Properties

Label 50.20.a.f.1.1
Level $50$
Weight $20$
Character 50.1
Self dual yes
Analytic conductor $114.408$
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] = [50,20,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, 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("50.1");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(114.408348278\)
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: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1711.01\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-512.000 q^{2} -51072.2 q^{3} +262144. q^{4} +2.61490e7 q^{6} -2.72650e7 q^{7} -1.34218e8 q^{8} +1.44611e9 q^{9} +8.90778e9 q^{11} -1.33883e10 q^{12} +3.88993e10 q^{13} +1.39597e10 q^{14} +6.87195e10 q^{16} +4.92493e11 q^{17} -7.40410e11 q^{18} +7.02926e11 q^{19} +1.39248e12 q^{21} -4.56078e12 q^{22} +5.93310e12 q^{23} +6.85480e12 q^{24} -1.99164e13 q^{26} -1.44969e13 q^{27} -7.14735e12 q^{28} +1.42477e14 q^{29} +2.10392e14 q^{31} -3.51844e13 q^{32} -4.54940e14 q^{33} -2.52156e14 q^{34} +3.79090e14 q^{36} +1.00145e15 q^{37} -3.59898e14 q^{38} -1.98667e15 q^{39} -4.08507e15 q^{41} -7.12952e14 q^{42} -3.21023e15 q^{43} +2.33512e15 q^{44} -3.03774e15 q^{46} +6.44781e15 q^{47} -3.50966e15 q^{48} -1.06555e16 q^{49} -2.51527e16 q^{51} +1.01972e16 q^{52} -1.88410e15 q^{53} +7.42243e15 q^{54} +3.65944e15 q^{56} -3.59000e16 q^{57} -7.29480e16 q^{58} -4.28479e15 q^{59} -1.19883e17 q^{61} -1.07721e17 q^{62} -3.94282e16 q^{63} +1.80144e16 q^{64} +2.32929e17 q^{66} -1.93776e17 q^{67} +1.29104e17 q^{68} -3.03017e17 q^{69} +3.68964e17 q^{71} -1.94094e17 q^{72} +9.09825e17 q^{73} -5.12740e17 q^{74} +1.84268e17 q^{76} -2.42870e17 q^{77} +1.01718e18 q^{78} +5.25508e17 q^{79} -9.40371e17 q^{81} +2.09156e18 q^{82} +2.77280e18 q^{83} +3.65031e17 q^{84} +1.64364e18 q^{86} -7.27660e18 q^{87} -1.19558e18 q^{88} -3.88866e17 q^{89} -1.06059e18 q^{91} +1.55533e18 q^{92} -1.07452e19 q^{93} -3.30128e18 q^{94} +1.79694e18 q^{96} +9.73741e17 q^{97} +5.45562e18 q^{98} +1.28817e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1024 q^{2} - 33724 q^{3} + 524288 q^{4} + 17266688 q^{6} - 83061292 q^{7} - 268435456 q^{8} + 584812954 q^{9} + 3549480144 q^{11} - 8840544256 q^{12} - 30250225564 q^{13} + 42527381504 q^{14} + 137438953472 q^{16}+ \cdots + 17\!\cdots\!88 q^{99}+O(q^{100}) \) 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 \(-0.769483\pi\)
−0.749037 + 0.662529i \(0.769483\pi\)
\(4\) 262144. 0.500000
\(5\) 0 0
\(6\) 2.61490e7 1.05930
\(7\) −2.72650e7 −0.255372 −0.127686 0.991815i \(-0.540755\pi\)
−0.127686 + 0.991815i \(0.540755\pi\)
\(8\) −1.34218e8 −0.353553
\(9\) 1.44611e9 1.24422
\(10\) 0 0
\(11\) 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 \(-0.330131\pi\)
0.508686 + 0.860952i \(0.330131\pi\)
\(14\) 1.39597e10 0.180575
\(15\) 0 0
\(16\) 6.87195e10 0.250000
\(17\) 4.92493e11 1.00725 0.503623 0.863924i \(-0.332000\pi\)
0.503623 + 0.863924i \(0.332000\pi\)
\(18\) −7.40410e11 −0.879799
\(19\) 7.02926e11 0.499747 0.249873 0.968278i \(-0.419611\pi\)
0.249873 + 0.968278i \(0.419611\pi\)
\(20\) 0 0
\(21\) 1.39248e12 0.382566
\(22\) −4.56078e12 −0.805423
\(23\) 5.93310e12 0.686858 0.343429 0.939179i \(-0.388412\pi\)
0.343429 + 0.939179i \(0.388412\pi\)
\(24\) 6.85480e12 0.529649
\(25\) 0 0
\(26\) −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 \(0.134639\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) 3.79090e14 0.622112
\(37\) 1.00145e15 1.26681 0.633404 0.773821i \(-0.281657\pi\)
0.633404 + 0.773821i \(0.281657\pi\)
\(38\) −3.59898e14 −0.353374
\(39\) −1.98667e15 −1.52410
\(40\) 0 0
\(41\) −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 \(-0.661920\pi\)
−0.487030 + 0.873385i \(0.661920\pi\)
\(44\) 2.33512e15 0.569520
\(45\) 0 0
\(46\) −3.03774e15 −0.485682
\(47\) 6.44781e15 0.840393 0.420196 0.907433i \(-0.361961\pi\)
0.420196 + 0.907433i \(0.361961\pi\)
\(48\) −3.50966e15 −0.374518
