Properties

Label 50.26.a.i.1.5
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,26,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, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
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: \(197.998389976\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 25534923283x^{3} - 31863478542482x^{2} + 141941149085067124800x + 2515032055818200956928000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{5}\cdot 5^{12} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-136720.\) 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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.22241e6 q^{3} +1.67772e7 q^{4} -5.00697e9 q^{6} -6.80801e10 q^{7} -6.87195e10 q^{8} +6.46987e11 q^{9} -1.63529e13 q^{11} +2.05086e13 q^{12} -1.23277e14 q^{13} +2.78856e14 q^{14} +2.81475e14 q^{16} -1.24327e15 q^{17} -2.65006e15 q^{18} +1.08089e16 q^{19} -8.32215e16 q^{21} +6.69815e16 q^{22} -1.26942e17 q^{23} -8.40031e16 q^{24} +5.04942e17 q^{26} -2.44849e17 q^{27} -1.14219e18 q^{28} -1.19617e18 q^{29} +3.33827e18 q^{31} -1.15292e18 q^{32} -1.99899e19 q^{33} +5.09242e18 q^{34} +1.08546e19 q^{36} -5.02410e19 q^{37} -4.42734e19 q^{38} -1.50694e20 q^{39} -2.63431e20 q^{41} +3.40875e20 q^{42} -1.36754e20 q^{43} -2.74356e20 q^{44} +5.19955e20 q^{46} +7.03907e20 q^{47} +3.44077e20 q^{48} +3.29383e21 q^{49} -1.51978e21 q^{51} -2.06824e21 q^{52} +3.75972e21 q^{53} +1.00290e21 q^{54} +4.67843e21 q^{56} +1.32129e22 q^{57} +4.89952e21 q^{58} -6.89657e21 q^{59} +1.54607e22 q^{61} -1.36736e22 q^{62} -4.40470e22 q^{63} +4.72237e21 q^{64} +8.18786e22 q^{66} +8.66895e22 q^{67} -2.08585e22 q^{68} -1.55175e23 q^{69} +4.41872e22 q^{71} -4.44606e22 q^{72} +8.03151e22 q^{73} +2.05787e23 q^{74} +1.81344e23 q^{76} +1.11331e24 q^{77} +6.17244e23 q^{78} -5.07310e23 q^{79} -8.47490e23 q^{81} +1.07901e24 q^{82} -5.22032e23 q^{83} -1.39623e24 q^{84} +5.60143e23 q^{86} -1.46221e24 q^{87} +1.12376e24 q^{88} -1.15373e24 q^{89} +8.39270e24 q^{91} -2.12974e24 q^{92} +4.08072e24 q^{93} -2.88320e24 q^{94} -1.40934e24 q^{96} +4.89110e24 q^{97} -1.34915e25 q^{98} -1.05801e25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 20480 q^{2} - 723995 q^{3} + 83886080 q^{4} + 2965483520 q^{6} - 49218886190 q^{7} - 343597383680 q^{8} + 975375361390 q^{9} - 8837033983815 q^{11} - 12146620497920 q^{12} - 67609989586220 q^{13}+ \cdots - 23\!\cdots\!70 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) 1.22241e6 1.32801 0.664003 0.747730i \(-0.268856\pi\)
0.664003 + 0.747730i \(0.268856\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −5.00697e9 −0.939041
\(7\) −6.80801e10 −1.85907 −0.929533 0.368739i \(-0.879789\pi\)
−0.929533 + 0.368739i \(0.879789\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 6.46987e11 0.763597
\(10\) 0 0
\(11\) −1.63529e13 −1.57104 −0.785519 0.618838i \(-0.787604\pi\)
−0.785519 + 0.618838i \(0.787604\pi\)
\(12\) 2.05086e13 0.664003
\(13\) −1.23277e14 −1.46754 −0.733769 0.679399i \(-0.762241\pi\)
−0.733769 + 0.679399i \(0.762241\pi\)
\(14\) 2.78856e14 1.31456
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −1.24327e15 −0.517550 −0.258775 0.965938i \(-0.583319\pi\)
−0.258775 + 0.965938i \(0.583319\pi\)
\(18\) −2.65006e15 −0.539945
\(19\) 1.08089e16 1.12037 0.560187 0.828366i \(-0.310729\pi\)
0.560187 + 0.828366i \(0.310729\pi\)
\(20\) 0 0
\(21\) −8.32215e16 −2.46885
\(22\) 6.69815e16 1.11089
\(23\) −1.26942e17 −1.20784 −0.603918 0.797046i \(-0.706395\pi\)
−0.603918 + 0.797046i \(0.706395\pi\)
\(24\) −8.40031e16 −0.469521
\(25\) 0 0
\(26\) 5.04942e17 1.03771
\(27\) −2.44849e17 −0.313944
\(28\) −1.14219e18 −0.929533
\(29\) −1.19617e18 −0.627797 −0.313898 0.949457i \(-0.601635\pi\)
−0.313898 + 0.949457i \(0.601635\pi\)
\(30\) 0 0
\(31\) 3.33827e18 0.761203 0.380602 0.924739i \(-0.375717\pi\)
0.380602 + 0.924739i \(0.375717\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) −1.99899e19 −2.08635
\(34\) 5.09242e18 0.365963
\(35\) 0 0
\(36\) 1.08546e19 0.381799
\(37\) −5.02410e19 −1.25469 −0.627346 0.778741i \(-0.715859\pi\)
−0.627346 + 0.778741i \(0.715859\pi\)
\(38\) −4.42734e19 −0.792224
\(39\) −1.50694e20 −1.94890
\(40\) 0 0
\(41\) −2.63431e20 −1.82335 −0.911673 0.410917i \(-0.865209\pi\)
−0.911673 + 0.410917i \(0.865209\pi\)
\(42\) 3.40875e20 1.74574
\(43\) −1.36754e20 −0.521895 −0.260947 0.965353i \(-0.584035\pi\)
−0.260947 + 0.965353i \(0.584035\pi\)
\(44\) −2.74356e20 −0.785519
\(45\) 0 0
\(46\) 5.19955e20 0.854069
\(47\) 7.03907e20 0.883674 0.441837 0.897095i \(-0.354327\pi\)
0.441837 + 0.897095i \(0.354327\pi\)
\(48\) 3.44077e20 0.332001
\(49\) 3.29383e21 2.45613
\(50\) 0 0
\(51\) −1.51978e21 −0.687309
\(52\) −2.06824e21 −0.733769
\(53\) 3.75972e21 1.05125 0.525626 0.850716i \(-0.323831\pi\)
0.525626 + 0.850716i \(0.323831\pi\)
\(54\) 1.00290e21 0.221992
\(55\) 0 0
\(56\) 4.67843e21 0.657279
\(57\) 1.32129e22 1.48786
\(58\) 4.89952e21 0.443919
\(59\) −6.89657e21 −0.504642 −0.252321 0.967644i \(-0.581194\pi\)
−0.252321 + 0.967644i \(0.581194\pi\)
\(60\) 0 0
\(61\) 1.54607e22 0.745773 0.372886 0.927877i \(-0.378368\pi\)
