Properties

Label 338.10.a.a.1.1
Level $338$
Weight $10$
Character 338.1
Self dual yes
Analytic conductor $174.082$
Analytic rank $1$
Dimension $1$
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] = [338,10,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(174.082112623\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.0000 q^{2} -156.000 q^{3} +256.000 q^{4} -870.000 q^{5} +2496.00 q^{6} +952.000 q^{7} -4096.00 q^{8} +4653.00 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} -156.000 q^{3} +256.000 q^{4} -870.000 q^{5} +2496.00 q^{6} +952.000 q^{7} -4096.00 q^{8} +4653.00 q^{9} +13920.0 q^{10} +56148.0 q^{11} -39936.0 q^{12} -15232.0 q^{14} +135720. q^{15} +65536.0 q^{16} -247662. q^{17} -74448.0 q^{18} -315380. q^{19} -222720. q^{20} -148512. q^{21} -898368. q^{22} +204504. q^{23} +638976. q^{24} -1.19622e6 q^{25} +2.34468e6 q^{27} +243712. q^{28} -3.84045e6 q^{29} -2.17152e6 q^{30} +1.30941e6 q^{31} -1.04858e6 q^{32} -8.75909e6 q^{33} +3.96259e6 q^{34} -828240. q^{35} +1.19117e6 q^{36} -4.30708e6 q^{37} +5.04608e6 q^{38} +3.56352e6 q^{40} -1.51204e6 q^{41} +2.37619e6 q^{42} +3.36706e7 q^{43} +1.43739e7 q^{44} -4.04811e6 q^{45} -3.27206e6 q^{46} +1.05811e7 q^{47} -1.02236e7 q^{48} -3.94473e7 q^{49} +1.91396e7 q^{50} +3.86353e7 q^{51} +1.66162e7 q^{53} -3.75149e7 q^{54} -4.88488e7 q^{55} -3.89939e6 q^{56} +4.91993e7 q^{57} +6.14472e7 q^{58} -1.12235e8 q^{59} +3.47443e7 q^{60} -3.31972e7 q^{61} -2.09505e7 q^{62} +4.42966e6 q^{63} +1.67772e7 q^{64} +1.40145e8 q^{66} +1.21372e8 q^{67} -6.34015e7 q^{68} -3.19026e7 q^{69} +1.32518e7 q^{70} +3.87173e8 q^{71} -1.90587e7 q^{72} -2.55240e8 q^{73} +6.89132e7 q^{74} +1.86611e8 q^{75} -8.07373e7 q^{76} +5.34529e7 q^{77} +4.92102e8 q^{79} -5.70163e7 q^{80} -4.57355e8 q^{81} +2.41927e7 q^{82} +4.57420e8 q^{83} -3.80191e7 q^{84} +2.15466e8 q^{85} -5.38730e8 q^{86} +5.99110e8 q^{87} -2.29982e8 q^{88} +3.18095e7 q^{89} +6.47698e7 q^{90} +5.23530e7 q^{92} -2.04268e8 q^{93} -1.69297e8 q^{94} +2.74381e8 q^{95} +1.63578e8 q^{96} +6.73532e8 q^{97} +6.31157e8 q^{98} +2.61257e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) −156.000 −1.11193 −0.555967 0.831204i \(-0.687652\pi\)
−0.555967 + 0.831204i \(0.687652\pi\)
\(4\) 256.000 0.500000
\(5\) −870.000 −0.622521 −0.311261 0.950325i \(-0.600751\pi\)
−0.311261 + 0.950325i \(0.600751\pi\)
\(6\) 2496.00 0.786256
\(7\) 952.000 0.149863 0.0749317 0.997189i \(-0.476126\pi\)
0.0749317 + 0.997189i \(0.476126\pi\)
\(8\) −4096.00 −0.353553
\(9\) 4653.00 0.236397
\(10\) 13920.0 0.440189
\(11\) 56148.0 1.15629 0.578146 0.815934i \(-0.303777\pi\)
0.578146 + 0.815934i \(0.303777\pi\)
\(12\) −39936.0 −0.555967
\(13\) 0 0
\(14\) −15232.0 −0.105969
\(15\) 135720. 0.692203
\(16\) 65536.0 0.250000
\(17\) −247662. −0.719183 −0.359591 0.933110i \(-0.617084\pi\)
−0.359591 + 0.933110i \(0.617084\pi\)
\(18\) −74448.0 −0.167158
\(19\) −315380. −0.555192 −0.277596 0.960698i \(-0.589538\pi\)
−0.277596 + 0.960698i \(0.589538\pi\)
\(20\) −222720. −0.311261
\(21\) −148512. −0.166638
\(22\) −898368. −0.817621
\(23\) 204504. 0.152380 0.0761898 0.997093i \(-0.475725\pi\)
0.0761898 + 0.997093i \(0.475725\pi\)
\(24\) 638976. 0.393128
\(25\) −1.19622e6 −0.612467
\(26\) 0 0
\(27\) 2.34468e6 0.849076
\(28\) 243712. 0.0749317