\(49\) −1.06555e16 −0.934785
\(50\) 0 0
\(51\) −2.51527e16 −1.50893
\(52\) 1.01972e16 0.508686
\(53\) −1.88410e15 −0.0784300 −0.0392150 0.999231i \(-0.512486\pi\)
−0.0392150 + 0.999231i \(0.512486\pi\)
\(54\) 7.42243e15 0.258705
\(55\) 0 0
\(56\) 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 \(0.510250\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) 2.32929e17 1.20658
\(67\) −1.93776e17 −0.870141 −0.435071 0.900396i \(-0.643277\pi\)
−0.435071 + 0.900396i \(0.643277\pi\)
\(68\) 1.29104e17 0.503623
\(69\) −3.03017e17 −1.02896
\(70\) 0 0
\(71\) 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 \(-0.140319\pi\)
0.904400 + 0.426685i \(0.140319\pi\)
\(74\) −5.12740e17 −0.895769
\(75\) 0 0
\(76\) 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 \(0.420668\pi\)
\(80\) 0 0
\(81\) −9.40371e17 −0.696132
\(82\) 2.09156e18 1.37796
\(83\) 2.77280e18 1.62808 0.814042 0.580807i \(-0.197263\pi\)
0.814042 + 0.580807i \(0.197263\pi\)
\(84\) 3.65031e17 0.191283
\(85\) 0 0
\(86\) 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 \(0.518736\pi\)
\(90\) 0 0
\(91\) −1.06059e18 −0.259809
\(92\) 1.55533e18 0.343429
\(93\) −1.07452e19 −2.14105
\(94\) −3.30128e18 −0.594247
\(95\) 0 0
\(96\) 1.79694e18 0.264824
\(97\) 9.73741e17 0.130050 0.0650252 0.997884i \(-0.479287\pi\)
0.0650252 + 0.997884i \(0.479287\pi\)
\(98\) 5.45562e18 0.660993
\(99\) 1.28817e19 1.41722
\(100\) 0 0
\(101\) −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 \(-0.553227\pi\)
−0.166440 + 0.986052i \(0.553227\pi\)
\(104\) −5.22098e18 −0.359695
\(105\) 0 0
\(106\) 9.64657e17 0.0554584
\(107\) 8.71914e18 0.458488 0.229244 0.973369i \(-0.426375\pi\)
0.229244 + 0.973369i \(0.426375\pi\)
\(108\) −3.80028e18 −0.182932
\(109\) 4.17114e19 1.83951 0.919757 0.392487i \(-0.128385\pi\)
0.919757 + 0.392487i \(0.128385\pi\)
\(110\) 0 0
\(111\) −5.11461e19 −1.89777
\(112\) −1.87363e18 −0.0638431
\(113\) 8.70592e18 0.272627 0.136314 0.990666i \(-0.456475\pi\)
0.136314 + 0.990666i \(0.456475\pi\)
\(114\) 1.83808e19 0.529381
\(115\) 0 0
\(116\) 3.73494e19 0.911869
\(117\) 5.62528e19 1.26584
\(118\) 2.19381e18 0.0455323
\(119\) −1.34278e19 −0.257222
\(120\) 0 0
\(121\) 1.81895e19 0.297413
\(122\) 6.13801e19 0.928130
\(123\) 2.08634e20 2.91935
\(124\) 5.51530e19 0.714600
\(125\) 0 0
\(126\) 2.01873e19 0.224676
\(127\) −6.07257e19 −0.626956 −0.313478 0.949595i \(-0.601494\pi\)
−0.313478 + 0.949595i \(0.601494\pi\)
\(128\) −9.22337e18 −0.0883883
\(129\) 1.63954e20 1.45921
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) −6.61013e19 −0.356115
\(137\) −1.21216e20 −0.609135 −0.304567 0.952491i \(-0.598512\pi\)
−0.304567 + 0.952491i \(0.598512\pi\)
\(138\) 1.55144e20 0.727587
\(139\) 2.22493e20 0.974263 0.487131 0.873329i \(-0.338043\pi\)
0.487131 + 0.873329i \(0.338043\pi\)
\(140\) 0 0
\(141\) −3.29304e20 −1.25897
\(142\) −1.88910e20 −0.675328
\(143\) 3.46506e20 1.15883
\(144\) 9.93761e19 0.311056
\(145\) 0 0
\(146\) −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 \(0.371592\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) −5.20795e20 −0.762049
\(157\) 3.07714e19 0.0423741 0.0211871 0.999776i \(-0.493255\pi\)
0.0211871 + 0.999776i \(0.493255\pi\)
\(158\) −2.69060e20 −0.348825
\(159\) 9.62250e19 0.117494
\(160\) 0 0
\(161\) −1.61766e20 −0.175404
\(162\) 4.81470e20 0.492239
\(163\) −5.76318e20 −0.555751 −0.277875 0.960617i \(-0.589630\pi\)
−0.277875 + 0.960617i \(0.589630\pi\)
\(164\) −1.07088e21 −0.974368
\(165\) 0 0
\(166\) −1.41967e21 −1.15123
\(167\) −6.89550e20 −0.528153 −0.264076 0.964502i \(-0.585067\pi\)
−0.264076 + 0.964502i \(0.585067\pi\)
\(168\) −1.86896e20 −0.135258
\(169\) 5.12350e19 0.0350464
\(170\) 0 0
\(171\) 1.01651e21 0.621797
\(172\) −8.41542e20 −0.487030
\(173\) 4.14244e19 0.0226892 0.0113446 0.999936i \(-0.496389\pi\)
0.0113446 + 0.999936i \(0.496389\pi\)
\(174\) 3.72562e21 1.93188
\(175\) 0 0