0.372886 + 0.927877i \(0.378368\pi\)
\(62\) −1.36736e22 −0.538252
\(63\) −4.40470e22 −1.41958
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) 8.18786e22 1.47527
\(67\) 8.66895e22 1.29429 0.647145 0.762367i \(-0.275963\pi\)
0.647145 + 0.762367i \(0.275963\pi\)
\(68\) −2.08585e22 −0.258775
\(69\) −1.55175e23 −1.60401
\(70\) 0 0
\(71\) 4.41872e22 0.319571 0.159785 0.987152i \(-0.448920\pi\)
0.159785 + 0.987152i \(0.448920\pi\)
\(72\) −4.44606e22 −0.269972
\(73\) 8.03151e22 0.410451 0.205226 0.978715i \(-0.434207\pi\)
0.205226 + 0.978715i \(0.434207\pi\)
\(74\) 2.05787e23 0.887201
\(75\) 0 0
\(76\) 1.81344e23 0.560187
\(77\) 1.11331e24 2.92066
\(78\) 6.17244e23 1.37808
\(79\) −5.07310e23 −0.965906 −0.482953 0.875646i \(-0.660436\pi\)
−0.482953 + 0.875646i \(0.660436\pi\)
\(80\) 0 0
\(81\) −8.47490e23 −1.18052
\(82\) 1.07901e24 1.28930
\(83\) −5.22032e23 −0.536070 −0.268035 0.963409i \(-0.586374\pi\)
−0.268035 + 0.963409i \(0.586374\pi\)
\(84\) −1.39623e24 −1.23442
\(85\) 0 0
\(86\) 5.60143e23 0.369035
\(87\) −1.46221e24 −0.833717
\(88\) 1.12376e24 0.555446
\(89\) −1.15373e24 −0.495142 −0.247571 0.968870i \(-0.579632\pi\)
−0.247571 + 0.968870i \(0.579632\pi\)
\(90\) 0 0
\(91\) 8.39270e24 2.72825
\(92\) −2.12974e24 −0.603918
\(93\) 4.08072e24 1.01088
\(94\) −2.88320e24 −0.624852
\(95\) 0 0
\(96\) −1.40934e24 −0.234760
\(97\) 4.89110e24 0.715748 0.357874 0.933770i \(-0.383502\pi\)
0.357874 + 0.933770i \(0.383502\pi\)
\(98\) −1.34915e25 −1.73674
\(99\) −1.05801e25 −1.19964
\(100\) 0 0
\(101\) 1.39348e25 1.23051 0.615255 0.788328i \(-0.289053\pi\)
0.615255 + 0.788328i \(0.289053\pi\)
\(102\) 6.22500e24 0.486001
\(103\) −1.11068e25 −0.767579 −0.383789 0.923421i \(-0.625381\pi\)
−0.383789 + 0.923421i \(0.625381\pi\)
\(104\) 8.47152e24 0.518853
\(105\) 0 0
\(106\) −1.53998e25 −0.743348
\(107\) −2.28767e25 −0.981967 −0.490983 0.871169i \(-0.663362\pi\)
−0.490983 + 0.871169i \(0.663362\pi\)
\(108\) −4.10789e24 −0.156972
\(109\) 3.92414e25 1.33633 0.668164 0.744014i \(-0.267081\pi\)
0.668164 + 0.744014i \(0.267081\pi\)
\(110\) 0 0
\(111\) −6.14149e25 −1.66624
\(112\) −1.91628e25 −0.464766
\(113\) −2.95166e25 −0.640599 −0.320299 0.947316i \(-0.603784\pi\)
−0.320299 + 0.947316i \(0.603784\pi\)
\(114\) −5.41201e25 −1.05208
\(115\) 0 0
\(116\) −2.00685e25 −0.313898
\(117\) −7.97585e25 −1.12061
\(118\) 2.82484e25 0.356835
\(119\) 8.46417e25 0.962160
\(120\) 0 0
\(121\) 1.59071e26 1.46816
\(122\) −6.33271e25 −0.527341
\(123\) −3.22020e26 −2.42141
\(124\) 5.60069e25 0.380602
\(125\) 0 0
\(126\) 1.80416e26 1.00379
\(127\) −3.71288e25 −0.187139 −0.0935696 0.995613i \(-0.529828\pi\)
−0.0935696 + 0.995613i \(0.529828\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) −1.67168e26 −0.693079
\(130\) 0 0
\(131\) −2.02734e26 −0.693482 −0.346741 0.937961i \(-0.612712\pi\)
−0.346741 + 0.937961i \(0.612712\pi\)
\(132\) −3.35375e26 −1.04317
\(133\) −7.35874e26 −2.08285
\(134\) −3.55080e26 −0.915201
\(135\) 0 0
\(136\) 8.54366e25 0.182982
\(137\) −4.21478e26 −0.823697 −0.411848 0.911252i \(-0.635117\pi\)
−0.411848 + 0.911252i \(0.635117\pi\)
\(138\) 6.35596e26 1.13421
\(139\) −2.61876e26 −0.426983 −0.213491 0.976945i \(-0.568484\pi\)
−0.213491 + 0.976945i \(0.568484\pi\)
\(140\) 0 0
\(141\) 8.60460e26 1.17352
\(142\) −1.80991e26 −0.225971
\(143\) 2.01593e27 2.30556
\(144\) 1.82111e26 0.190899
\(145\) 0 0
\(146\) −3.28971e26 −0.290233
\(147\) 4.02640e27 3.26175
\(148\) −8.42905e26 −0.627346
\(149\) 5.33091e26 0.364732 0.182366 0.983231i \(-0.441625\pi\)
0.182366 + 0.983231i \(0.441625\pi\)
\(150\) 0 0
\(151\) −1.15195e26 −0.0667148 −0.0333574 0.999443i \(-0.510620\pi\)
−0.0333574 + 0.999443i \(0.510620\pi\)
\(152\) −7.42784e26 −0.396112
\(153\) −8.04377e26 −0.395200
\(154\) −4.56011e27 −2.06522
\(155\) 0 0
\(156\) −2.52823e27 −0.974449
\(157\) −7.11181e26 −0.253066 −0.126533 0.991962i \(-0.540385\pi\)
−0.126533 + 0.991962i \(0.540385\pi\)
\(158\) 2.07794e27 0.682999
\(159\) 4.59590e27 1.39607
\(160\) 0 0
\(161\) 8.64224e27 2.24545
\(162\) 3.47132e27 0.834751
\(163\) −4.92749e27 −1.09719 −0.548594 0.836089i \(-0.684837\pi\)
−0.548594 + 0.836089i \(0.684837\pi\)
\(164\) −4.41964e27 −0.911673
\(165\) 0 0
\(166\) 2.13825e27 0.379059
\(167\) −1.54847e26 −0.0254651 −0.0127326 0.999919i \(-0.504053\pi\)
−0.0127326 + 0.999919i \(0.504053\pi\)
\(168\) 5.71894e27 0.872870
\(169\) 8.14076e27 1.15367
\(170\) 0 0
\(171\) 6.99325e27 0.855515
\(172\) −2.29434e27 −0.260947
\(173\) 1.24940e28 1.32168 0.660840 0.750527i \(-0.270200\pi\)
0.660840 + 0.750527i \(0.270200\pi\)
\(174\) 5.98921e27 0.589527
\(175\) 0 0
\(176\) −4.60293e27 −0.392759
\(177\) −8.43041e27 −0.670167
\(178\) 4.72569e27 0.350118
\(179\) 1.81393e28 1.25302 0.626509 0.779414i \(-0.284483\pi\)
0.626509 + 0.779414i \(0.284483\pi\)
\(180\) 0 0
\(181\) −2.28334e28 −1.37274 −0.686369 0.727254i \(-0.740796\pi\)
−0.686369 + 0.727254i \(0.740796\pi\)
\(182\) −3.43765e28 −1.92916
\(183\) 1.88993e28 0.990390
\(184\) 8.72340e27 0.427035
\(185\) 0 0
\(186\) −1.67146e28 −0.714801
\(187\) 2.03310e28 0.813091
\(188\) 1.18096e28 0.441837