\(29\) −3.84045e6 −1.00830 −0.504152 0.863615i \(-0.668195\pi\)
−0.504152 + 0.863615i \(0.668195\pi\)
\(30\) −2.17152e6 −0.489461
\(31\) 1.30941e6 0.254652 0.127326 0.991861i \(-0.459361\pi\)
0.127326 + 0.991861i \(0.459361\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) −8.75909e6 −1.28572
\(34\) 3.96259e6 0.508539
\(35\) −828240. −0.0932932
\(36\) 1.19117e6 0.118198
\(37\) −4.30708e6 −0.377811 −0.188906 0.981995i \(-0.560494\pi\)
−0.188906 + 0.981995i \(0.560494\pi\)
\(38\) 5.04608e6 0.392580
\(39\) 0 0
\(40\) 3.56352e6 0.220095
\(41\) −1.51204e6 −0.0835673 −0.0417837 0.999127i \(-0.513304\pi\)
−0.0417837 + 0.999127i \(0.513304\pi\)
\(42\) 2.37619e6 0.117831
\(43\) 3.36706e7 1.50191 0.750953 0.660355i \(-0.229594\pi\)
0.750953 + 0.660355i \(0.229594\pi\)
\(44\) 1.43739e7 0.578146
\(45\) −4.04811e6 −0.147162
\(46\) −3.27206e6 −0.107749
\(47\) 1.05811e7 0.316293 0.158146 0.987416i \(-0.449448\pi\)
0.158146 + 0.987416i \(0.449448\pi\)
\(48\) −1.02236e7 −0.277983
\(49\) −3.94473e7 −0.977541
\(50\) 1.91396e7 0.433080
\(51\) 3.86353e7 0.799684
\(52\) 0 0
\(53\) 1.66162e7 0.289262 0.144631 0.989486i \(-0.453801\pi\)
0.144631 + 0.989486i \(0.453801\pi\)
\(54\) −3.75149e7 −0.600388
\(55\) −4.88488e7 −0.719816
\(56\) −3.89939e6 −0.0529847
\(57\) 4.91993e7 0.617336
\(58\) 6.14472e7 0.712978
\(59\) −1.12235e8 −1.20585 −0.602927 0.797796i \(-0.705999\pi\)
−0.602927 + 0.797796i \(0.705999\pi\)
\(60\) 3.47443e7 0.346101
\(61\) −3.31972e7 −0.306985 −0.153493 0.988150i \(-0.549052\pi\)
−0.153493 + 0.988150i \(0.549052\pi\)
\(62\) −2.09505e7 −0.180066
\(63\) 4.42966e6 0.0354273
\(64\) 1.67772e7 0.125000
\(65\) 0 0
\(66\) 1.40145e8 0.909141
\(67\) 1.21372e8 0.735839 0.367919 0.929858i \(-0.380070\pi\)
0.367919 + 0.929858i \(0.380070\pi\)
\(68\) −6.34015e7 −0.359591
\(69\) −3.19026e7 −0.169436
\(70\) 1.32518e7 0.0659683
\(71\) 3.87173e8 1.80818 0.904091 0.427340i \(-0.140549\pi\)
0.904091 + 0.427340i \(0.140549\pi\)
\(72\) −1.90587e7 −0.0835789
\(73\) −2.55240e8 −1.05195 −0.525976 0.850499i \(-0.676300\pi\)
−0.525976 + 0.850499i \(0.676300\pi\)
\(74\) 6.89132e7 0.267153
\(75\) 1.86611e8 0.681023
\(76\) −8.07373e7 −0.277596
\(77\) 5.34529e7 0.173286
\(78\) 0 0
\(79\) 4.92102e8 1.42145 0.710727 0.703467i \(-0.248366\pi\)
0.710727 + 0.703467i \(0.248366\pi\)
\(80\) −5.70163e7 −0.155630
\(81\) −4.57355e8 −1.18051
\(82\) 2.41927e7 0.0590910
\(83\) 4.57420e8 1.05795 0.528974 0.848638i \(-0.322577\pi\)
0.528974 + 0.848638i \(0.322577\pi\)
\(84\) −3.80191e7 −0.0833191
\(85\) 2.15466e8 0.447707
\(86\) −5.38730e8 −1.06201
\(87\) 5.99110e8 1.12117
\(88\) −2.29982e8 −0.408811
\(89\) 3.18095e7 0.0537405 0.0268703 0.999639i \(-0.491446\pi\)
0.0268703 + 0.999639i \(0.491446\pi\)
\(90\) 6.47698e7 0.104059
\(91\) 0 0
\(92\) 5.23530e7 0.0761898
\(93\) −2.04268e8 −0.283156
\(94\) −1.69297e8 −0.223653
\(95\) 2.74381e8 0.345619
\(96\) 1.63578e8 0.196564
\(97\) 6.73532e8 0.772477 0.386238 0.922399i \(-0.373774\pi\)
0.386238 + 0.922399i \(0.373774\pi\)
\(98\) 6.31157e8 0.691226
\(99\) 2.61257e8 0.273344
\(100\) −3.06234e8 −0.306234
\(101\) 1.05772e9 1.01140 0.505701 0.862709i \(-0.331234\pi\)
0.505701 + 0.862709i \(0.331234\pi\)
\(102\) −6.18164e8 −0.565462
\(103\) 7.95866e8 0.696743 0.348371 0.937357i \(-0.386735\pi\)
0.348371 + 0.937357i \(0.386735\pi\)