\(176\) 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 \(0.242584\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 5.50154e21 1.51395
\(187\) 4.38702e21 1.14729
\(188\) 1.69025e21 0.420196
\(189\) 3.95258e20 0.0934316
\(190\) 0 0
\(191\) −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 \(-0.417198\pi\)
0.257207 + 0.966356i \(0.417198\pi\)
\(194\) −4.98555e20 −0.0919596
\(195\) 0 0
\(196\) −2.79328e21 −0.467393
\(197\) −1.02356e22 −1.63186 −0.815929 0.578153i \(-0.803774\pi\)
−0.815929 + 0.578153i \(0.803774\pi\)
\(198\) −6.59541e21 −1.00213
\(199\) −2.53082e21 −0.366570 −0.183285 0.983060i \(-0.558673\pi\)
−0.183285 + 0.983060i \(0.558673\pi\)
\(200\) 0 0
\(201\) 9.89659e21 1.30354
\(202\) 8.57400e21 1.07732
\(203\) −3.88462e21 −0.465732
\(204\) −6.59363e21 −0.754463
\(205\) 0 0
\(206\) 2.25689e21 0.235382
\(207\) 8.57993e21 0.854605
\(208\) 2.67314e21 0.254343
\(209\) 6.26151e21 0.569232
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 1.94574e21 0.129353
\(217\) −5.73633e21 −0.364978
\(218\) −2.13562e22 −1.30073
\(219\) −4.64668e22 −2.70972
\(220\) 0 0
\(221\) 1.91576e22 1.02474
\(222\) 2.61868e22 1.34193
\(223\) −1.40599e22 −0.690375 −0.345187 0.938534i \(-0.612185\pi\)
−0.345187 + 0.938534i \(0.612185\pi\)
\(224\) 9.59301e20 0.0451439
\(225\) 0 0
\(226\) −4.45743e21 −0.192777
\(227\) −3.41441e22 −1.41602 −0.708011 0.706201i \(-0.750408\pi\)
−0.708011 + 0.706201i \(0.750408\pi\)
\(228\) −9.41097e21 −0.374329
\(229\) 2.26978e22 0.866057 0.433028 0.901380i \(-0.357445\pi\)
0.433028 + 0.901380i \(0.357445\pi\)
\(230\) 0 0
\(231\) 1.24039e22 0.435758
\(232\) −1.91229e22 −0.644789
\(233\) −3.14636e22 −1.01842 −0.509211 0.860642i \(-0.670063\pi\)
−0.509211 + 0.860642i \(0.670063\pi\)
\(234\) −2.88014e22 −0.895083
\(235\) 0 0
\(236\) −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 \(0.550701\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) −1.06820e23 −2.06429
\(247\) 2.73433e22 0.508429
\(248\) −2.82383e22 −0.505299
\(249\) −1.41613e23 −2.43899
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 4.72237e21 0.0625000
\(257\) 2.11998e22 0.270375 0.135187 0.990820i \(-0.456836\pi\)
0.135187 + 0.990820i \(0.456836\pi\)
\(258\) −8.39442e22 −1.03182
\(259\) −2.73044e22 −0.323508
\(260\) 0 0
\(261\) 2.06037e23 2.26914
\(262\) −9.97536e22 −1.05942
\(263\) 1.06654e23 1.09244 0.546222 0.837641i \(-0.316066\pi\)
0.546222 + 0.837641i \(0.316066\pi\)
\(264\) 6.10611e22 0.603291
\(265\) 0 0
\(266\) 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 \(0.374723\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) −7.94340e22 −0.514482
\(277\) 7.04507e21 0.0440887 0.0220444 0.999757i \(-0.492982\pi\)
0.0220444 + 0.999757i \(0.492982\pi\)
\(278\) −1.13916e23 −0.688908
\(279\) 3.04251e23 1.77824
\(280\) 0 0
\(281\) −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 \(-0.317143\pi\)
0.543383 + 0.839485i \(0.317143\pi\)
\(284\) 9.67218e22 0.477529
\(285\) 0 0
\(286\) −1.77411e23 −0.819415
\(287\) 1.11379e23 0.497653
\(288\) −5.08806e22 −0.219950
\(289\) 3.47675e21 0.0145427
\(290\) 0 0
\(291\) −4.97311e22 −0.194825
\(292\) 2.38505e23 0.904400
\(293\) −5.10289e23 −1.87315 −0.936577 0.350462i \(-0.886025\pi\)
−0.936577 + 0.350462i \(0.886025\pi\)
\(294\) −2.78631e23 −0.990216
\(295\) 0 0
\(296\) −1.34412e23 −0.447885
\(297\) −1.29135e23 −0.416734
\(298\) −1.77610e23 −0.555152
\(299\) 2.30793e23 0.698790
\(300\) 0 0
\(301\) 8.75268e22 0.248748
\(302\) 5.38650e21 0.0148334
\(303\) 8.55260e23 2.28241
\(304\) 4.83047e22 0.124937
\(305\) 0 0
\(306\) −3.64646e23 −0.886173
\(307\) 4.42100e23 1.04161 0.520806 0.853675i \(-0.325631\pi\)
0.520806 + 0.853675i \(0.325631\pi\)
\(308\) −6.36670e22 −0.145440
\(309\) 2.25126e23 0.498678
\(310\) 0 0
\(311\) −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 \(-0.282327\pi\)
0.631774 + 0.775153i \(0.282327\pi\)
\(314\) −1.57550e22 −0.0299630
\(315\) 0 0
\(316\) 1.37759e23 0.246656