\(189\) 1.66694e28 0.583642
\(190\) 0 0
\(191\) −3.88305e28 −1.19195 −0.595973 0.803005i \(-0.703233\pi\)
−0.595973 + 0.803005i \(0.703233\pi\)
\(192\) 5.77265e27 0.166001
\(193\) −5.16030e28 −1.39062 −0.695310 0.718710i \(-0.744733\pi\)
−0.695310 + 0.718710i \(0.744733\pi\)
\(194\) −2.00340e28 −0.506110
\(195\) 0 0
\(196\) 5.52613e28 1.22806
\(197\) −4.39578e28 −0.916659 −0.458330 0.888782i \(-0.651552\pi\)
−0.458330 + 0.888782i \(0.651552\pi\)
\(198\) 4.33362e28 0.848274
\(199\) 6.89463e28 1.26721 0.633604 0.773658i \(-0.281575\pi\)
0.633604 + 0.773658i \(0.281575\pi\)
\(200\) 0 0
\(201\) 1.05970e29 1.71882
\(202\) −5.70771e28 −0.870102
\(203\) 8.14356e28 1.16712
\(204\) −2.54976e28 −0.343655
\(205\) 0 0
\(206\) 4.54934e28 0.542760
\(207\) −8.21300e28 −0.922301
\(208\) −3.46993e28 −0.366885
\(209\) −1.76758e29 −1.76015
\(210\) 0 0
\(211\) 2.04578e29 1.80854 0.904269 0.426962i \(-0.140416\pi\)
0.904269 + 0.426962i \(0.140416\pi\)
\(212\) 6.30776e28 0.525626
\(213\) 5.40147e28 0.424392
\(214\) 9.37031e28 0.694355
\(215\) 0 0
\(216\) 1.68259e28 0.110996
\(217\) −2.27270e29 −1.41513
\(218\) −1.60733e29 −0.944926
\(219\) 9.81777e28 0.545081
\(220\) 0 0
\(221\) 1.53266e29 0.759525
\(222\) 2.51556e29 1.17821
\(223\) 3.97032e29 1.75798 0.878991 0.476839i \(-0.158217\pi\)
0.878991 + 0.476839i \(0.158217\pi\)
\(224\) 7.84910e28 0.328639
\(225\) 0 0
\(226\) 1.20900e29 0.452972
\(227\) 3.06368e29 1.08623 0.543113 0.839660i \(-0.317246\pi\)
0.543113 + 0.839660i \(0.317246\pi\)
\(228\) 2.21676e29 0.743931
\(229\) 2.92488e28 0.0929319 0.0464660 0.998920i \(-0.485204\pi\)
0.0464660 + 0.998920i \(0.485204\pi\)
\(230\) 0 0
\(231\) 1.36091e30 3.87865
\(232\) 8.22004e28 0.221960
\(233\) 3.00032e29 0.767748 0.383874 0.923385i \(-0.374590\pi\)
0.383874 + 0.923385i \(0.374590\pi\)
\(234\) 3.26691e29 0.792390
\(235\) 0 0
\(236\) −1.15705e29 −0.252321
\(237\) −6.20139e29 −1.28273
\(238\) −3.46692e29 −0.680350
\(239\) −7.62279e29 −1.41952 −0.709758 0.704445i \(-0.751196\pi\)
−0.709758 + 0.704445i \(0.751196\pi\)
\(240\) 0 0
\(241\) −5.68961e29 −0.954705 −0.477352 0.878712i \(-0.658403\pi\)
−0.477352 + 0.878712i \(0.658403\pi\)
\(242\) −6.51553e29 −1.03814
\(243\) −8.28519e29 −1.25379
\(244\) 2.59388e29 0.372886
\(245\) 0 0
\(246\) 1.31899e30 1.71220
\(247\) −1.33249e30 −1.64419
\(248\) −2.29404e29 −0.269126
\(249\) −6.38136e29 −0.711903
\(250\) 0 0
\(251\) 7.84848e29 0.792253 0.396127 0.918196i \(-0.370354\pi\)
0.396127 + 0.918196i \(0.370354\pi\)
\(252\) −7.38986e29 −0.709789
\(253\) 2.07587e30 1.89756
\(254\) 1.52080e29 0.132327
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −1.93819e30 −1.45624 −0.728119 0.685450i \(-0.759605\pi\)
−0.728119 + 0.685450i \(0.759605\pi\)
\(258\) 6.84722e29 0.490081
\(259\) 3.42041e30 2.33255
\(260\) 0 0
\(261\) −7.73909e29 −0.479384
\(262\) 8.30399e29 0.490366
\(263\) −7.76661e29 −0.437305 −0.218653 0.975803i \(-0.570166\pi\)
−0.218653 + 0.975803i \(0.570166\pi\)
\(264\) 1.37369e30 0.737634
\(265\) 0 0
\(266\) 3.01414e30 1.47280
\(267\) −1.41033e30 −0.657551
\(268\) 1.45441e30 0.647145
\(269\) 1.26958e30 0.539209 0.269604 0.962971i \(-0.413107\pi\)
0.269604 + 0.962971i \(0.413107\pi\)
\(270\) 0 0
\(271\) −2.59506e29 −0.100469 −0.0502344 0.998737i \(-0.515997\pi\)
−0.0502344 + 0.998737i \(0.515997\pi\)
\(272\) −3.49948e29 −0.129388
\(273\) 1.02593e31 3.62313
\(274\) 1.72637e30 0.582442
\(275\) 0 0
\(276\) −2.60340e30 −0.802006
\(277\) −4.20101e30 −1.23696 −0.618482 0.785799i \(-0.712252\pi\)
−0.618482 + 0.785799i \(0.712252\pi\)
\(278\) 1.07264e30 0.301922
\(279\) 2.15982e30 0.581253
\(280\) 0 0
\(281\) 7.34654e30 1.80823 0.904115 0.427290i \(-0.140532\pi\)
0.904115 + 0.427290i \(0.140532\pi\)
\(282\) −3.52444e30 −0.829806
\(283\) −7.45011e30 −1.67816 −0.839078 0.544011i \(-0.816905\pi\)
−0.839078 + 0.544011i \(0.816905\pi\)
\(284\) 7.41339e29 0.159785
\(285\) 0 0
\(286\) −8.25727e30 −1.63028
\(287\) 1.79344e31 3.38972
\(288\) −7.45926e29 −0.134986
\(289\) −4.22492e30 −0.732142
\(290\) 0 0
\(291\) 5.97891e30 0.950517
\(292\) 1.34746e30 0.205226
\(293\) −3.40180e30 −0.496436 −0.248218 0.968704i \(-0.579845\pi\)
−0.248218 + 0.968704i \(0.579845\pi\)
\(294\) −1.64921e31 −2.30640
\(295\) 0 0
\(296\) 3.45254e30 0.443601
\(297\) 4.00400e30 0.493217
\(298\) −2.18354e30 −0.257904
\(299\) 1.56490e31 1.77255
\(300\) 0 0
\(301\) 9.31020e30 0.970237
\(302\) 4.71839e29 0.0471745
\(303\) 1.70340e31 1.63412
\(304\) 3.04245e30 0.280093
\(305\) 0 0
\(306\) 3.29473e30 0.279449
\(307\) −2.05490e31 −1.67325 −0.836624 0.547778i \(-0.815474\pi\)
−0.836624 + 0.547778i \(0.815474\pi\)
\(308\) 1.86782e31 1.46033
\(309\) −1.35770e31 −1.01935
\(310\) 0 0
\(311\) −7.23094e28 −0.00500829 −0.00250415 0.999997i \(-0.500797\pi\)
−0.00250415 + 0.999997i \(0.500797\pi\)
\(312\) 1.03556e31 0.689040
\(313\) −1.14602e31 −0.732635 −0.366318 0.930490i \(-0.619382\pi\)
−0.366318 + 0.930490i \(0.619382\pi\)
\(314\) 2.91300e30 0.178945
\(315\) 0 0
\(316\) −8.51125e30 −0.482953
\(317\) 2.43704e31 1.32929 0.664647 0.747158i \(-0.268582\pi\)
0.664647 + 0.747158i \(0.268582\pi\)