\(104\) 0 0
\(105\) 1.29205e8 0.103736
\(106\) −2.65859e8 −0.204539
\(107\) −1.97413e9 −1.45596 −0.727981 0.685598i \(-0.759541\pi\)
−0.727981 + 0.685598i \(0.759541\pi\)
\(108\) 6.00238e8 0.424538
\(109\) 1.34465e9 0.912408 0.456204 0.889875i \(-0.349209\pi\)
0.456204 + 0.889875i \(0.349209\pi\)
\(110\) 7.81580e8 0.508987
\(111\) 6.71904e8 0.420101
\(112\) 6.23903e7 0.0374659
\(113\) 2.70680e9 1.56172 0.780861 0.624705i \(-0.214781\pi\)
0.780861 + 0.624705i \(0.214781\pi\)
\(114\) −7.87188e8 −0.436523
\(115\) −1.77918e8 −0.0948595
\(116\) −9.83155e8 −0.504152
\(117\) 0 0
\(118\) 1.79576e9 0.852667
\(119\) −2.35774e8 −0.107779
\(120\) −5.55909e8 −0.244731
\(121\) 7.94650e8 0.337009
\(122\) 5.31155e8 0.217071
\(123\) 2.35879e8 0.0929213
\(124\) 3.35208e8 0.127326
\(125\) 2.73993e9 1.00380
\(126\) −7.08745e7 −0.0250509
\(127\) 1.19960e9 0.409185 0.204593 0.978847i \(-0.434413\pi\)
0.204593 + 0.978847i \(0.434413\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −5.25261e9 −1.67002
\(130\) 0 0
\(131\) 2.78615e9 0.826576 0.413288 0.910600i \(-0.364380\pi\)
0.413288 + 0.910600i \(0.364380\pi\)
\(132\) −2.24233e9 −0.642860
\(133\) −3.00242e8 −0.0832029
\(134\) −1.94196e9 −0.520317
\(135\) −2.03987e9 −0.528568
\(136\) 1.01442e9 0.254269
\(137\) −2.88233e9 −0.699039 −0.349519 0.936929i \(-0.613655\pi\)
−0.349519 + 0.936929i \(0.613655\pi\)
\(138\) 5.10442e8 0.119809
\(139\) 2.15641e9 0.489965 0.244982 0.969528i \(-0.421218\pi\)
0.244982 + 0.969528i \(0.421218\pi\)
\(140\) −2.12029e8 −0.0466466
\(141\) −1.65065e9 −0.351697
\(142\) −6.19476e9 −1.27858
\(143\) 0 0
\(144\) 3.04939e8 0.0590992
\(145\) 3.34119e9 0.627690
\(146\) 4.08384e9 0.743843
\(147\) 6.15378e9 1.08696
\(148\) −1.10261e9 −0.188906
\(149\) −7.54548e9 −1.25415 −0.627074 0.778960i \(-0.715747\pi\)
−0.627074 + 0.778960i \(0.715747\pi\)
\(150\) −2.98578e9 −0.481556
\(151\) 4.31308e9 0.675136 0.337568 0.941301i \(-0.390396\pi\)
0.337568 + 0.941301i \(0.390396\pi\)
\(152\) 1.29180e9 0.196290
\(153\) −1.15237e9 −0.170013
\(154\) −8.55246e8 −0.122532
\(155\) −1.13918e9 −0.158526
\(156\) 0 0
\(157\) −4.23157e9 −0.555845 −0.277922 0.960604i \(-0.589646\pi\)
−0.277922 + 0.960604i \(0.589646\pi\)
\(158\) −7.87363e9 −1.00512
\(159\) −2.59213e9 −0.321640
\(160\) 9.12261e8 0.110047
\(161\) 1.94688e8 0.0228361
\(162\) 7.31768e9 0.834749
\(163\) −8.28448e8 −0.0919223 −0.0459612 0.998943i \(-0.514635\pi\)
−0.0459612 + 0.998943i \(0.514635\pi\)
\(164\) −3.87083e8 −0.0417837
\(165\) 7.62041e9 0.800388
\(166\) −7.31872e9 −0.748082
\(167\) 2.85500e9 0.284041 0.142021 0.989864i \(-0.454640\pi\)
0.142021 + 0.989864i \(0.454640\pi\)
\(168\) 6.08305e8 0.0589155
\(169\) 0 0
\(170\) −3.44746e9 −0.316576
\(171\) −1.46746e9 −0.131246
\(172\) 8.61967e9 0.750953
\(173\) −1.76690e10 −1.49970 −0.749851 0.661607i \(-0.769875\pi\)
−0.749851 + 0.661607i \(0.769875\pi\)
\(174\) −9.58576e9 −0.792784
\(175\) −1.13881e9 −0.0917865
\(176\) 3.67972e9 0.289073
\(177\) 1.75087e10 1.34083
\(178\) −5.08952e8 −0.0380003
\(179\) −5.86732e8 −0.0427170 −0.0213585 0.999772i \(-0.506799\pi\)
−0.0213585 + 0.999772i \(0.506799\pi\)
\(180\) −1.03632e9 −0.0735811
\(181\) −5.43396e9 −0.376325 −0.188162 0.982138i \(-0.560253\pi\)
−0.188162 + 0.982138i \(0.560253\pi\)
\(182\) 0 0
\(183\) 5.17877e9 0.341347
