
\(611\) 2.50815e26 0.854992
\(612\) −1.86699e26 −0.626619
\(613\) −7.21990e25 −0.238592 −0.119296 0.992859i \(-0.538064\pi\)
−0.119296 + 0.992859i \(0.538064\pi\)
\(614\) −2.26355e26 −0.736531
\(615\) −4.07488e26 −1.30557
\(616\) 3.25975e25 0.102841
\(617\) −1.44998e26 −0.450458 −0.225229 0.974306i \(-0.572313\pi\)
−0.225229 + 0.974306i \(0.572313\pi\)
\(618\) 1.15265e26 0.352619
\(619\) −2.13817e26 −0.644141 −0.322071 0.946716i \(-0.604379\pi\)
−0.322071 + 0.946716i \(0.604379\pi\)
\(620\) 1.07721e26 0.319579
\(621\) −8.60116e25 −0.251297
\(622\) −1.56136e26 −0.449256
\(623\) −1.06024e25 −0.0300448
\(624\) −1.36523e26 −0.381025
\(625\) 1.45519e25 0.0400000
\(626\) −3.30014e26 −0.893463
\(627\) 3.19789e26 0.852751
\(628\) −8.06654e24 −0.0211871
\(629\) 4.93205e26 1.27599
\(630\) 3.94282e25 0.100478
\(631\) 1.60890e26 0.403877 0.201939 0.979398i \(-0.435276\pi\)
0.201939 + 0.979398i \(0.435276\pi\)
\(632\) 7.05325e25 0.174412
\(633\) −7.31519e26 −1.78193
\(634\) 5.25656e26 1.26140
\(635\) 1.18605e26 0.280383
\(636\) 2.52248e25 0.0587469
\(637\) 4.14492e26 0.951024
\(638\) 6.49805e26 1.46888
\(639\) 5.33564e26 1.18831
\(640\) 1.80144e25 0.0395285
\(641\) −5.97417e26 −1.29160 −0.645798 0.763509i \(-0.723475\pi\)
−0.645798 + 0.763509i \(0.723475\pi\)
\(642\) −2.27997e26 −0.485675
\(643\) −8.08712e26 −1.69742 −0.848710 0.528858i \(-0.822620\pi\)
−0.848710 + 0.528858i \(0.822620\pi\)
\(644\) −4.24059e25 −0.0877022
\(645\) 3.20222e26 0.652580
\(646\) −1.77247e26 −0.355935
\(647\) −7.53176e26 −1.49041 −0.745205 0.666835i \(-0.767649\pi\)
−0.745205 + 0.666835i \(0.767649\pi\)
\(648\) −1.26214e26 −0.246120
\(649\) −3.81679e25 −0.0733456
\(650\) −7.59752e25 −0.143878
\(651\) 2.92967e26 0.546764
\(652\) 1.51078e26 0.277875
\(653\) 1.75162e25 0.0317515 0.0158758 0.999874i \(-0.494946\pi\)
0.0158758 + 0.999874i \(0.494946\pi\)
\(654\) 1.09071e27 1.94859
\(655\) 3.80530e26 0.670033
\(656\) −2.80724e26 −0.487184
\(657\) −1.31571e27 −2.25055
\(658\) −9.00092e25 −0.151754
\(659\) −1.10678e27 −1.83929 −0.919644 0.392754i \(-0.871523\pi\)
−0.919644 + 0.392754i \(0.871523\pi\)
\(660\) 2.32929e26 0.381555
\(661\) −1.90870e26 −0.308194 −0.154097 0.988056i \(-0.549247\pi\)
−0.154097 + 0.988056i \(0.549247\pi\)
\(662\) 4.30015e26 0.684437
\(663\) 9.78423e26 1.53514
\(664\) −3.72159e26 −0.575614
\(665\) 3.74321e25 0.0570741
\(666\) −7.41480e26 −1.11454
\(667\) −8.45327e26 −1.25265
\(668\) 1.80761e26 0.264076
\(669\) 7.18069e26 1.03423
\(670\) 1.93776e26 0.275163
\(671\) −1.06789e27 −1.49508
\(672\) 4.89937e25 0.0676288
\(673\) −1.19576e27 −1.62742 −0.813710 0.581270i \(-0.802556\pi\)
−0.813710 + 0.581270i \(0.802556\pi\)