\(318\) −1.88248e31 −0.987170
\(319\) 1.95609e31 0.986292
\(320\) 0 0
\(321\) −2.79646e31 −1.30406
\(322\) −3.53986e31 −1.58777
\(323\) −1.34384e31 −0.579850
\(324\) −1.42185e31 −0.590258
\(325\) 0 0
\(326\) 2.01830e31 0.775828
\(327\) 4.79689e31 1.77465
\(328\) 1.81029e31 0.644650
\(329\) −4.79220e31 −1.64281
\(330\) 0 0
\(331\) −3.61088e31 −1.14753 −0.573764 0.819021i \(-0.694517\pi\)
−0.573764 + 0.819021i \(0.694517\pi\)
\(332\) −8.75825e30 −0.268035
\(333\) −3.25053e31 −0.958079
\(334\) 6.34252e29 0.0180066
\(335\) 0 0
\(336\) −2.34248e31 −0.617212
\(337\) −5.01811e31 −1.27399 −0.636995 0.770868i \(-0.719823\pi\)
−0.636995 + 0.770868i \(0.719823\pi\)
\(338\) −3.33446e31 −0.815767
\(339\) −3.60813e31 −0.850718
\(340\) 0 0
\(341\) −5.45905e31 −1.19588
\(342\) −2.86443e31 −0.604940
\(343\) −1.32944e32 −2.70703
\(344\) 9.39763e30 0.184518
\(345\) 0 0
\(346\) −5.11756e31 −0.934569
\(347\) −2.69651e31 −0.474988 −0.237494 0.971389i \(-0.576326\pi\)
−0.237494 + 0.971389i \(0.576326\pi\)
\(348\) −2.45318e31 −0.416859
\(349\) 4.66669e31 0.765052 0.382526 0.923945i \(-0.375054\pi\)
0.382526 + 0.923945i \(0.375054\pi\)
\(350\) 0 0
\(351\) 3.01842e31 0.460725
\(352\) 1.88536e31 0.277723
\(353\) −6.63592e31 −0.943446 −0.471723 0.881747i \(-0.656368\pi\)
−0.471723 + 0.881747i \(0.656368\pi\)
\(354\) 3.45310e31 0.473879
\(355\) 0 0
\(356\) −1.93564e31 −0.247571
\(357\) 1.03466e32 1.27775
\(358\) −7.42985e31 −0.886017
\(359\) −7.18124e31 −0.827025 −0.413513 0.910498i \(-0.635698\pi\)
−0.413513 + 0.910498i \(0.635698\pi\)
\(360\) 0 0
\(361\) 2.37566e31 0.255238
\(362\) 9.35254e31 0.970672
\(363\) 1.94449e32 1.94972
\(364\) 1.40806e32 1.36413
\(365\) 0 0
\(366\) −7.74114e31 −0.700311
\(367\) 8.09654e31 0.707903 0.353951 0.935264i \(-0.384838\pi\)
0.353951 + 0.935264i \(0.384838\pi\)
\(368\) −3.57310e31 −0.301959
\(369\) −1.70437e32 −1.39230
\(370\) 0 0
\(371\) −2.55962e32 −1.95435
\(372\) 6.84632e31 0.505441
\(373\) 1.63975e32 1.17063 0.585313 0.810807i \(-0.300972\pi\)
0.585313 + 0.810807i \(0.300972\pi\)
\(374\) −8.32758e31 −0.574942
\(375\) 0 0
\(376\) −4.83721e31 −0.312426
\(377\) 1.47460e32 0.921316
\(378\) −6.82777e31 −0.412697
\(379\) −5.11219e31 −0.298963 −0.149481 0.988765i \(-0.547760\pi\)
−0.149481 + 0.988765i \(0.547760\pi\)
\(380\) 0 0
\(381\) −4.53865e31 −0.248522
\(382\) 1.59050e32 0.842833
\(383\) 2.90570e32 1.49028 0.745138 0.666911i \(-0.232384\pi\)
0.745138 + 0.666911i \(0.232384\pi\)
\(384\) −2.36448e31 −0.117380
\(385\) 0 0
\(386\) 2.11366e32 0.983317
\(387\) −8.84778e31 −0.398518
\(388\) 8.20591e31 0.357874
\(389\) 9.70428e31 0.409820 0.204910 0.978781i \(-0.434310\pi\)
0.204910 + 0.978781i \(0.434310\pi\)
\(390\) 0 0
\(391\) 1.57823e32 0.625116
\(392\) −2.26350e32 −0.868371
\(393\) −2.47823e32 −0.920948
\(394\) 1.80051e32 0.648176
\(395\) 0 0
\(396\) −1.77505e32 −0.599820
\(397\) 4.94379e32 1.61875 0.809373 0.587295i \(-0.199807\pi\)
0.809373 + 0.587295i \(0.199807\pi\)
\(398\) −2.82404e32 −0.896051
\(399\) −8.99536e32 −2.76603
\(400\) 0 0
\(401\) 4.47162e31 0.129170 0.0645848 0.997912i \(-0.479428\pi\)
0.0645848 + 0.997912i \(0.479428\pi\)
\(402\) −4.34052e32 −1.21539
\(403\) −4.11531e32 −1.11709
\(404\) 2.33788e32 0.615255
\(405\) 0 0
\(406\) −3.33560e32 −0.825275
\(407\) 8.21587e32 1.97117
\(408\) 1.04438e32 0.243001
\(409\) 3.47804e32 0.784863 0.392431 0.919781i \(-0.371634\pi\)
0.392431 + 0.919781i \(0.371634\pi\)
\(410\) 0 0
\(411\) −5.15217e32 −1.09387
\(412\) −1.86341e32 −0.383789
\(413\) 4.69519e32 0.938162
\(414\) 3.36404e32 0.652165
\(415\) 0 0
\(416\) 1.42128e32 0.259427
\(417\) −3.20119e32 −0.567035
\(418\) 7.23999e32 1.24461
\(419\) −1.64856e32 −0.275061 −0.137531 0.990498i \(-0.543917\pi\)
−0.137531 + 0.990498i \(0.543917\pi\)
\(420\) 0 0
\(421\) 4.59846e32 0.722914 0.361457 0.932389i \(-0.382279\pi\)
0.361457 + 0.932389i \(0.382279\pi\)
\(422\) −8.37951e32 −1.27883
\(423\) 4.55419e32 0.674771
\(424\) −2.58366e32 −0.371674
\(425\) 0 0
\(426\) −2.21244e32 −0.300090
\(427\) −1.05257e33 −1.38644
\(428\) −3.83808e32 −0.490983
\(429\) 2.46429e33 3.06179
\(430\) 0 0
\(431\) 7.08014e32 0.829996 0.414998 0.909822i \(-0.363782\pi\)
0.414998 + 0.909822i \(0.363782\pi\)
\(432\) −6.89190e31 −0.0784860
\(433\) 1.74438e32 0.192993 0.0964964 0.995333i \(-0.469236\pi\)
0.0964964 + 0.995333i \(0.469236\pi\)
\(434\) 9.30898e32 1.00065
\(435\) 0 0
\(436\) 6.58361e32 0.668164
\(437\) −1.37211e33 −1.35323
\(438\) −4.02136e32 −0.385431
\(439\) −9.83917e32 −0.916541 −0.458271 0.888813i \(-0.651531\pi\)
−0.458271 + 0.888813i \(0.651531\pi\)
\(440\) 0 0
\(441\) 2.13107e33 1.87549
\(442\) −6.27777e32 −0.537065
\(443\) −9.66613e32 −0.803908 −0.401954 0.915660i \(-0.631669\pi\)
−0.401954 + 0.915660i \(0.631669\pi\)
\(444\) −1.03037e33 −0.833118
\(445\) 0 0
\(446\) −1.62624e33 −1.24308
\(447\) 6.51654e32 0.484365
\(448\) −3.21499e32 −0.232383
\(449\) 5.74227e32 0.403650 0.201825 0.979422i \(-0.435313\pi\)
0.201825 + 0.979422i \(0.435313\pi\)
\(450\) 0 0
\(451\) 4.30787e33 2.86454
\(452\) −4.95207e32 −0.320299
\(453\) −1.40815e32 −0.0885977
\(454\) −1.25488e33 −0.768078