\(184\) −8.37648e8 −0.0538743
\(185\) 3.74716e9 0.235196
\(186\) 3.26828e9 0.200222
\(187\) −1.39057e10 −0.831585
\(188\) 2.70875e9 0.158146
\(189\) 2.23214e9 0.127245
\(190\) −4.39009e9 −0.244389
\(191\) 3.23292e10 1.75770 0.878851 0.477096i \(-0.158311\pi\)
0.878851 + 0.477096i \(0.158311\pi\)
\(192\) −2.61725e9 −0.138992
\(193\) 1.29399e10 0.671311 0.335655 0.941985i \(-0.391042\pi\)
0.335655 + 0.941985i \(0.391042\pi\)
\(194\) −1.07765e10 −0.546224
\(195\) 0 0
\(196\) −1.00985e10 −0.488770
\(197\) −8.81090e9 −0.416795 −0.208397 0.978044i \(-0.566825\pi\)
−0.208397 + 0.978044i \(0.566825\pi\)
\(198\) −4.18011e9 −0.193283
\(199\) −2.48534e10 −1.12343 −0.561716 0.827330i \(-0.689859\pi\)
−0.561716 + 0.827330i \(0.689859\pi\)
\(200\) 4.89974e9 0.216540
\(201\) −1.89341e10 −0.818204
\(202\) −1.69235e10 −0.715170
\(203\) −3.65611e9 −0.151108
\(204\) 9.89063e9 0.399842
\(205\) 1.31548e9 0.0520224
\(206\) −1.27339e10 −0.492672
\(207\) 9.51557e8 0.0360220
\(208\) 0 0
\(209\) −1.77080e10 −0.641963
\(210\) −2.06729e9 −0.0733523
\(211\) −4.65163e10 −1.61560 −0.807801 0.589456i \(-0.799342\pi\)
−0.807801 + 0.589456i \(0.799342\pi\)
\(212\) 4.25375e9 0.144631
\(213\) −6.03989e10 −2.01058
\(214\) 3.15862e10 1.02952
\(215\) −2.92934e10 −0.934969
\(216\) −9.60381e9 −0.300194
\(217\) 1.24656e9 0.0381631
\(218\) −2.15144e10 −0.645170
\(219\) 3.98175e10 1.16970
\(220\) −1.25053e10 −0.359908
\(221\) 0 0
\(222\) −1.07505e10 −0.297056
\(223\) −4.66347e10 −1.26281 −0.631404 0.775454i \(-0.717521\pi\)
−0.631404 + 0.775454i \(0.717521\pi\)
\(224\) −9.98244e8 −0.0264924
\(225\) −5.56603e9 −0.144785
\(226\) −4.33088e10 −1.10430
\(227\) −2.65867e10 −0.664582 −0.332291 0.943177i \(-0.607822\pi\)
−0.332291 + 0.943177i \(0.607822\pi\)
\(228\) 1.25950e10 0.308668
\(229\) −3.99907e10 −0.960947 −0.480474 0.877009i \(-0.659535\pi\)
−0.480474 + 0.877009i \(0.659535\pi\)
\(230\) 2.84670e9 0.0670758
\(231\) −8.33865e9 −0.192682
\(232\) 1.57305e10 0.356489
\(233\) 6.53338e9 0.145223 0.0726116 0.997360i \(-0.476867\pi\)
0.0726116 + 0.997360i \(0.476867\pi\)
\(234\) 0 0
\(235\) −9.20553e9 −0.196899
\(236\) −2.87322e10 −0.602927
\(237\) −7.67679e10 −1.58056
\(238\) 3.77239e9 0.0762114
\(239\) 5.66773e10 1.12362 0.561809 0.827267i \(-0.310106\pi\)
0.561809 + 0.827267i \(0.310106\pi\)
\(240\) 8.89455e9 0.173051
\(241\) −4.61491e9 −0.0881225 −0.0440613 0.999029i \(-0.514030\pi\)
−0.0440613 + 0.999029i \(0.514030\pi\)
\(242\) −1.27144e10 −0.238302
\(243\) 2.51971e10 0.463577
\(244\) −8.49849e9 −0.153493
\(245\) 3.43192e10 0.608540
\(246\) −3.77406e9 −0.0657053
\(247\) 0 0
\(248\) −5.36334e9 −0.0900331
\(249\) −7.13576e10 −1.17637
\(250\) −4.38390e10 −0.709790
\(251\) 6.80194e10 1.08169 0.540843 0.841124i \(-0.318105\pi\)
0.540843 + 0.841124i \(0.318105\pi\)
\(252\) 1.13399e9 0.0177136
\(253\) 1.14825e10 0.176195
\(254\) −1.91936e10 −0.289338
\(255\) −3.36127e10 −0.497820
\(256\) 4.29497e9 0.0625000
\(257\) −9.35958e10 −1.33831 −0.669156 0.743122i \(-0.733344\pi\)
−0.669156 + 0.743122i \(0.733344\pi\)
\(258\) 8.40418e10 1.18088
\(259\) −4.10034e9 −0.0566201
\(260\) 0 0
\(261\) −1.78696e10 −0.238360
\(262\) −4.45783e10 −0.584478
\(263\) −9.40401e10 −1.21203 −0.606013 0.795454i \(-0.707232\pi\)
−0.606013 + 0.795454i \(0.707232\pi\)