\(317\) −1.02667e24 −1.78389 −0.891947 0.452140i \(-0.850661\pi\)
−0.891947 + 0.452140i \(0.850661\pi\)
\(318\) −4.92672e22 −0.0830807
\(319\) 1.26915e24 2.07731
\(320\) 0 0
\(321\) −4.45306e23 −0.686849
\(322\) 8.28240e22 0.124030
\(323\) 3.46186e23 0.503368
\(324\) −2.46513e23 −0.348066
\(325\) 0 0
\(326\) 2.95075e23 0.392975
\(327\) −2.13030e24 −2.75573
\(328\) 5.48289e23 0.688982
\(329\) −1.75799e23 −0.214613
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) 9.56907e22 0.0956416
\(337\) 7.88161e23 0.765827 0.382913 0.923784i \(-0.374921\pi\)
0.382913 + 0.923784i \(0.374921\pi\)
\(338\) −2.62323e22 −0.0247815
\(339\) −4.44631e23 −0.408416
\(340\) 0 0
\(341\) 1.87413e24 1.62792
\(342\) −5.20453e23 −0.439677
\(343\) 6.01313e23 0.494090
\(344\) 4.30870e23 0.344382
\(345\) 0 0
\(346\) −2.12093e22 −0.0160437
\(347\) 3.27984e23 0.241392 0.120696 0.992690i \(-0.461487\pi\)
0.120696 + 0.992690i \(0.461487\pi\)
\(348\) −1.90752e24 −1.36605
\(349\) −1.57618e24 −1.09841 −0.549204 0.835689i \(-0.685069\pi\)
−0.549204 + 0.835689i \(0.685069\pi\)
\(350\) 0 0
\(351\) −5.63920e23 −0.372220
\(352\) −3.13415e23 −0.201356
\(353\) 1.44980e23 0.0906667 0.0453333 0.998972i \(-0.485565\pi\)
0.0453333 + 0.998972i \(0.485565\pi\)
\(354\) −1.12043e23 −0.0682108
\(355\) 0 0
\(356\) −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 \(0.434699\pi\)
\(360\) 0 0
\(361\) −1.48432e24 −0.750253
\(362\) −5.38786e23 −0.265268
\(363\) −9.28978e23 −0.445546
\(364\) −2.78027e23 −0.129904
\(365\) 0 0
\(366\) −3.13482e24 −1.39041
\(367\) 2.08854e24 0.902639 0.451320 0.892362i \(-0.350953\pi\)
0.451320 + 0.892362i \(0.350953\pi\)
\(368\) 4.07719e23 0.171714
\(369\) −5.90747e24 −2.42466
\(370\) 0 0
\(371\) 5.13698e22 0.0200288
\(372\) −2.81679e24 −1.07052
\(373\) −2.90964e24 −1.07797 −0.538984 0.842316i \(-0.681192\pi\)
−0.538984 + 0.842316i \(0.681192\pi\)
\(374\) −2.24615e24 −0.811258
\(375\) 0 0
\(376\) −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 \(0.240662\pi\)
\(380\) 0 0
\(381\) 3.10140e24 0.939226
\(382\) 3.39805e24 1.00375
\(383\) −4.74338e24 −1.36678 −0.683392 0.730052i \(-0.739496\pi\)
−0.683392 + 0.730052i \(0.739496\pi\)
\(384\) 4.71058e23 0.132412
\(385\) 0 0
\(386\) −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 \(0.563846\pi\)
\(390\) 0 0
\(391\) 2.92201e24 0.691834
\(392\) 1.43016e24 0.330496
\(393\) −9.95047e24 −2.24447
\(394\) 5.24060e24 1.15390
\(395\) 0 0
\(396\) 3.37685e24 0.708610
\(397\) −5.17477e24 −1.06018 −0.530092 0.847940i \(-0.677843\pi\)
−0.530092 + 0.847940i \(0.677843\pi\)
\(398\) 1.29578e24 0.259204
\(399\) 9.78812e23 0.191186
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 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 \(0.774728\pi\)
\(410\) 0 0
\(411\) 6.19076e24 0.912529
\(412\) −1.15553e24 −0.166440
\(413\) 1.16825e23 0.0164440
\(414\) −4.39292e24 −0.604297
\(415\) 0 0
\(416\) −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 \(0.472666\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) 9.64804e24 1.01169
\(427\) 3.26861e24 0.335195
\(428\) 2.28567e24 0.229244
\(429\) −1.76969e25 −1.73601
\(430\) 0 0
\(431\) −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 \(-0.846618\pi\)
−0.886133 + 0.463432i \(0.846618\pi\)
\(434\) 2.93700e24 0.258078
\(435\) 0 0
\(436\) 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 \(0.114643\pi\)
\(440\) 0 0
\(441\) −1.54091e25 −1.16308
\(442\) −9.80870e24 −0.724603
\(443\) 1.08663e25 0.785679 0.392840 0.919607i \(-0.371493\pi\)
0.392840 + 0.919607i \(0.371493\pi\)
\(444\) −1.34076e25 −0.948886
\(445\) 0 0
\(446\) 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 \(0.325248\pi\)
\(450\) 0 0
\(451\) −3.63889e25 −2.21969
\(452\) 2.28220e24 0.136314
\(453\) 5.37306e23 0.0314260
\(454\) 1.74818e25 1.00128
\(455\) 0 0
\(456\) 4.81841e24 0.264690
\(457\) 1.90127e25 1.02291 0.511457 0.859309i \(-0.329106\pi\)