\(674\) −4.03538e26 −0.541521
\(675\) 5.53014e25 0.0731729
\(676\) 1.34309e25 0.0175232
\(677\) 4.22916e26 0.544079 0.272040 0.962286i \(-0.412302\pi\)
0.272040 + 0.962286i \(0.412302\pi\)
\(678\) −2.27651e26 −0.288793
\(679\) −2.65490e25 −0.0332113
\(680\) −1.29104e26 −0.159259
\(681\) 1.74382e27 2.12131
\(682\) 9.59552e26 1.15111
\(683\) 7.22585e24 0.00854854 0.00427427 0.999991i \(-0.498639\pi\)
0.00427427 + 0.999991i \(0.498639\pi\)
\(684\) 2.66472e26 0.310898
\(685\) 2.36749e26 0.272413
\(686\) −3.07872e26 −0.349375
\(687\) 1.15923e27 1.29742
\(688\) 2.20605e26 0.243515
\(689\) −7.32900e25 −0.0797925
\(690\) −3.03017e26 −0.325387
\(691\) −1.59913e26 −0.169372 −0.0846861 0.996408i \(-0.526989\pi\)
−0.0846861 + 0.996408i \(0.526989\pi\)
\(692\) −1.08592e25 −0.0113446
\(693\) 3.51218e26 0.361919
\(694\) −1.67928e26 −0.170690
\(695\) 4.34557e26 0.435703
\(696\) 9.76648e26 0.965941
\(697\) 2.01187e27 1.96285
\(698\) −8.07003e26 −0.776691
\(699\) 1.60692e27 1.52567
\(700\) 2.72650e25 0.0255372
\(701\) −3.07030e26 −0.283700 −0.141850 0.989888i \(-0.545305\pi\)
−0.141850 + 0.989888i \(0.545305\pi\)
\(702\) −2.88727e26 −0.263199
\(703\) −7.03942e26 −0.633084
\(704\) 1.60468e26 0.142380
\(705\) −6.43172e26 −0.563028
\(706\) −7.42296e25 −0.0641110
\(707\) −4.56582e26 −0.389076
\(708\) −5.73659e25 −0.0482323
\(709\) −2.50359e26 −0.207694 −0.103847 0.994593i \(-0.533115\pi\)
−0.103847 + 0.994593i \(0.533115\pi\)
\(710\) 3.68964e26 0.302016
\(711\) 7.59944e26 0.613791
\(712\) −5.21927e25 −0.0415958
\(713\) −1.24828e27 −0.981657
\(714\) −3.51124e26 −0.272475
\(715\) −6.76770e26 −0.518244
\(716\) 9.57294e26 0.723388
\(717\) −6.37264e26 −0.475212
\(718\) 3.91488e26 0.288095
\(719\) −4.99484e26 −0.362741 −0.181371 0.983415i \(-0.558053\pi\)
−0.181371 + 0.983415i \(0.558053\pi\)
\(720\) 1.94094e26 0.139108
\(721\) 1.20184e26 0.0850083
\(722\) −7.59969e26 −0.530509
\(723\) −5.93786e25 −0.0409087
\(724\) 2.75858e26 0.187573
\(725\) 5.43505e26 0.364748
\(726\) 4.75637e26 0.315048
\(727\) 1.17305e27 0.766900 0.383450 0.923562i \(-0.374736\pi\)
0.383450 + 0.923562i \(0.374736\pi\)
\(728\) −1.42350e26 −0.0918562
\(729\) −2.22041e27 −1.41423
\(730\) −9.09825e26 −0.571993
\(731\) −1.58101e27 −0.981117
\(732\) −1.60503e27 −0.983166
\(733\) 7.29500e26 0.441101 0.220550 0.975376i \(-0.429215\pi\)
0.220550 + 0.975376i \(0.429215\pi\)
\(734\) −1.06933e27 −0.638262
\(735\) −1.06289e27 −0.626267
\(736\) −2.08752e26 −0.121420
\(737\) 1.72612e27 0.991126
\(738\) −3.02463e27 −1.71450
\(739\) 2.41117e27 1.34929 0.674647 0.738141i \(-0.264296\pi\)
0.674647 + 0.738141i \(0.264296\pi\)
\(740\) −5.12740e26 −0.283267
\(741\) −1.39648e27 −0.761663