\(264\) 3.58772e10 0.454570
\(265\) −1.44561e10 −0.180071
\(266\) 4.80387e9 0.0588334
\(267\) −4.96228e9 −0.0597559
\(268\) 3.10713e10 0.367919
\(269\) 1.22724e11 1.42904 0.714522 0.699613i \(-0.246644\pi\)
0.714522 + 0.699613i \(0.246644\pi\)
\(270\) 3.26379e10 0.373754
\(271\) 1.64257e11 1.84996 0.924982 0.380012i \(-0.124080\pi\)
0.924982 + 0.380012i \(0.124080\pi\)
\(272\) −1.62308e10 −0.179796
\(273\) 0 0
\(274\) 4.61173e10 0.494295
\(275\) −6.71656e10 −0.708190
\(276\) −8.16707e9 −0.0847180
\(277\) 6.50134e10 0.663505 0.331752 0.943367i \(-0.392360\pi\)
0.331752 + 0.943367i \(0.392360\pi\)
\(278\) −3.45026e10 −0.346458
\(279\) 6.09268e9 0.0601990
\(280\) 3.39247e9 0.0329841
\(281\) −5.20964e10 −0.498459 −0.249230 0.968444i \(-0.580177\pi\)
−0.249230 + 0.968444i \(0.580177\pi\)
\(282\) 2.64104e10 0.248687
\(283\) −9.06992e10 −0.840552 −0.420276 0.907396i \(-0.638067\pi\)
−0.420276 + 0.907396i \(0.638067\pi\)
\(284\) 9.91162e10 0.904091
\(285\) −4.28034e10 −0.384305
\(286\) 0 0
\(287\) −1.43946e9 −0.0125237
\(288\) −4.87902e9 −0.0417895
\(289\) −5.72514e10 −0.482776
\(290\) −5.34591e10 −0.443844
\(291\) −1.05071e11 −0.858943
\(292\) −6.53415e10 −0.525976
\(293\) −7.25569e10 −0.575141 −0.287571 0.957759i \(-0.592848\pi\)
−0.287571 + 0.957759i \(0.592848\pi\)
\(294\) −9.84605e10 −0.768597
\(295\) 9.76445e10 0.750670
\(296\) 1.76418e10 0.133576
\(297\) 1.31649e11 0.981779
\(298\) 1.20728e11 0.886816
\(299\) 0 0
\(300\) 4.77724e10 0.340512
\(301\) 3.20544e10 0.225081
\(302\) −6.90093e10 −0.477394
\(303\) −1.65004e11 −1.12461
\(304\) −2.06687e10 −0.138798
\(305\) 2.88816e10 0.191105
\(306\) 1.84379e10 0.120217
\(307\) −1.81977e11 −1.16921 −0.584607 0.811317i \(-0.698751\pi\)
−0.584607 + 0.811317i \(0.698751\pi\)
\(308\) 1.36839e10 0.0866429
\(309\) −1.24155e11 −0.774732
\(310\) 1.82270e10 0.112095
\(311\) −8.98295e10 −0.544499 −0.272250 0.962227i \(-0.587768\pi\)
−0.272250 + 0.962227i \(0.587768\pi\)
\(312\) 0 0
\(313\) 5.51394e9 0.0324722 0.0162361 0.999868i \(-0.494832\pi\)
0.0162361 + 0.999868i \(0.494832\pi\)
\(314\) 6.77052e10 0.393041
\(315\) −3.85380e9 −0.0220542
\(316\) 1.25978e11 0.710727
\(317\) 1.94806e10 0.108351 0.0541757 0.998531i \(-0.482747\pi\)
0.0541757 + 0.998531i \(0.482747\pi\)
\(318\) 4.14741e10 0.227434
\(319\) −2.15634e11 −1.16589
\(320\) −1.45962e10 −0.0778152
\(321\) 3.07965e11 1.61893
\(322\) −3.11500e9 −0.0161476
\(323\) 7.81076e10 0.399284
\(324\) −1.17083e11 −0.590257
\(325\) 0 0
\(326\) 1.32552e10 0.0649989
\(327\) −2.09765e11 −1.01454
\(328\) 6.19332e9 0.0295455
\(329\) 1.00732e10 0.0474007
\(330\) −1.21927e11 −0.565960
\(331\) 1.06801e11 0.489046 0.244523 0.969644i \(-0.421369\pi\)
0.244523 + 0.969644i \(0.421369\pi\)
\(332\) 1.17100e11 0.528974
\(333\) −2.00408e10 −0.0893134
\(334\) −4.56800e10 −0.200848
\(335\) −1.05594e11 −0.458075
\(336\) −9.73288e9 −0.0416596
\(337\) −1.75776e11 −0.742380 −0.371190 0.928557i \(-0.621050\pi\)
−0.371190 + 0.928557i \(0.621050\pi\)
\(338\) 0 0
\(339\) −4.22261e11 −1.73653
\(340\) 5.51593e10 0.223853
\(341\) 7.35206e10 0.294452
\(342\) 2.34794e10 0.0928046
\(343\) −7.59705e10 −0.296361
\(344\) −1.37915e11 −0.531004
\(345\) 2.77553e10 0.105477
\(346\) 2.82704e11 1.06045
\(347\) 8.33026e10 0.308444 0.154222 0.988036i \(-0.450713\pi\)