0.511457 + 0.859309i \(0.329106\pi\)
\(458\) −1.16213e25 −0.612395
\(459\) −7.13963e24 −0.368515
\(460\) 0 0
\(461\) −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 \(-0.671591\pi\)
−0.513337 + 0.858187i \(0.671591\pi\)
\(464\) 9.79092e24 0.455935
\(465\) 0 0
\(466\) 1.61094e25 0.720133
\(467\) −6.57907e24 −0.288174 −0.144087 0.989565i \(-0.546024\pi\)
−0.144087 + 0.989565i \(0.546024\pi\)
\(468\) 1.47463e25 0.632919
\(469\) 5.28330e24 0.222210
\(470\) 0 0
\(471\) −1.57157e24 −0.0634796
\(472\) 5.75094e23 0.0227662
\(473\) −2.85960e25 −1.10949
\(474\) 1.37415e25 0.522565
\(475\) 0 0
\(476\) −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 \(0.397316\pi\)
\(480\) 0 0
\(481\) 3.89555e25 1.28882
\(482\) 5.95271e23 0.0193094
\(483\) 8.26174e24 0.262769
\(484\) 4.76826e24 0.148706
\(485\) 0 0
\(486\) −3.32165e25 −0.996116
\(487\) 5.85949e25 1.72320 0.861598 0.507591i \(-0.169464\pi\)
0.861598 + 0.507591i \(0.169464\pi\)
\(488\) 1.60904e25 0.464065
\(489\) 2.94339e25 0.832555
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) 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 \(0.558740\pi\)
\(500\) 0 0
\(501\) 3.52169e25 0.791211
\(502\) −5.61434e25 −1.23769
\(503\) 2.64249e24 0.0571632 0.0285816 0.999591i \(-0.490901\pi\)
0.0285816 + 0.999591i \(0.490901\pi\)
\(504\) 5.29197e24 0.112338
\(505\) 0 0
\(506\) −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 \(0.221500\pi\)
\(510\) 0 0
\(511\) −2.48064e25 −0.461918
\(512\) −2.41785e24 −0.0441942
\(513\) −1.01903e25 −0.182840
\(514\) −1.08543e25 −0.191184
\(515\) 0 0
\(516\) 4.29794e25 0.729606
\(517\) 5.74356e25 0.957241
\(518\) 1.39798e25 0.228755
\(519\) −2.11564e24 −0.0339900
\(520\) 0 0
\(521\) 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 \(-0.339523\pi\)
0.483066 + 0.875584i \(0.339523\pi\)
\(524\) 5.10739e25 0.749120
\(525\) 0 0
\(526\) −5.46070e25 −0.772474
\(527\) 1.03617e26 1.43955
\(528\) −3.12633e25 −0.426591
\(529\) −3.94138e25 −0.528226
\(530\) 0 0
\(531\) −6.19628e24 −0.0801186
\(532\) −5.02405e24 −0.0638108
\(533\) −1.58906e26 −1.98259
\(534\) −1.01685e25 −0.124627
\(535\) 0 0
\(536\) 2.60082e25 0.307641
\(537\) −1.86505e26 −2.16738
\(538\) −4.75004e25 −0.542333
\(539\) −9.49170e25 −1.06476
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) −2.77333e25 −0.275215
\(547\) −5.22682e25 −0.509750 −0.254875 0.966974i \(-0.582034\pi\)
−0.254875 + 0.966974i \(0.582034\pi\)
\(548\) −3.17760e25 −0.304567
\(549\) −1.73364e26 −1.63314
\(550\) 0 0
\(551\) 1.00150e26 0.911408
\(552\) 4.06702e25 0.363793
\(553\) −1.43280e25 −0.125978
\(554\) −3.60708e24 −0.0311754
\(555\) 0 0
\(556\) 5.83252e25 0.487131
\(557\) −5.10671e25 −0.419292 −0.209646 0.977777i \(-0.567231\pi\)
−0.209646 + 0.977777i \(0.567231\pi\)
\(558\) −1.55776e26 −1.25741
\(559\) −1.24876e26 −0.990981
\(560\) 0 0
\(561\) −2.24055e26 −1.71873
\(562\) 5.13328e24 0.0387169
\(563\) −1.95962e26 −1.45325 −0.726627 0.687032i \(-0.758913\pi\)
−0.726627 + 0.687032i \(0.758913\pi\)
\(564\) −8.63250e25 −0.629485
\(565\) 0 0
\(566\) −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 \(0.521863\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 2.60509e25 0.155528
\(577\) 1.24271e26 0.729792 0.364896 0.931048i \(-0.381105\pi\)
0.364896 + 0.931048i \(0.381105\pi\)
\(578\) −1.78010e24 −0.0102832
\(579\) −1.35610e26 −0.770629
\(580\) 0 0
\(581\) −7.56003e25 −0.415767
\(582\) 2.54623e25 0.137762
\(583\) −1.67831e25 −0.0893349
\(584\) −1.22115e26 −0.639508
\(585\) 0 0
\(586\) 2.61268e26 1.32452
\(587\) −3.67268e26 −1.83198 −0.915991 0.401199i \(-0.868593\pi\)
−0.915991 + 0.401199i \(0.868593\pi\)
\(588\) 1.42659e26 0.700188
\(589\) 1.47890e26 0.714238
\(590\) 0 0
\(591\) 5.22753e26 2.44464
\(592\) 6.88188e25 0.316702
\(593\) 1.40519e26 0.636378 0.318189 0.948027i \(-0.396925\pi\)
0.318189 + 0.948027i \(0.396925\pi\)
\(594\) 6.61173e25 0.294676
\(595\) 0 0