\(742\) 2.63013e25 0.0141625
\(743\) 1.98675e27 1.05621 0.528105 0.849179i \(-0.322903\pi\)
0.528105 + 0.849179i \(0.322903\pi\)
\(744\) 1.44220e27 0.756974
\(745\) 6.77527e26 0.351109
\(746\) 1.48974e27 0.762238
\(747\) −4.00978e27 −2.02570
\(748\) −1.15003e27 −0.573646
\(749\) −2.37727e26 −0.117085
\(750\) 1.94825e26 0.0947465
\(751\) 4.77748e26 0.229414 0.114707 0.993399i \(-0.463407\pi\)
0.114707 + 0.993399i \(0.463407\pi\)
\(752\) −4.43090e26 −0.210098
\(753\) 5.60033e27 2.62217
\(754\) −2.83763e27 −1.31198
\(755\) −2.05479e25 −0.00938147
\(756\) 1.03615e26 0.0467158
\(757\) 8.34040e25 0.0371344 0.0185672 0.999828i \(-0.494090\pi\)
0.0185672 + 0.999828i \(0.494090\pi\)
\(758\) 2.34009e27 1.02890
\(759\) −2.69920e27 −1.17203
\(760\) 1.84268e26 0.0790169
\(761\) −2.96552e27 −1.25587 −0.627937 0.778264i \(-0.716100\pi\)
−0.627937 + 0.778264i \(0.716100\pi\)
\(762\) 1.58792e27 0.664133
\(763\) 1.13726e27 0.469761
\(764\) −1.73980e27 −0.709761
\(765\) −1.39102e27 −0.560465
\(766\) 2.42861e27 0.966462
\(767\) 1.66675e26 0.0655111
\(768\) 2.41182e26 0.0936296
\(769\) 9.68856e26 0.371500 0.185750 0.982597i \(-0.440528\pi\)
0.185750 + 0.982597i \(0.440528\pi\)
\(770\) 2.42870e26 0.0919841
\(771\) −1.08272e27 −0.405041
\(772\) −6.96060e26 −0.257207
\(773\) 5.16072e27 1.88367 0.941836 0.336074i \(-0.109099\pi\)
0.941836 + 0.336074i \(0.109099\pi\)
\(774\) 2.37688e27 0.856976
\(775\) 8.02582e26 0.285840
\(776\) −1.30693e26 −0.0459798
\(777\) −1.39450e27 −0.484638
\(778\) −8.20709e26 −0.281762
\(779\) −2.87150e27 −0.973875
\(780\) −1.01718e27 −0.340799
\(781\) 3.28665e27 1.08785
\(782\) 1.49607e27 0.489201
\(783\) 2.06547e27 0.667241
\(784\) −7.32242e26 −0.233696
\(785\) −6.01004e25 −0.0189503
\(786\) 5.09464e27 1.58708
\(787\) −4.95523e27 −1.52512 −0.762559 0.646919i \(-0.776057\pi\)
−0.762559 + 0.646919i \(0.776057\pi\)
\(788\) 2.68319e27 0.815929
\(789\) −5.44708e27 −1.63656
\(790\) 5.25508e26 0.155999
\(791\) −2.37367e26 −0.0696214
\(792\) 1.72895e27 0.501063
\(793\) 4.66336e27 1.33538
\(794\) 2.64948e27 0.749663
\(795\) 1.87939e26 0.0525449
\(796\) −6.63439e26 −0.183285
\(797\) 9.51644e26 0.259789 0.129894 0.991528i \(-0.458536\pi\)
0.129894 + 0.991528i \(0.458536\pi\)
\(798\) 5.01152e26 0.135189
\(799\) 3.17550e27 0.846481
\(800\) 1.34218e26 0.0353553
\(801\) −5.62344e26 −0.146384
\(802\) 5.37049e26 0.138152
\(803\) −8.10452e27 −2.06030
\(804\) 2.59433e27 0.651768
\(805\) −3.15949e26 −0.0784433
\(806\) −4.19026e27 −1.02815
\(807\) 4.73819e27 1.14898
\(808\) −2.24762e27 −0.538661
\(809\) 6.64620e26 0.157421 0.0787105 0.996898i \(-0.474920\pi\)
0.0787105 + 0.996898i \(0.474920\pi\)
\(810\) −9.40371e26 −0.220136