0.154222 + 0.988036i \(0.450713\pi\)
\(348\) 1.53372e11 0.560583
\(349\) −3.21368e11 −1.15955 −0.579773 0.814778i \(-0.696859\pi\)
−0.579773 + 0.814778i \(0.696859\pi\)
\(350\) 1.82209e10 0.0649028
\(351\) 0 0
\(352\) −5.88754e10 −0.204405
\(353\) −4.07688e11 −1.39747 −0.698735 0.715381i \(-0.746253\pi\)
−0.698735 + 0.715381i \(0.746253\pi\)
\(354\) −2.80139e11 −0.948110
\(355\) −3.36840e11 −1.12563
\(356\) 8.14323e9 0.0268703
\(357\) 3.67808e10 0.119843
\(358\) 9.38771e9 0.0302055
\(359\) 5.60079e11 1.77961 0.889804 0.456343i \(-0.150841\pi\)
0.889804 + 0.456343i \(0.150841\pi\)
\(360\) 1.65811e10 0.0520297
\(361\) −2.23223e11 −0.691762
\(362\) 8.69433e10 0.266102
\(363\) −1.23965e11 −0.374732
\(364\) 0 0
\(365\) 2.22059e11 0.654863
\(366\) −8.28603e10 −0.241369
\(367\) 3.76056e11 1.08207 0.541036 0.841000i \(-0.318032\pi\)
0.541036 + 0.841000i \(0.318032\pi\)
\(368\) 1.34024e10 0.0380949
\(369\) −7.03553e9 −0.0197551
\(370\) −5.99545e10 −0.166308
\(371\) 1.58186e10 0.0433497
\(372\) −5.22925e10 −0.141578
\(373\) −8.12245e10 −0.217269 −0.108634 0.994082i \(-0.534648\pi\)
−0.108634 + 0.994082i \(0.534648\pi\)
\(374\) 2.22492e11 0.588019
\(375\) −4.27430e11 −1.11615
\(376\) −4.33401e10 −0.111826
\(377\) 0 0
\(378\) −3.57142e10 −0.0899762
\(379\) −2.02729e11 −0.504708 −0.252354 0.967635i \(-0.581205\pi\)
−0.252354 + 0.967635i \(0.581205\pi\)
\(380\) 7.02414e10 0.172809
\(381\) −1.87138e11 −0.454987
\(382\) −5.17268e11 −1.24288
\(383\) 4.76816e10 0.113229 0.0566143 0.998396i \(-0.481969\pi\)
0.0566143 + 0.998396i \(0.481969\pi\)
\(384\) 4.18759e10 0.0982820
\(385\) −4.65040e10 −0.107874
\(386\) −2.07039e11 −0.474688
\(387\) 1.56669e11 0.355046
\(388\) 1.72424e11 0.386238
\(389\) −1.89795e11 −0.420253 −0.210126 0.977674i \(-0.567388\pi\)
−0.210126 + 0.977674i \(0.567388\pi\)
\(390\) 0 0
\(391\) −5.06479e10 −0.109589
\(392\) 1.61576e11 0.345613
\(393\) −4.34639e11 −0.919098
\(394\) 1.40974e11 0.294718
\(395\) −4.28129e11 −0.884886
\(396\) 6.68817e10 0.136672
\(397\) −4.00237e11 −0.808649 −0.404325 0.914616i \(-0.632493\pi\)
−0.404325 + 0.914616i \(0.632493\pi\)
\(398\) 3.97654e11 0.794387
\(399\) 4.68377e10 0.0925162
\(400\) −7.83958e10 −0.153117
\(401\) −8.76186e11 −1.69218 −0.846090 0.533040i \(-0.821049\pi\)
−0.846090 + 0.533040i \(0.821049\pi\)
\(402\) 3.02945e11 0.578558
\(403\) 0 0
\(404\) 2.70776e11 0.505701
\(405\) 3.97899e11 0.734895
\(406\) 5.84977e10 0.106849
\(407\) −2.41834e11 −0.436860
\(408\) −1.58250e11 −0.282731
\(409\) −5.72300e11 −1.01127 −0.505637 0.862746i \(-0.668742\pi\)
−0.505637 + 0.862746i \(0.668742\pi\)
\(410\) −2.10476e10 −0.0367854
\(411\) 4.49643e11 0.777285
\(412\) 2.03742e11 0.348371
\(413\) −1.06848e11 −0.180713
\(414\) −1.52249e10 −0.0254714
\(415\) −3.97956e11 −0.658595
\(416\) 0 0
\(417\) −3.36400e11 −0.544809
\(418\) 2.83327e11 0.453937
\(419\) 4.16693e11 0.660469 0.330235 0.943899i \(-0.392872\pi\)
0.330235 + 0.943899i \(0.392872\pi\)
\(420\) 3.30766e10 0.0518679
\(421\) 1.19043e12 1.84687 0.923434 0.383757i \(-0.125370\pi\)
0.923434 + 0.383757i \(0.125370\pi\)
\(422\) 7.44261e11 1.14240
\(423\) 4.92337e10 0.0747706
\(424\) −6.80600e10 −0.102269
\(425\) 2.96259e11 0.440476
\(426\) 9.66383e11 1.42169
\(427\) −3.16038e10 −0.0460059
\(428\) −5.05379e11 −0.727981
\(429\) 0 0