\(596\) 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 \(0.360934\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) −4.37893e26 −1.61391
\(607\) −2.30237e26 −0.835376 −0.417688 0.908591i \(-0.637159\pi\)
−0.417688 + 0.908591i \(0.637159\pi\)
\(608\) −2.47320e25 −0.0883436
\(609\) 1.98396e26 0.697701
\(610\) 0 0
\(611\) 2.50815e26 0.854992
\(612\) 1.86699e26 0.626619
\(613\) 7.21990e25 0.238592 0.119296 0.992859i \(-0.461936\pi\)
0.119296 + 0.992859i \(0.461936\pi\)
\(614\) −2.26355e26 −0.736531
\(615\) 0 0
\(616\) 3.25975e25 0.102841
\(617\) 1.44998e26 0.450458 0.225229 0.974306i \(-0.427687\pi\)
0.225229 + 0.974306i \(0.427687\pi\)
\(618\) −1.15265e26 −0.352619
\(619\) −2.13817e26 −0.644141 −0.322071 0.946716i \(-0.604379\pi\)
−0.322071 + 0.946716i \(0.604379\pi\)
\(620\) 0 0
\(621\) −8.60116e25 −0.251297
\(622\) 1.56136e26 0.449256
\(623\) 1.06024e25 0.0300448
\(624\) −1.36523e26 −0.381025
\(625\) 0 0
\(626\) −3.30014e26 −0.893463
\(627\) −3.19789e26 −0.852751
\(628\) 8.06654e24 0.0211871
\(629\) 4.93205e26 1.27599
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) 2.52248e25 0.0587469
\(637\) −4.14492e26 −0.951024
\(638\) −6.49805e26 −1.46888
\(639\) 5.33564e26 1.18831
\(640\) 0 0
\(641\) −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 \(-0.177380\pi\)
0.848710 + 0.528858i \(0.177380\pi\)
\(644\) −4.24059e25 −0.0877022
\(645\) 0 0
\(646\) −1.77247e26 −0.355935
\(647\) 7.53176e26 1.49041 0.745205 0.666835i \(-0.232351\pi\)
0.745205 + 0.666835i \(0.232351\pi\)
\(648\) 1.26214e26 0.246120
\(649\) −3.81679e25 −0.0733456
\(650\) 0 0
\(651\) 2.92967e26 0.546764
\(652\) −1.51078e26 −0.277875
\(653\) −1.75162e25 −0.0317515 −0.0158758 0.999874i \(-0.505054\pi\)
−0.0158758 + 0.999874i \(0.505054\pi\)
\(654\) 1.09071e27 1.94859
\(655\) 0 0
\(656\) −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 \(0.871523\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) −7.41480e26 −1.11454
\(667\) 8.45327e26 1.25265
\(668\) −1.80761e26 −0.264076
\(669\) 7.18069e26 1.03423
\(670\) 0 0
\(671\) −1.06789e27 −1.49508
\(672\) −4.89937e25 −0.0676288
\(673\) 1.19576e27 1.62742 0.813710 0.581270i \(-0.197444\pi\)
0.813710 + 0.581270i \(0.197444\pi\)
\(674\) −4.03538e26 −0.541521
\(675\) 0 0
\(676\) 1.34309e25 0.0175232
\(677\) −4.22916e26 −0.544079 −0.272040 0.962286i \(-0.587698\pi\)
−0.272040 + 0.962286i \(0.587698\pi\)
\(678\) 2.27651e26 0.288793
\(679\) −2.65490e25 −0.0332113
\(680\) 0 0
\(681\) 1.74382e27 2.12131
\(682\) −9.59552e26 −1.15111
\(683\) −7.22585e24 −0.00854854 −0.00427427 0.999991i \(-0.501361\pi\)
−0.00427427 + 0.999991i \(0.501361\pi\)
\(684\) 2.66472e26 0.310898
\(685\) 0 0
\(686\) −3.07872e26 −0.349375
\(687\) −1.15923e27 −1.29742
\(688\) −2.20605e26 −0.243515
\(689\) −7.32900e25 −0.0797925
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 9.76648e26 0.965941
\(697\) −2.01187e27 −1.96285
\(698\) 8.07003e26 0.776691
\(699\) 1.60692e27 1.52567
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 \(0.533115\pi\)
\(710\) 0 0
\(711\) 7.59944e26 0.613791
\(712\) 5.21927e25 0.0415958
\(713\) 1.24828e27 0.981657
\(714\) −3.51124e26 −0.272475
\(715\) 0 0
\(716\) 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 \(0.558053\pi\)
\(720\) 0 0
\(721\) 1.20184e26 0.0850083
\(722\) 7.59969e26 0.530509
\(723\) 5.93786e25 0.0409087
\(724\) 2.75858e26 0.187573
\(725\) 0 0
\(726\) 4.75637e26 0.315048
\(727\) −1.17305e27 −0.766900 −0.383450 0.923562i \(-0.625264\pi\)
−0.383450 + 0.923562i \(0.625264\pi\)
\(728\) 1.42350e26 0.0918562
\(729\) −2.22041e27 −1.41423
\(730\) 0 0
\(731\) −1.58101e27 −0.981117
\(732\) 1.60503e27 0.983166
\(733\) −7.29500e26 −0.441101 −0.220550 0.975376i \(-0.570785\pi\)
−0.220550 + 0.975376i \(0.570785\pi\)
\(734\) −1.06933e27 −0.638262
\(735\) 0 0
\(736\) −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 \(0.264296\pi\)
\(740\) 0 0
\(741\) −1.39648e27 −0.761663
\(742\) −2.63013e25 −0.0141625