\(811\) −3.63702e27 −0.841487 −0.420743 0.907180i \(-0.638231\pi\)
−0.420743 + 0.907180i \(0.638231\pi\)
\(812\) 1.01833e27 0.232866
\(813\) −4.83296e27 −1.09233
\(814\) −4.56738e27 −1.02032
\(815\) 1.12562e27 0.248539
\(816\) −1.72848e27 −0.377232
\(817\) 2.25655e27 0.486783
\(818\) −5.03964e27 −1.07459
\(819\) −1.53373e27 −0.323260
\(820\) −2.09156e27 −0.435751
\(821\) −4.44788e26 −0.0915996 −0.0457998 0.998951i \(-0.514584\pi\)
−0.0457998 + 0.998951i \(0.514584\pi\)
\(822\) 3.16967e27 0.645255
\(823\) −5.12412e27 −1.03115 −0.515574 0.856845i \(-0.672421\pi\)
−0.515574 + 0.856845i \(0.672421\pi\)
\(824\) 5.91631e26 0.117691
\(825\) 1.73546e27 0.341273
\(826\) −5.98142e25 −0.0116277
\(827\) −5.00813e27 −0.962439 −0.481219 0.876600i \(-0.659806\pi\)
−0.481219 + 0.876600i \(0.659806\pi\)
\(828\) −2.24918e27 −0.427302
\(829\) −3.98409e27 −0.748275 −0.374138 0.927373i \(-0.622061\pi\)
−0.374138 + 0.927373i \(0.622061\pi\)
\(830\) −2.77280e27 −0.514845
\(831\) −3.59807e26 −0.0660481
\(832\) −7.00747e26 −0.127172
\(833\) 5.24777e27 0.941558
\(834\) 5.81797e27 1.03203
\(835\) 1.34678e27 0.236197
\(836\) 1.64142e27 0.284616
\(837\) 3.05004e27 0.522893
\(838\) 7.15567e26 0.121292
\(839\) 9.17325e27 1.53739 0.768696 0.639615i \(-0.220906\pi\)
0.768696 + 0.639615i \(0.220906\pi\)
\(840\) 3.65031e26 0.0604890
\(841\) 1.41963e28 2.32602
\(842\) 1.51838e27 0.245988
\(843\) −5.12047e26 −0.0820254
\(844\) −3.75474e27 −0.594740
\(845\) 1.00068e26 0.0156732
\(846\) −4.77402e27 −0.739376
\(847\) 4.95936e26 0.0759509
\(848\) 1.29474e26 0.0196075
\(849\) −1.08716e28 −1.62806
\(850\) −9.61900e26 −0.142446
\(851\) 5.94167e27 0.870118
\(852\) 4.93980e27 0.715374
\(853\) −7.79931e27 −1.11697 −0.558484 0.829516i \(-0.688617\pi\)
−0.558484 + 0.829516i \(0.688617\pi\)
\(854\) −1.67353e27 −0.237019
\(855\) 1.98537e27 0.278076
\(856\) −1.17026e27 −0.162100
\(857\) 5.48738e26 0.0751705 0.0375852 0.999293i \(-0.488033\pi\)
0.0375852 + 0.999293i \(0.488033\pi\)
\(858\) −9.06079e27 −1.22754
\(859\) 2.88785e27 0.386936 0.193468 0.981107i \(-0.438026\pi\)
0.193468 + 0.981107i \(0.438026\pi\)
\(860\) 1.64364e27 0.217806
\(861\) −5.68839e27 −0.745521
\(862\) −8.89559e27 −1.15307
\(863\) −1.27182e27 −0.163051 −0.0815257 0.996671i \(-0.525979\pi\)
−0.0815257 + 0.996671i \(0.525979\pi\)
\(864\) 5.10065e26 0.0646763
\(865\) −8.09071e25 −0.0101469
\(866\) 1.01026e28 1.25318
\(867\) 1.77566e26 0.0217860
\(868\) 1.50374e27 0.182489
\(869\) 4.68111e27 0.561903
\(870\) 7.27660e27 0.863964
\(871\) −7.53776e27 −0.885258
\(872\) 5.59841e27 0.650367
\(873\) −1.40814e27 −0.161812
\(874\) −2.13531e27 −0.242718
\(875\) 2.03140e26 0.0228412
\(876\) −1.21810e28 −1.35486