\(455\) 0 0
\(456\) −9.07984e32 −0.526039
\(457\) 3.19404e33 1.80048 0.900238 0.435398i \(-0.143392\pi\)
0.900238 + 0.435398i \(0.143392\pi\)
\(458\) −1.19803e32 −0.0657128
\(459\) 3.04413e32 0.162482
\(460\) 0 0
\(461\) 2.15114e33 1.08745 0.543723 0.839264i \(-0.317014\pi\)
0.543723 + 0.839264i \(0.317014\pi\)
\(462\) −5.57430e33 −2.74262
\(463\) 6.57140e32 0.314699 0.157350 0.987543i \(-0.449705\pi\)
0.157350 + 0.987543i \(0.449705\pi\)
\(464\) −3.36693e32 −0.156949
\(465\) 0 0
\(466\) −1.22893e33 −0.542880
\(467\) 1.92599e32 0.0828312 0.0414156 0.999142i \(-0.486813\pi\)
0.0414156 + 0.999142i \(0.486813\pi\)
\(468\) −1.33813e33 −0.560304
\(469\) −5.90183e33 −2.40617
\(470\) 0 0
\(471\) −8.69352e32 −0.336074
\(472\) 4.73929e32 0.178418
\(473\) 2.23632e33 0.819916
\(474\) 2.54009e33 0.907026
\(475\) 0 0
\(476\) 1.42005e33 0.481080
\(477\) 2.43249e33 0.802734
\(478\) 3.12230e33 1.00375
\(479\) 3.90669e33 1.22353 0.611766 0.791039i \(-0.290459\pi\)
0.611766 + 0.791039i \(0.290459\pi\)
\(480\) 0 0
\(481\) 6.19355e33 1.84131
\(482\) 2.33046e33 0.675078
\(483\) 1.05643e34 2.98196
\(484\) 2.66876e33 0.734079
\(485\) 0 0
\(486\) 3.39361e33 0.886562
\(487\) 4.17680e33 1.06349 0.531743 0.846906i \(-0.321537\pi\)
0.531743 + 0.846906i \(0.321537\pi\)
\(488\) −1.06245e33 −0.263670
\(489\) −6.02339e33 −1.45707
\(490\) 0 0
\(491\) −1.40550e33 −0.323083 −0.161541 0.986866i \(-0.551647\pi\)
−0.161541 + 0.986866i \(0.551647\pi\)
\(492\) −5.40260e33 −1.21071
\(493\) 1.48716e33 0.324916
\(494\) 5.45788e33 1.16262
\(495\) 0 0
\(496\) 9.39640e32 0.190301
\(497\) −3.00827e33 −0.594103
\(498\) 2.61380e33 0.503392
\(499\) 8.69572e32 0.163323 0.0816617 0.996660i \(-0.473977\pi\)
0.0816617 + 0.996660i \(0.473977\pi\)
\(500\) 0 0
\(501\) −1.89285e32 −0.0338178
\(502\) −3.21474e33 −0.560208
\(503\) −7.81824e33 −1.32895 −0.664476 0.747310i \(-0.731345\pi\)
−0.664476 + 0.747310i \(0.731345\pi\)
\(504\) 3.02688e33 0.501897
\(505\) 0 0
\(506\) −8.50278e33 −1.34177
\(507\) 9.95131e33 1.53208
\(508\) −6.22918e32 −0.0935696
\(509\) −7.92846e33 −1.16203 −0.581014 0.813894i \(-0.697344\pi\)
−0.581014 + 0.813894i \(0.697344\pi\)
\(510\) 0 0
\(511\) −5.46786e33 −0.763056
\(512\) −3.24519e32 −0.0441942
\(513\) −2.64656e33 −0.351734
\(514\) 7.93883e33 1.02972
\(515\) 0 0
\(516\) −2.80462e33 −0.346539
\(517\) −1.15109e34 −1.38828
\(518\) −1.40100e34 −1.64936
\(519\) 1.52728e34 1.75520
\(520\) 0 0
\(521\) −3.29787e33 −0.361212 −0.180606 0.983556i \(-0.557806\pi\)
−0.180606 + 0.983556i \(0.557806\pi\)
\(522\) 3.16993e33 0.338976
\(523\) −1.60818e33 −0.167905 −0.0839524 0.996470i \(-0.526754\pi\)
−0.0839524 + 0.996470i \(0.526754\pi\)
\(524\) −3.40131e33 −0.346741
\(525\) 0 0
\(526\) 3.18120e33 0.309221
\(527\) −4.15036e33 −0.393961
\(528\) −5.62665e33 −0.521586
\(529\) 5.06855e33 0.458868
\(530\) 0 0
\(531\) −4.46199e33 −0.385343
\(532\) −1.23459e34 −1.04142
\(533\) 3.24750e34 2.67583
\(534\) 5.77671e33 0.464959
\(535\) 0 0
\(536\) −5.95726e33 −0.457601
\(537\) 2.21736e34 1.66401
\(538\) −5.20020e33 −0.381278
\(539\) −5.38637e34 −3.85866
\(540\) 0 0
\(541\) −1.05559e34 −0.721990 −0.360995 0.932568i \(-0.617563\pi\)
−0.360995 + 0.932568i \(0.617563\pi\)
\(542\) 1.06294e33 0.0710422
\(543\) −2.79116e34 −1.82300
\(544\) 1.43339e33 0.0914908
\(545\) 0 0
\(546\) −4.20220e34 −2.56194
\(547\) 2.41768e34 1.44065 0.720324 0.693638i \(-0.243993\pi\)
0.720324 + 0.693638i \(0.243993\pi\)
\(548\) −7.07122e33 −0.411848
\(549\) 1.00029e34 0.569470
\(550\) 0 0
\(551\) −1.29294e34 −0.703367
\(552\) 1.06635e34 0.567104
\(553\) 3.45377e34 1.79568
\(554\) 1.72073e34 0.874665
\(555\) 0 0
\(556\) −4.39355e33 −0.213491
\(557\) −1.98926e34 −0.945153 −0.472576 0.881290i \(-0.656676\pi\)
−0.472576 + 0.881290i \(0.656676\pi\)
\(558\) −8.84662e33 −0.411008
\(559\) 1.68585e34 0.765901
\(560\) 0 0
\(561\) 2.48527e34 1.07979
\(562\) −3.00914e34 −1.27861
\(563\) −3.43624e34 −1.42800 −0.714001 0.700145i \(-0.753119\pi\)
−0.714001 + 0.700145i \(0.753119\pi\)
\(564\) 1.44361e34 0.586762
\(565\) 0 0
\(566\) 3.05157e34 1.18664
\(567\) 5.76972e34 2.19466
\(568\) −3.03652e33 −0.112985
\(569\) −4.74393e34 −1.72677 −0.863386 0.504544i \(-0.831661\pi\)
−0.863386 + 0.504544i \(0.831661\pi\)
\(570\) 0 0
\(571\) −4.26964e33 −0.148744 −0.0743721 0.997231i \(-0.523695\pi\)
−0.0743721 + 0.997231i \(0.523695\pi\)
\(572\) 3.38218e34 1.15278
\(573\) −4.74667e34 −1.58291
\(574\) −7.34594e34 −2.39689
\(575\) 0 0
\(576\) 3.05531e33 0.0954497
\(577\) −1.96194e34 −0.599775 −0.299887 0.953975i \(-0.596949\pi\)
−0.299887 + 0.953975i \(0.596949\pi\)
\(578\) 1.73053e34 0.517702
\(579\) −6.30798e34 −1.84675
\(580\) 0 0
\(581\) 3.55400e34 0.996589
\(582\) −2.44896e34 −0.672117
\(583\) −6.14824e34 −1.65156
\(584\) −5.51921e33 −0.145116
\(585\) 0 0
\(586\) 1.39338e34 0.351033
\(587\) 4.23298e34 1.04393 0.521963 0.852968i \(-0.325200\pi\)
0.521963 + 0.852968i \(0.325200\pi\)
\(588\) 6.75518e34 1.63087
\(589\) 3.60832e34 0.852832
\(590\) 0 0
\(591\) −5.37343e34 −1.21733
\(592\) −1.41416e34 −0.313673
\(593\) −2.29135e34 −0.497631 −0.248816 0.968551i \(-0.580041\pi\)