\(430\) 4.68695e11 0.661123
\(431\) 4.36455e11 0.609245 0.304622 0.952473i \(-0.401470\pi\)
0.304622 + 0.952473i \(0.401470\pi\)
\(432\) 1.53661e11 0.212269
\(433\) 4.48430e11 0.613055 0.306527 0.951862i \(-0.400833\pi\)
0.306527 + 0.951862i \(0.400833\pi\)
\(434\) −1.99449e10 −0.0269854
\(435\) −5.21226e11 −0.697950
\(436\) 3.44230e11 0.456204
\(437\) −6.44965e10 −0.0845998
\(438\) −6.37079e11 −0.827104
\(439\) −6.59703e11 −0.847731 −0.423866 0.905725i \(-0.639327\pi\)
−0.423866 + 0.905725i \(0.639327\pi\)
\(440\) 2.00085e11 0.254493
\(441\) −1.83548e11 −0.231088
\(442\) 0 0
\(443\) 9.48507e11 1.17010 0.585051 0.810997i \(-0.301075\pi\)
0.585051 + 0.810997i \(0.301075\pi\)
\(444\) 1.72007e11 0.210051
\(445\) −2.76743e10 −0.0334546
\(446\) 7.46156e11 0.892941
\(447\) 1.17709e12 1.39453
\(448\) 1.59719e10 0.0187329
\(449\) 6.11763e11 0.710354 0.355177 0.934799i \(-0.384421\pi\)
0.355177 + 0.934799i \(0.384421\pi\)
\(450\) 8.90566e10 0.102379
\(451\) −8.48981e10 −0.0966282
\(452\) 6.92941e11 0.780861
\(453\) −6.72841e11 −0.750707
\(454\) 4.25388e11 0.469930
\(455\) 0 0
\(456\) −2.01520e11 −0.218261
\(457\) 3.79033e11 0.406494 0.203247 0.979128i \(-0.434851\pi\)
0.203247 + 0.979128i \(0.434851\pi\)
\(458\) 6.39852e11 0.679492
\(459\) −5.80688e11 −0.610641
\(460\) −4.55471e10 −0.0474297
\(461\) 8.90062e11 0.917839 0.458919 0.888478i \(-0.348237\pi\)
0.458919 + 0.888478i \(0.348237\pi\)
\(462\) 1.33418e11 0.136247
\(463\) −1.32852e12 −1.34355 −0.671776 0.740755i \(-0.734468\pi\)
−0.671776 + 0.740755i \(0.734468\pi\)
\(464\) −2.51688e11 −0.252076
\(465\) 1.77713e11 0.176271
\(466\) −1.04534e11 −0.102688
\(467\) 1.65638e12 1.61151 0.805755 0.592249i \(-0.201760\pi\)
0.805755 + 0.592249i \(0.201760\pi\)
\(468\) 0 0
\(469\) 1.15546e11 0.110275
\(470\) 1.47289e11 0.139229
\(471\) 6.60125e11 0.618062
\(472\) 4.59715e11 0.426334
\(473\) 1.89054e12 1.73664
\(474\) 1.22829e12 1.11763
\(475\) 3.77265e11 0.340037
\(476\) −6.03582e10 −0.0538896
\(477\) 7.73152e10 0.0683805
\(478\) −9.06837e11 −0.794518
\(479\) 1.07973e12 0.937143 0.468571 0.883426i \(-0.344769\pi\)
0.468571 + 0.883426i \(0.344769\pi\)
\(480\) −1.42313e11 −0.122365
\(481\) 0 0
\(482\) 7.38386e10 0.0623120
\(483\) −3.03713e10 −0.0253923
\(484\) 2.03430e11 0.168505
\(485\) −5.85973e11 −0.480883
\(486\) −4.03153e11 −0.327798
\(487\) −1.60549e12 −1.29338 −0.646690 0.762753i \(-0.723847\pi\)
−0.646690 + 0.762753i \(0.723847\pi\)
\(488\) 1.35976e11 0.108536
\(489\) 1.29238e11 0.102212
\(490\) −5.49106e11 −0.430303
\(491\) 7.93629e11 0.616242 0.308121 0.951347i \(-0.400300\pi\)
0.308121 + 0.951347i \(0.400300\pi\)
\(492\) 6.03849e10 0.0464607
\(493\) 9.51134e11 0.725154
\(494\) 0 0
\(495\) −2.27293e11 −0.170162
\(496\) 8.58134e10 0.0636630
\(497\) 3.68588e11 0.270980
\(498\) 1.14172e12 0.831817
\(499\) 1.96951e12 1.42202 0.711010 0.703182i \(-0.248238\pi\)
0.711010 + 0.703182i \(0.248238\pi\)
\(500\) 7.01423e11 0.501898
\(501\) −4.45380e11 −0.315835
\(502\) −1.08831e12 −0.764867
\(503\) −5.42230e11 −0.377683 −0.188842 0.982008i \(-0.560473\pi\)
−0.188842 + 0.982008i \(0.560473\pi\)
\(504\) −1.81439e10 −0.0125254
\(505\) −9.20216e11 −0.629620
\(506\) −1.83720e11 −0.124589
\(507\) 0 0
\(508\) 3.07098e11 0.204593
\(509\) 1.69215e12 1.11740 0.558699 0.829370i \(-0.311301\pi\)