\(877\) −7.93254e27 −0.872803 −0.436401 0.899752i \(-0.643747\pi\)
−0.436401 + 0.899752i \(0.643747\pi\)
\(878\) 1.21595e28 1.32348
\(879\) 2.60616e28 2.80612
\(880\) 1.19558e27 0.127349
\(881\) −4.65349e27 −0.490351 −0.245176 0.969479i \(-0.578846\pi\)
−0.245176 + 0.969479i \(0.578846\pi\)
\(882\) −7.88945e27 −0.822423
\(883\) 1.39322e28 1.43678 0.718392 0.695639i \(-0.244878\pi\)
0.718392 + 0.695639i \(0.244878\pi\)
\(884\) 5.02206e27 0.512372
\(885\) −4.27409e26 −0.0431403
\(886\) −5.56353e27 −0.555559
\(887\) −2.98771e27 −0.295165 −0.147582 0.989050i \(-0.547149\pi\)
−0.147582 + 0.989050i \(0.547149\pi\)
\(888\) −6.86471e27 −0.670964
\(889\) 1.65569e27 0.160107
\(890\) −3.88866e26 −0.0372045
\(891\) −8.37662e27 −0.792922
\(892\) 3.68571e27 0.345187
\(893\) −4.53233e27 −0.419984
\(894\) 9.07092e27 0.831658
\(895\) 7.13239e27 0.647018
\(896\) 2.51475e26 0.0225719
\(897\) 1.17871e28 1.04684
\(898\) 8.39793e27 0.737984
\(899\) 2.99759e28 2.60649
\(900\) 1.44611e27 0.124422
\(901\) −9.27904e26 −0.0789982
\(902\) −1.86311e28 −1.56956
\(903\) 4.47019e27 0.372642
\(904\) −1.16849e27 −0.0963883
\(905\) 2.05530e27 0.167770
\(906\) −2.75101e26 −0.0222215
\(907\) −1.01372e28 −0.810308 −0.405154 0.914248i \(-0.632782\pi\)
−0.405154 + 0.914248i \(0.632782\pi\)
\(908\) 8.95068e27 0.708011
\(909\) −2.42167e28 −1.89565
\(910\) −1.06059e27 −0.0821587
\(911\) 1.01665e28 0.779377 0.389688 0.920947i \(-0.372583\pi\)
0.389688 + 0.920947i \(0.372583\pi\)
\(912\) 2.46703e27 0.187164
\(913\) −2.46995e28 −1.85445
\(914\) −9.73448e27 −0.723309
\(915\) −1.19584e28 −0.879371
\(916\) 5.95009e27 0.433028
\(917\) 5.31207e27 0.382609
\(918\) −3.65549e27 −0.260580
\(919\) −8.89132e27 −0.627291 −0.313645 0.949540i \(-0.601550\pi\)
−0.313645 + 0.949540i \(0.601550\pi\)
\(920\) −1.55533e27 −0.108602
\(921\) −2.25790e28 −1.56041
\(922\) −1.40243e28 −0.959265
\(923\) −1.43524e28 −0.971650
\(924\) 3.25162e27 0.217879
\(925\) −3.82021e27 −0.253362
\(926\) 1.10592e28 0.725968
\(927\) 6.37446e27 0.414177
\(928\) 5.01295e27 0.322394
\(929\) 7.61744e26 0.0484909 0.0242454 0.999706i \(-0.492282\pi\)
0.0242454 + 0.999706i \(0.492282\pi\)
\(930\) 1.07452e28 0.677058
\(931\) −7.49004e27 −0.467156
\(932\) 8.24800e27 0.509211
\(933\) −1.55746e28 −0.951791
\(934\) 3.36849e27 0.203770
\(935\) −8.56840e27 −0.513085
\(936\) −7.55012e27 −0.447541
\(937\) −1.46040e28 −0.856931 −0.428465 0.903558i \(-0.640946\pi\)
−0.428465 + 0.903558i \(0.640946\pi\)
\(938\) 2.70505e27 0.157126
\(939\) −3.29191e28 −1.89289
\(940\) −3.30128e27 −0.187918
\(941\) −1.23989e28 −0.698687 −0.349344 0.936995i \(-0.613595\pi\)
−0.349344 + 0.936995i \(0.613595\pi\)
\(942\) −8.04642e26 −0.0448868