−0.248816 + 0.968551i \(0.580041\pi\)
\(594\) −1.64004e34 −0.348757
\(595\) 0 0
\(596\) 8.94379e33 0.182366
\(597\) 8.42804e34 1.68286
\(598\) −6.40984e34 −1.25338
\(599\) 1.21937e34 0.233508 0.116754 0.993161i \(-0.462751\pi\)
0.116754 + 0.993161i \(0.462751\pi\)
\(600\) 0 0
\(601\) −2.38798e33 −0.0438631 −0.0219315 0.999759i \(-0.506982\pi\)
−0.0219315 + 0.999759i \(0.506982\pi\)
\(602\) −3.81346e34 −0.686061
\(603\) 5.60870e34 0.988317
\(604\) −1.93265e33 −0.0333574
\(605\) 0 0
\(606\) −6.97714e34 −1.15550
\(607\) 2.92481e34 0.474504 0.237252 0.971448i \(-0.423753\pi\)
0.237252 + 0.971448i \(0.423753\pi\)
\(608\) −1.24619e34 −0.198056
\(609\) 9.95473e34 1.54993
\(610\) 0 0
\(611\) −8.67754e34 −1.29682
\(612\) −1.34952e34 −0.197600
\(613\) −6.94539e34 −0.996416 −0.498208 0.867057i \(-0.666008\pi\)
−0.498208 + 0.867057i \(0.666008\pi\)
\(614\) 8.41686e34 1.18316
\(615\) 0 0
\(616\) −7.65059e34 −1.03261
\(617\) 4.20916e34 0.556713 0.278356 0.960478i \(-0.410210\pi\)
0.278356 + 0.960478i \(0.410210\pi\)
\(618\) 5.56114e34 0.720788
\(619\) 5.93194e34 0.753466 0.376733 0.926322i \(-0.377047\pi\)
0.376733 + 0.926322i \(0.377047\pi\)
\(620\) 0 0
\(621\) 3.10817e34 0.379193
\(622\) 2.96179e32 0.00354140
\(623\) 7.85462e34 0.920502
\(624\) −4.24167e34 −0.487225
\(625\) 0 0
\(626\) 4.69410e34 0.518051
\(627\) −2.16069e35 −2.33749
\(628\) −1.19316e34 −0.126533
\(629\) 6.24630e34 0.649366
\(630\) 0 0
\(631\) 1.46762e35 1.46638 0.733192 0.680022i \(-0.238030\pi\)
0.733192 + 0.680022i \(0.238030\pi\)
\(632\) 3.48621e34 0.341499
\(633\) 2.50077e35 2.40175
\(634\) −9.98210e34 −0.939953
\(635\) 0 0
\(636\) 7.71065e34 0.698034
\(637\) −4.06053e35 −3.60446
\(638\) −8.01215e34 −0.697414
\(639\) 2.85886e34 0.244024
\(640\) 0 0
\(641\) 1.22792e35 1.00796 0.503982 0.863714i \(-0.331868\pi\)
0.503982 + 0.863714i \(0.331868\pi\)
\(642\) 1.14543e35 0.922107
\(643\) −1.26806e34 −0.100116 −0.0500579 0.998746i \(-0.515941\pi\)
−0.0500579 + 0.998746i \(0.515941\pi\)
\(644\) 1.44993e35 1.12272
\(645\) 0 0
\(646\) 5.50436e34 0.410016
\(647\) −3.01852e34 −0.220541 −0.110271 0.993902i \(-0.535172\pi\)
−0.110271 + 0.993902i \(0.535172\pi\)
\(648\) 5.82391e34 0.417376
\(649\) 1.12779e35 0.792811
\(650\) 0 0
\(651\) −2.77816e35 −1.87930
\(652\) −8.26696e34 −0.548594
\(653\) −1.81062e35 −1.17872 −0.589361 0.807870i \(-0.700620\pi\)
−0.589361 + 0.807870i \(0.700620\pi\)
\(654\) −1.96481e35 −1.25487
\(655\) 0 0
\(656\) −7.41493e34 −0.455836
\(657\) 5.19629e34 0.313420
\(658\) 1.96289e35 1.16164
\(659\) −2.69289e35 −1.56369 −0.781847 0.623471i \(-0.785722\pi\)
−0.781847 + 0.623471i \(0.785722\pi\)
\(660\) 0 0
\(661\) 3.11605e35 1.74215 0.871075 0.491150i \(-0.163423\pi\)
0.871075 + 0.491150i \(0.163423\pi\)
\(662\) 1.47902e35 0.811424
\(663\) 1.87353e35 1.00865
\(664\) 3.58738e34 0.189529
\(665\) 0 0
\(666\) 1.33142e35 0.677464
\(667\) 1.51845e35 0.758276
\(668\) −2.59789e33 −0.0127326
\(669\) 4.85334e35 2.33461
\(670\) 0 0
\(671\) −2.52828e35 −1.17164
\(672\) 9.59479e34 0.436435
\(673\) 1.24919e35 0.557752 0.278876 0.960327i \(-0.410038\pi\)
0.278876 + 0.960327i \(0.410038\pi\)
\(674\) 2.05542e35 0.900848
\(675\) 0 0
\(676\) 1.36579e35 0.576834
\(677\) −2.88499e35 −1.19615 −0.598075 0.801440i \(-0.704067\pi\)
−0.598075 + 0.801440i \(0.704067\pi\)
\(678\) 1.47789e35 0.601549
\(679\) −3.32987e35 −1.33062
\(680\) 0 0
\(681\) 3.74506e35 1.44251
\(682\) 2.23603e35 0.845614
\(683\) −3.81708e34 −0.141733 −0.0708667 0.997486i \(-0.522577\pi\)
−0.0708667 + 0.997486i \(0.522577\pi\)
\(684\) 1.17327e35 0.427757
\(685\) 0 0
\(686\) 5.44540e35 1.91416
\(687\) 3.57539e34 0.123414
\(688\) −3.84927e34 −0.130474
\(689\) −4.63486e35 −1.54275
\(690\) 0 0
\(691\) −2.01041e35 −0.645372 −0.322686 0.946506i \(-0.604586\pi\)
−0.322686 + 0.946506i \(0.604586\pi\)
\(692\) 2.09615e35 0.660840
\(693\) 7.20296e35 2.23021
\(694\) 1.10449e35 0.335867
\(695\) 0 0
\(696\) 1.00482e35 0.294764
\(697\) 3.27515e35 0.943673
\(698\) −1.91148e35 −0.540974
\(699\) 3.66761e35 1.01957
\(700\) 0 0
\(701\) 1.78724e35 0.479411 0.239705 0.970846i \(-0.422949\pi\)
0.239705 + 0.970846i \(0.422949\pi\)
\(702\) −1.23635e35 −0.325781
\(703\) −5.43052e35 −1.40572
\(704\) −7.72244e34 −0.196380
\(705\) 0 0
\(706\) 2.71807e35 0.667117
\(707\) −9.48686e35 −2.28760
\(708\) −1.41439e35 −0.335083
\(709\) 5.59205e33 0.0130165 0.00650824 0.999979i \(-0.497928\pi\)
0.00650824 + 0.999979i \(0.497928\pi\)
\(710\) 0 0
\(711\) −3.28223e35 −0.737564
\(712\) 7.92839e34 0.175059
\(713\) −4.23768e35 −0.919409
\(714\) −4.23799e35 −0.903508
\(715\) 0 0
\(716\) 3.04327e35 0.626509
\(717\) −9.31815e35 −1.88512
\(718\) 2.94144e35 0.584795
\(719\) 3.90750e35 0.763463 0.381732 0.924273i \(-0.375328\pi\)
0.381732 + 0.924273i \(0.375328\pi\)
\(720\) 0 0
\(721\) 7.56151e35 1.42698
\(722\) −9.73071e34 −0.180480
\(723\) −6.95501e35 −1.26785
\(724\) −3.83080e35 −0.686369
\(725\) 0 0
\(726\) −7.96462e35 −1.37866
\(727\) 9.31579e35 1.58504 0.792518 0.609848i \(-0.208770\pi\)
0.792518 + 0.609848i \(0.208770\pi\)
\(728\) −5.76742e35 −0.964582
\(729\) −2.94718e35 −0.484520