\(743\) −1.98675e27 −1.05621 −0.528105 0.849179i \(-0.677097\pi\)
−0.528105 + 0.849179i \(0.677097\pi\)
\(744\) 1.44220e27 0.756974
\(745\) 0 0
\(746\) 1.48974e27 0.762238
\(747\) 4.00978e27 2.02570
\(748\) 1.15003e27 0.573646
\(749\) −2.37727e26 −0.117085
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 1.03615e26 0.0467158
\(757\) −8.34040e25 −0.0371344 −0.0185672 0.999828i \(-0.505910\pi\)
−0.0185672 + 0.999828i \(0.505910\pi\)
\(758\) −2.34009e27 −1.02890
\(759\) −2.69920e27 −1.17203
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) 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 \(0.440528\pi\)
\(770\) 0 0
\(771\) −1.08272e27 −0.405041
\(772\) 6.96060e26 0.257207
\(773\) −5.16072e27 −1.88367 −0.941836 0.336074i \(-0.890901\pi\)
−0.941836 + 0.336074i \(0.890901\pi\)
\(774\) 2.37688e27 0.856976
\(775\) 0 0
\(776\) −1.30693e26 −0.0459798
\(777\) 1.39450e27 0.484638
\(778\) 8.20709e26 0.281762
\(779\) −2.87150e27 −0.973875
\(780\) 0 0
\(781\) 3.28665e27 1.08785
\(782\) −1.49607e27 −0.489201
\(783\) −2.06547e27 −0.667241
\(784\) −7.32242e26 −0.233696
\(785\) 0 0
\(786\) 5.09464e27 1.58708
\(787\) 4.95523e27 1.52512 0.762559 0.646919i \(-0.223943\pi\)
0.762559 + 0.646919i \(0.223943\pi\)
\(788\) −2.68319e27 −0.815929
\(789\) −5.44708e27 −1.63656
\(790\) 0 0
\(791\) −2.37367e26 −0.0696214
\(792\) −1.72895e27 −0.501063
\(793\) −4.66336e27 −1.33538
\(794\) 2.64948e27 0.749663
\(795\) 0 0
\(796\) −6.63439e26 −0.183285
\(797\) −9.51644e26 −0.259789 −0.129894 0.991528i \(-0.541464\pi\)
−0.129894 + 0.991528i \(0.541464\pi\)
\(798\) −5.01152e26 −0.135189
\(799\) 3.17550e27 0.846481
\(800\) 0 0
\(801\) −5.62344e26 −0.146384
\(802\) −5.37049e26 −0.138152
\(803\) 8.10452e27 2.06030
\(804\) 2.59433e27 0.651768
\(805\) 0 0
\(806\) −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 \(0.474920\pi\)
\(810\) 0 0
\(811\) −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
\(815\) 0 0
\(816\) −1.72848e27 −0.377232
\(817\) −2.25655e27 −0.486783
\(818\) 5.03964e27 1.07459
\(819\) −1.53373e27 −0.323260
\(820\) 0 0
\(821\) −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 \(-0.327579\pi\)
0.515574 + 0.856845i \(0.327579\pi\)
\(824\) 5.91631e26 0.117691
\(825\) 0 0
\(826\) −5.98142e25 −0.0116277
\(827\) 5.00813e27 0.962439 0.481219 0.876600i \(-0.340194\pi\)
0.481219 + 0.876600i \(0.340194\pi\)
\(828\) 2.24918e27 0.427302
\(829\) −3.98409e27 −0.748275 −0.374138 0.927373i \(-0.622061\pi\)
−0.374138 + 0.927373i \(0.622061\pi\)
\(830\) 0 0
\(831\) −3.59807e26 −0.0660481
\(832\) 7.00747e26 0.127172
\(833\) −5.24777e27 −0.941558
\(834\) 5.81797e27 1.03203
\(835\) 0 0
\(836\) 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 \(0.220906\pi\)
\(840\) 0 0
\(841\) 1.41963e28 2.32602
\(842\) −1.51838e27 −0.245988
\(843\) 5.12047e26 0.0820254
\(844\) −3.75474e27 −0.594740
\(845\) 0 0
\(846\) −4.77402e27 −0.739376
\(847\) −4.95936e26 −0.0759509
\(848\) −1.29474e26 −0.0196075
\(849\) −1.08716e28 −1.62806
\(850\) 0 0
\(851\) 5.94167e27 0.870118
\(852\) −4.93980e27 −0.715374
\(853\) 7.79931e27 1.11697 0.558484 0.829516i \(-0.311383\pi\)
0.558484 + 0.829516i \(0.311383\pi\)
\(854\) −1.67353e27 −0.237019
\(855\) 0 0
\(856\) −1.17026e27 −0.162100
\(857\) −5.48738e26 −0.0751705 −0.0375852 0.999293i \(-0.511967\pi\)
−0.0375852 + 0.999293i \(0.511967\pi\)
\(858\) 9.06079e27 1.22754
\(859\) 2.88785e27 0.386936 0.193468 0.981107i \(-0.438026\pi\)
0.193468 + 0.981107i \(0.438026\pi\)
\(860\) 0 0
\(861\) −5.68839e27 −0.745521
\(862\) 8.89559e27 1.15307
\(863\) 1.27182e27 0.163051 0.0815257 0.996671i \(-0.474021\pi\)
0.0815257 + 0.996671i \(0.474021\pi\)
\(864\) 5.10065e26 0.0646763
\(865\) 0 0
\(866\) 1.01026e28 1.25318
\(867\) −1.77566e26 −0.0217860
\(868\) −1.50374e27 −0.182489
\(869\) 4.68111e27 0.561903
\(870\) 0 0
\(871\) −7.53776e27 −0.885258
\(872\) −5.59841e27 −0.650367
\(873\) 1.40814e27 0.161812
\(874\) −2.13531e27 −0.242718