\(943\) 2.42371e28 1.33850
\(944\) −2.94448e26 −0.0160981
\(945\) 7.71989e26 0.0417839
\(946\) 1.46412e28 0.784530
\(947\) −1.35611e28 −0.719400 −0.359700 0.933068i \(-0.617121\pi\)
−0.359700 + 0.933068i \(0.617121\pi\)
\(948\) 7.03565e27 0.369509
\(949\) 3.53916e28 1.84022
\(950\) 1.37290e27 0.0706749
\(951\) 5.24345e28 2.67240
\(952\) −1.80225e27 −0.0909419
\(953\) −1.00835e28 −0.503764 −0.251882 0.967758i \(-0.581050\pi\)
−0.251882 + 0.967758i \(0.581050\pi\)
\(954\) 1.39500e27 0.0690026
\(955\) −1.29625e28 −0.634830
\(956\) −3.27095e27 −0.158608
\(957\) 6.48183e28 3.11196
\(958\) 9.43148e27 0.448340
\(959\) 3.30494e27 0.155556
\(960\) 1.79694e27 0.0837448
\(961\) 2.25941e28 1.04261
\(962\) 1.99452e28 0.911331
\(963\) −1.26089e28 −0.570461
\(964\) −3.04779e26 −0.0136538
\(965\) −5.18605e27 −0.230053
\(966\) −4.23001e27 −0.185806
\(967\) 1.24965e28 0.543547 0.271774 0.962361i \(-0.412390\pi\)
0.271774 + 0.962361i \(0.412390\pi\)
\(968\) 2.44135e27 0.105151
\(969\) −1.76805e28 −0.754082
\(970\) −9.73741e26 −0.0411256
\(971\) −9.87663e27 −0.413072 −0.206536 0.978439i \(-0.566219\pi\)
−0.206536 + 0.978439i \(0.566219\pi\)
\(972\) −1.70069e28 −0.704360
\(973\) 6.06627e27 0.248800
\(974\) −3.00006e28 −1.21848
\(975\) −7.57856e27 −0.304820
\(976\) −8.23829e27 −0.328144
\(977\) −2.84735e27 −0.112316 −0.0561581 0.998422i \(-0.517885\pi\)
−0.0561581 + 0.998422i \(0.517885\pi\)
\(978\) 1.50701e28 0.588705
\(979\) −3.46393e27 −0.134009
\(980\) −5.45562e27 −0.209024
\(981\) 6.03194e28 2.28877
\(982\) 6.01941e26 0.0226201
\(983\) −2.10459e28 −0.783265 −0.391632 0.920122i \(-0.628090\pi\)
−0.391632 + 0.920122i \(0.628090\pi\)
\(984\) −2.80023e28 −1.03215
\(985\) 1.99913e28 0.729789
\(986\) −3.59264e28 −1.29892
\(987\) −8.97846e27 −0.321506
\(988\) −7.16789e27 −0.254214
\(989\) −1.90466e28 −0.669041
\(990\) 1.28817e28 0.448164
\(991\) 5.03784e28 1.73598 0.867989 0.496583i \(-0.165412\pi\)
0.867989 + 0.496583i \(0.165412\pi\)
\(992\) 7.40251e27 0.252649
\(993\) 4.28943e28 1.45004
\(994\) 5.15062e27 0.172460
\(995\) −4.94301e27 −0.163935
\(996\) −3.71230e28 −1.21949
\(997\) −1.44919e28 −0.471544 −0.235772 0.971808i \(-0.575762\pi\)
−0.235772 + 0.971808i \(0.575762\pi\)
\(998\) −8.05144e27 −0.259498
\(999\) −1.45179e28 −0.463480
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.1.2 2
3.2 odd 2 90.20.a.g.1.1 2
4.3 odd 2 80.20.a.d.1.1 2
5.2 odd 4 50.20.b.f.49.3 4
5.3 odd 4 50.20.b.f.49.2 4
5.4 even 2 50.20.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.20.a.d.1.2 2 1.1 even 1 trivial
50.20.a.f.1.1 2 5.4 even 2
50.20.b.f.49.2 4 5.3 odd 4
50.20.b.f.49.3 4 5.2 odd 4
80.20.a.d.1.1 2 4.3 odd 2
90.20.a.g.1.1 2 3.2 odd 2