\(730\) 0 0
\(731\) 1.70021e35 0.270107
\(732\) 3.17077e35 0.495195
\(733\) −2.49884e35 −0.383653 −0.191826 0.981429i \(-0.561441\pi\)
−0.191826 + 0.981429i \(0.561441\pi\)
\(734\) −3.31634e35 −0.500563
\(735\) 0 0
\(736\) 1.46354e35 0.213517
\(737\) −1.41763e36 −2.03338
\(738\) 6.98109e35 0.984506
\(739\) −1.12909e35 −0.156558 −0.0782788 0.996932i \(-0.524942\pi\)
−0.0782788 + 0.996932i \(0.524942\pi\)
\(740\) 0 0
\(741\) −1.62885e36 −2.18349
\(742\) 1.04842e36 1.38193
\(743\) −3.71494e35 −0.481495 −0.240748 0.970588i \(-0.577393\pi\)
−0.240748 + 0.970588i \(0.577393\pi\)
\(744\) −2.80425e35 −0.357401
\(745\) 0 0
\(746\) −6.71643e35 −0.827758
\(747\) −3.37748e35 −0.409342
\(748\) 3.41098e35 0.406545
\(749\) 1.55745e36 1.82554
\(750\) 0 0
\(751\) −6.99646e35 −0.793194 −0.396597 0.917993i \(-0.629809\pi\)
−0.396597 + 0.917993i \(0.629809\pi\)
\(752\) 1.98132e35 0.220918
\(753\) 9.59403e35 1.05212
\(754\) −6.03998e35 −0.651469
\(755\) 0 0
\(756\) 2.79666e35 0.291821
\(757\) −7.64525e35 −0.784681 −0.392341 0.919820i \(-0.628335\pi\)
−0.392341 + 0.919820i \(0.628335\pi\)
\(758\) 2.09395e35 0.211399
\(759\) 2.53756e36 2.51996
\(760\) 0 0
\(761\) −7.74903e35 −0.744628 −0.372314 0.928107i \(-0.621436\pi\)
−0.372314 + 0.928107i \(0.621436\pi\)
\(762\) 1.85903e35 0.175731
\(763\) −2.67156e36 −2.48432
\(764\) −6.51468e35 −0.595973
\(765\) 0 0
\(766\) −1.19018e36 −1.05378
\(767\) 8.50187e35 0.740581
\(768\) 9.68490e34 0.0830003
\(769\) −1.04301e36 −0.879447 −0.439723 0.898133i \(-0.644923\pi\)
−0.439723 + 0.898133i \(0.644923\pi\)
\(770\) 0 0
\(771\) −2.36926e36 −1.93389
\(772\) −8.65755e35 −0.695310
\(773\) 1.42237e36 1.12400 0.562002 0.827136i \(-0.310031\pi\)
0.562002 + 0.827136i \(0.310031\pi\)
\(774\) 3.62405e35 0.281794
\(775\) 0 0
\(776\) −3.36114e35 −0.253055
\(777\) 4.18114e36 3.09764
\(778\) −3.97487e35 −0.289786
\(779\) −2.84741e36 −2.04283
\(780\) 0 0
\(781\) −7.22590e35 −0.502058
\(782\) −6.46443e35 −0.442024
\(783\) 2.92882e35 0.197093
\(784\) 9.27131e35 0.614031
\(785\) 0 0
\(786\) 1.01508e36 0.651209
\(787\) 2.40737e36 1.52005 0.760026 0.649892i \(-0.225186\pi\)
0.760026 + 0.649892i \(0.225186\pi\)
\(788\) −7.37489e35 −0.458330
\(789\) −9.49395e35 −0.580743
\(790\) 0 0
\(791\) 2.00949e36 1.19092
\(792\) 7.27061e35 0.424137
\(793\) −1.90595e36 −1.09445
\(794\) −2.02498e36 −1.14463
\(795\) 0 0
\(796\) 1.15673e36 0.633604
\(797\) −9.73330e35 −0.524845 −0.262423 0.964953i \(-0.584521\pi\)
−0.262423 + 0.964953i \(0.584521\pi\)
\(798\) 3.68450e36 1.95588
\(799\) −8.75143e35 −0.457345
\(800\) 0 0
\(801\) −7.46450e35 −0.378089
\(802\) −1.83158e35 −0.0913368
\(803\) −1.31339e36 −0.644834
\(804\) 1.77788e36 0.859412
\(805\) 0 0
\(806\) 1.68563e36 0.789905
\(807\) 1.55194e36 0.716072
\(808\) −9.57595e35 −0.435051
\(809\) 3.20638e36 1.43436 0.717179 0.696889i \(-0.245433\pi\)
0.717179 + 0.696889i \(0.245433\pi\)
\(810\) 0 0
\(811\) 2.99227e36 1.29790 0.648948 0.760833i \(-0.275209\pi\)
0.648948 + 0.760833i \(0.275209\pi\)
\(812\) 1.36626e36 0.583558
\(813\) −3.17222e35 −0.133423
\(814\) −3.36522e36 −1.39383
\(815\) 0 0
\(816\) −4.27779e35 −0.171827
\(817\) −1.47816e36 −0.584717
\(818\) −1.42461e36 −0.554982
\(819\) 5.42997e36 2.08328
\(820\) 0 0
\(821\) −4.26235e36 −1.58621 −0.793103 0.609087i \(-0.791536\pi\)
−0.793103 + 0.609087i \(0.791536\pi\)
\(822\) 2.11033e36 0.773485
\(823\) 5.35214e36 1.93210 0.966051 0.258352i \(-0.0831796\pi\)
0.966051 + 0.258352i \(0.0831796\pi\)
\(824\) 7.63252e35 0.271380
\(825\) 0 0
\(826\) −1.92315e36 −0.663381
\(827\) 3.57743e36 1.21549 0.607747 0.794131i \(-0.292074\pi\)
0.607747 + 0.794131i \(0.292074\pi\)
\(828\) −1.37791e36 −0.461150
\(829\) −4.26555e36 −1.40619 −0.703095 0.711096i \(-0.748199\pi\)
−0.703095 + 0.711096i \(0.748199\pi\)
\(830\) 0 0
\(831\) −5.13534e36 −1.64269
\(832\) −5.82158e35 −0.183442
\(833\) −4.09511e36 −1.27117
\(834\) 1.31121e36 0.400955
\(835\) 0 0
\(836\) −2.96550e36 −0.880075
\(837\) −8.17374e35 −0.238975
\(838\) 6.75250e35 0.194498
\(839\) −5.62994e36 −1.59764 −0.798820 0.601570i \(-0.794542\pi\)
−0.798820 + 0.601570i \(0.794542\pi\)
\(840\) 0 0
\(841\) −2.19953e36 −0.605871
\(842\) −1.88353e36 −0.511177
\(843\) 8.98045e36 2.40134
\(844\) 3.43225e36 0.904269
\(845\) 0 0
\(846\) −1.86540e36 −0.477135
\(847\) −1.08295e37 −2.72940
\(848\) 1.05827e36 0.262813
\(849\) −9.10706e36 −2.22860
\(850\) 0 0
\(851\) 6.37771e36 1.51546
\(852\) 9.06217e35 0.212196
\(853\) −4.55647e36 −1.05139 −0.525697 0.850672i \(-0.676195\pi\)
−0.525697 + 0.850672i \(0.676195\pi\)
\(854\) 4.31131e36 0.980361
\(855\) 0 0
\(856\) 1.57208e36 0.347178
\(857\) 7.95266e35 0.173082 0.0865411 0.996248i \(-0.472419\pi\)
0.0865411 + 0.996248i \(0.472419\pi\)
\(858\) −1.00937e37 −2.16501
\(859\) 1.26545e36 0.267505 0.133752 0.991015i \(-0.457297\pi\)
0.133752 + 0.991015i \(0.457297\pi\)
\(860\) 0 0
\(861\) 2.19231e37 4.50156
\(862\) −2.90003e36 −0.586896
\(863\) −3.32635e36 −0.663487 −0.331744 0.943370i \(-0.607637\pi\)
−0.331744 + 0.943370i \(0.607637\pi\)
\(864\) 2.82292e35 0.0554979
\(865\) 0 0
\(866\) −7.14496e35 −0.136467