\(875\) 0 0
\(876\) −1.21810e28 −1.35486
\(877\) 7.93254e27 0.872803 0.436401 0.899752i \(-0.356253\pi\)
0.436401 + 0.899752i \(0.356253\pi\)
\(878\) −1.21595e28 −1.32348
\(879\) 2.60616e28 2.80612
\(880\) 0 0
\(881\) −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 \(-0.755122\pi\)
−0.718392 + 0.695639i \(0.755122\pi\)
\(884\) 5.02206e27 0.512372
\(885\) 0 0
\(886\) −5.56353e27 −0.555559
\(887\) 2.98771e27 0.295165 0.147582 0.989050i \(-0.452851\pi\)
0.147582 + 0.989050i \(0.452851\pi\)
\(888\) 6.86471e27 0.670964
\(889\) 1.65569e27 0.160107
\(890\) 0 0
\(891\) −8.37662e27 −0.792922
\(892\) −3.68571e27 −0.345187
\(893\) 4.53233e27 0.419984
\(894\) 9.07092e27 0.831658
\(895\) 0 0
\(896\) 2.51475e26 0.0225719
\(897\) −1.17871e28 −1.04684
\(898\) −8.39793e27 −0.737984
\(899\) 2.99759e28 2.60649
\(900\) 0 0
\(901\) −9.27904e26 −0.0789982
\(902\) 1.86311e28 1.56956
\(903\) −4.47019e27 −0.372642
\(904\) −1.16849e27 −0.0963883
\(905\) 0 0
\(906\) −2.75101e26 −0.0222215
\(907\) 1.01372e28 0.810308 0.405154 0.914248i \(-0.367218\pi\)
0.405154 + 0.914248i \(0.367218\pi\)
\(908\) −8.95068e27 −0.708011
\(909\) −2.42167e28 −1.89565
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) 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 \(0.601550\pi\)
\(920\) 0 0
\(921\) −2.25790e28 −1.56041
\(922\) 1.40243e28 0.959265
\(923\) 1.43524e28 0.971650
\(924\) 3.25162e27 0.217879
\(925\) 0 0
\(926\) 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 \(0.492282\pi\)
\(930\) 0 0
\(931\) −7.49004e27 −0.467156
\(932\) −8.24800e27 −0.509211
\(933\) 1.55746e28 0.951791
\(934\) 3.36849e27 0.203770
\(935\) 0 0
\(936\) −7.55012e27 −0.447541
\(937\) 1.46040e28 0.856931 0.428465 0.903558i \(-0.359054\pi\)
0.428465 + 0.903558i \(0.359054\pi\)
\(938\) −2.70505e27 −0.157126
\(939\) −3.29191e28 −1.89289
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 1.46412e28 0.784530
\(947\) 1.35611e28 0.719400 0.359700 0.933068i \(-0.382879\pi\)
0.359700 + 0.933068i \(0.382879\pi\)
\(948\) −7.03565e27 −0.369509
\(949\) 3.53916e28 1.84022
\(950\) 0 0
\(951\) 5.24345e28 2.67240
\(952\) 1.80225e27 0.0909419
\(953\) 1.00835e28 0.503764 0.251882 0.967758i \(-0.418950\pi\)
0.251882 + 0.967758i \(0.418950\pi\)
\(954\) 1.39500e27 0.0690026
\(955\) 0 0
\(956\) −3.27095e27 −0.158608
\(957\) −6.48183e28 −3.11196
\(958\) −9.43148e27 −0.448340
\(959\) 3.30494e27 0.155556
\(960\) 0 0
\(961\) 2.25941e28 1.04261
\(962\) −1.99452e28 −0.911331
\(963\) 1.26089e28 0.570461
\(964\) −3.04779e26 −0.0136538
\(965\) 0 0
\(966\) −4.23001e27 −0.185806
\(967\) −1.24965e28 −0.543547 −0.271774 0.962361i \(-0.587610\pi\)
−0.271774 + 0.962361i \(0.587610\pi\)
\(968\) −2.44135e27 −0.105151
\(969\) −1.76805e28 −0.754082
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −8.23829e27 −0.328144
\(977\) 2.84735e27 0.112316 0.0561581 0.998422i \(-0.482115\pi\)
0.0561581 + 0.998422i \(0.482115\pi\)
\(978\) −1.50701e28 −0.588705
\(979\) −3.46393e27 −0.134009
\(980\) 0 0
\(981\) 6.03194e28 2.28877
\(982\) −6.01941e26 −0.0226201
\(983\) 2.10459e28 0.783265 0.391632 0.920122i \(-0.371910\pi\)
0.391632 + 0.920122i \(0.371910\pi\)
\(984\) −2.80023e28 −1.03215
\(985\) 0 0
\(986\) −3.59264e28 −1.29892
\(987\) 8.97846e27 0.321506
\(988\) 7.16789e27 0.254214
\(989\) −1.90466e28 −0.669041
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −3.71230e28 −1.21949
\(997\) 1.44919e28 0.471544 0.235772 0.971808i \(-0.424238\pi\)
0.235772 + 0.971808i \(0.424238\pi\)
\(998\) 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 50.20.a.f.1.1 2
5.2 odd 4 50.20.b.f.49.2 4
5.3 odd 4 50.20.b.f.49.3 4
5.4 even 2 10.20.a.d.1.2 2
15.14 odd 2 90.20.a.g.1.1 2
20.19 odd 2 80.20.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.20.a.d.1.2 2 5.4 even 2
50.20.a.f.1.1 2 1.1 even 1 trivial
50.20.b.f.49.2 4 5.2 odd 4
50.20.b.f.49.3 4 5.3 odd 4
80.20.a.d.1.1 2 20.19 odd 2
90.20.a.g.1.1 2 15.14 odd 2