\(867\) −5.16456e36 −0.972288
\(868\) −3.81296e36 −0.707563
\(869\) 8.29600e36 1.51747
\(870\) 0 0
\(871\) −1.06868e37 −1.89942
\(872\) −2.69665e36 −0.472463
\(873\) 3.16448e36 0.546543
\(874\) 5.62016e36 0.956877
\(875\) 0 0
\(876\) 1.64715e36 0.272541
\(877\) −5.26043e36 −0.858078 −0.429039 0.903286i \(-0.641148\pi\)
−0.429039 + 0.903286i \(0.641148\pi\)
\(878\) 4.03012e36 0.648093
\(879\) −4.15838e36 −0.659269
\(880\) 0 0
\(881\) 5.34809e36 0.824137 0.412069 0.911153i \(-0.364806\pi\)
0.412069 + 0.911153i \(0.364806\pi\)
\(882\) −8.72885e36 −1.32617
\(883\) −3.32166e36 −0.497560 −0.248780 0.968560i \(-0.580030\pi\)
−0.248780 + 0.968560i \(0.580030\pi\)
\(884\) 2.57137e36 0.379762
\(885\) 0 0
\(886\) 3.95925e36 0.568449
\(887\) −8.17924e36 −1.15789 −0.578945 0.815366i \(-0.696536\pi\)
−0.578945 + 0.815366i \(0.696536\pi\)
\(888\) 4.22040e36 0.589104
\(889\) 2.52773e36 0.347904
\(890\) 0 0
\(891\) 1.38589e37 1.85464
\(892\) 6.66109e36 0.878991
\(893\) 7.60848e36 0.990045
\(894\) −2.66917e36 −0.342498
\(895\) 0 0
\(896\) 1.31686e36 0.164320
\(897\) 1.91295e37 2.35395
\(898\) −2.35204e36 −0.285423
\(899\) −3.99315e36 −0.477881
\(900\) 0 0
\(901\) −4.67433e36 −0.544076
\(902\) −1.76450e37 −2.02554
\(903\) 1.13808e37 1.28848
\(904\) 2.02837e36 0.226486
\(905\) 0 0
\(906\) 5.76779e35 0.0626480
\(907\) 1.71298e37 1.83511 0.917553 0.397615i \(-0.130162\pi\)
0.917553 + 0.397615i \(0.130162\pi\)
\(908\) 5.14000e36 0.543113
\(909\) 9.01567e36 0.939614
\(910\) 0 0
\(911\) 6.12131e36 0.620676 0.310338 0.950626i \(-0.399558\pi\)
0.310338 + 0.950626i \(0.399558\pi\)
\(912\) 3.71910e36 0.371966
\(913\) 8.53675e36 0.842186
\(914\) −1.30828e37 −1.27313
\(915\) 0 0
\(916\) 4.90714e35 0.0464660
\(917\) 1.38022e37 1.28923
\(918\) −1.24687e36 −0.114892
\(919\) −1.52693e36 −0.138795 −0.0693977 0.997589i \(-0.522108\pi\)
−0.0693977 + 0.997589i \(0.522108\pi\)
\(920\) 0 0
\(921\) −2.51192e37 −2.22208
\(922\) −8.81109e36 −0.768941
\(923\) −5.44726e36 −0.468983
\(924\) 2.28323e37 1.93933
\(925\) 0 0
\(926\) −2.69165e36 −0.222526
\(927\) −7.18595e36 −0.586121
\(928\) 1.37909e36 0.110980
\(929\) 1.93910e36 0.153959 0.0769793 0.997033i \(-0.475472\pi\)
0.0769793 + 0.997033i \(0.475472\pi\)
\(930\) 0 0
\(931\) 3.56028e37 2.75178
\(932\) 5.03370e36 0.383874
\(933\) −8.83914e34 −0.00665104
\(934\) −7.88887e35 −0.0585705
\(935\) 0 0
\(936\) 5.48097e36 0.396195
\(937\) 1.01514e37 0.724069 0.362035 0.932165i \(-0.382082\pi\)
0.362035 + 0.932165i \(0.382082\pi\)
\(938\) 2.41739e37 1.70142
\(939\) −1.40090e37 −0.972944
\(940\) 0 0
\(941\) 4.58463e36 0.310051 0.155026 0.987910i \(-0.450454\pi\)
0.155026 + 0.987910i \(0.450454\pi\)
\(942\) 3.56087e36 0.237640
\(943\) 3.34405e37 2.20230
\(944\) −1.94121e36 −0.126160
\(945\) 0 0
\(946\) −9.15996e36 −0.579768
\(947\) −2.00468e37 −1.25219 −0.626094 0.779748i \(-0.715347\pi\)
−0.626094 + 0.779748i \(0.715347\pi\)
\(948\) −1.04042e37 −0.641364
\(949\) −9.90099e36 −0.602353
\(950\) 0 0
\(951\) 2.97905e37 1.76531
\(952\) −5.81653e36 −0.340175
\(953\) −2.53894e37 −1.46552 −0.732758 0.680489i \(-0.761767\pi\)
−0.732758 + 0.680489i \(0.761767\pi\)
\(954\) −9.96348e36 −0.567619
\(955\) 0 0
\(956\) −1.27889e37 −0.709758
\(957\) 2.39114e37 1.30980
\(958\) −1.60018e37 −0.865168
\(959\) 2.86942e37 1.53131
\(960\) 0 0
\(961\) −8.08873e36 −0.420570
\(962\) −2.53688e37 −1.30200
\(963\) −1.48010e37 −0.749827
\(964\) −9.54557e36 −0.477352
\(965\) 0 0
\(966\) −4.32715e37 −2.10857
\(967\) 3.53299e37 1.69946 0.849731 0.527216i \(-0.176764\pi\)
0.849731 + 0.527216i \(0.176764\pi\)
\(968\) −1.09312e37 −0.519072
\(969\) −1.64272e37 −0.770043
\(970\) 0 0
\(971\) −2.42162e37 −1.10628 −0.553141 0.833088i \(-0.686571\pi\)
−0.553141 + 0.833088i \(0.686571\pi\)
\(972\) −1.39002e37 −0.626894
\(973\) 1.78285e37 0.793789
\(974\) −1.71082e37 −0.751998
\(975\) 0 0
\(976\) 4.35180e36 0.186443
\(977\) −2.74750e36 −0.116213 −0.0581066 0.998310i \(-0.518506\pi\)
−0.0581066 + 0.998310i \(0.518506\pi\)
\(978\) 2.46718e37 1.03030
\(979\) 1.88669e37 0.777887
\(980\) 0 0
\(981\) 2.53887e37 1.02042
\(982\) 5.75695e36 0.228454
\(983\) 2.03235e37 0.796303 0.398152 0.917320i \(-0.369652\pi\)
0.398152 + 0.917320i \(0.369652\pi\)
\(984\) 2.21290e37 0.856099
\(985\) 0 0
\(986\) −6.09141e36 −0.229751
\(987\) −5.85802e37 −2.18166
\(988\) −2.23555e37 −0.822096
\(989\) 1.73598e37 0.630363
\(990\) 0 0
\(991\) 1.72765e37 0.611695 0.305847 0.952081i \(-0.401060\pi\)
0.305847 + 0.952081i \(0.401060\pi\)
\(992\) −3.84877e36 −0.134563
\(993\) −4.41396e37 −1.52392
\(994\) 1.23219e37 0.420095
\(995\) 0 0
\(996\) −1.07061e37 −0.355952
\(997\) −7.90582e36 −0.259572 −0.129786 0.991542i \(-0.541429\pi\)
−0.129786 + 0.991542i \(0.541429\pi\)
\(998\) −3.56177e36 −0.115487
\(999\) 1.23015e37 0.393903
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 50.26.a.i.1.5 5
5.2 odd 4 50.26.b.h.49.1 10
5.3 odd 4 50.26.b.h.49.10 10
5.4 even 2 50.26.a.j.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.26.a.i.1.5 5 1.1 even 1 trivial
50.26.a.j.1.1 yes 5 5.4 even 2
50.26.b.h.49.1 10 5.2 odd 4
50.26.b.h.49.10 10 5.3 odd 4