0.558699 + 0.829370i \(0.311301\pi\)
\(510\) 5.37803e11 0.352012
\(511\) −2.42989e11 −0.157649
\(512\) −6.87195e10 −0.0441942
\(513\) −7.39465e11 −0.471400
\(514\) 1.49753e12 0.946329
\(515\) −6.92403e11 −0.433737
\(516\) −1.34467e12 −0.835010
\(517\) 5.94106e11 0.365727
\(518\) 6.56054e10 0.0400365
\(519\) 2.75637e12 1.66757
\(520\) 0 0
\(521\) 2.97596e12 1.76953 0.884764 0.466040i \(-0.154320\pi\)
0.884764 + 0.466040i \(0.154320\pi\)
\(522\) 2.85914e11 0.168546
\(523\) −1.07989e12 −0.631137 −0.315568 0.948903i \(-0.602195\pi\)
−0.315568 + 0.948903i \(0.602195\pi\)
\(524\) 7.13253e11 0.413288
\(525\) 1.77654e11 0.102060
\(526\) 1.50464e12 0.857032
\(527\) −3.24291e11 −0.183141
\(528\) −5.74036e11 −0.321430
\(529\) −1.75933e12 −0.976780
\(530\) 2.31298e11 0.127330
\(531\) −5.22230e11 −0.285060
\(532\) −7.68619e10 −0.0416015
\(533\) 0 0
\(534\) 7.93965e10 0.0422538
\(535\) 1.71750e12 0.906367
\(536\) −4.97141e11 −0.260158
\(537\) 9.15302e10 0.0474985
\(538\) −1.96359e12 −1.01049
\(539\) −2.21489e12 −1.13032
\(540\) −5.22207e11 −0.264284
\(541\) 2.02167e12 1.01467 0.507333 0.861750i \(-0.330631\pi\)
0.507333 + 0.861750i \(0.330631\pi\)
\(542\) −2.62812e12 −1.30812
\(543\) 8.47697e11 0.418448
\(544\) 2.59692e11 0.127135
\(545\) −1.16984e12 −0.567994
\(546\) 0 0
\(547\) −2.62612e12 −1.25421 −0.627107 0.778933i \(-0.715761\pi\)
−0.627107 + 0.778933i \(0.715761\pi\)
\(548\) −7.37876e11 −0.349519
\(549\) −1.54467e11 −0.0725703
\(550\) 1.07465e12 0.500766
\(551\) 1.21120e12 0.559802
\(552\) 1.30673e11 0.0599046
\(553\) 4.68481e11 0.213024
\(554\) −1.04021e12 −0.469169
\(555\) −5.84557e11 −0.261522
\(556\) 5.52041e11 0.244982
\(557\) −3.48482e12 −1.53402 −0.767012 0.641633i \(-0.778257\pi\)
−0.767012 + 0.641633i \(0.778257\pi\)
\(558\) −9.74828e10 −0.0425671
\(559\) 0 0
\(560\) −5.42795e10 −0.0233233
\(561\) 2.16929e12 0.924667
\(562\) 8.33543e11 0.352464
\(563\) −2.77091e12 −1.16235 −0.581173 0.813780i \(-0.697406\pi\)
−0.581173 + 0.813780i \(0.697406\pi\)
\(564\) −4.22566e11 −0.175848
\(565\) −2.35492e12 −0.972205
\(566\) 1.45119e12 0.594360
\(567\) −4.35402e11 −0.176916
\(568\) −1.58586e12 −0.639289
\(569\) −6.70382e11 −0.268113 −0.134056 0.990974i \(-0.542800\pi\)
−0.134056 + 0.990974i \(0.542800\pi\)
\(570\) 6.84854e11 0.271745
\(571\) 2.67123e12 1.05160 0.525798 0.850609i \(-0.323767\pi\)
0.525798 + 0.850609i \(0.323767\pi\)
\(572\) 0 0
\(573\) −5.04336e12 −1.95445
\(574\) 2.30314e10 0.00885559
\(575\) −2.44633e11 −0.0933274
\(576\) 7.80644e10 0.0295496
\(577\) 6.59284e11 0.247618 0.123809 0.992306i \(-0.460489\pi\)
0.123809 + 0.992306i \(0.460489\pi\)
\(578\) 9.16023e11 0.341374
\(579\) −2.01863e12 −0.746453
\(580\) 8.55345e11 0.313845
\(581\) 4.35464e11 0.158548
\(582\) 1.68114e12 0.607365
\(583\) 9.32967e11 0.334471
\(584\) 1.04546e12 0.371921
\(585\) 0 0
\(586\) 1.16091e12 0.406686
\(587\) −1.04947e12 −0.364835 −0.182418 0.983221i \(-0.558392\pi\)
−0.182418 + 0.983221i \(0.558392\pi\)
\(588\) 1.57537e12 0.543480
\(589\) −4.12961e11 −0.141381
\(590\) −1.56231e12 −0.530804
\(591\) 1.37450e12 0.463448
\(592\) −2.82269e11 −0.0944528
\(593\) 1.31188e12 0.435662 0.217831 0.975987i \(-0.430102\pi\)
0.217831 + 0.975987i \(0.430102\pi\)
\(594\) −2.10639e12 −0.694223
\(595\) 2.05124e11 0.0670949
\(596\) −1.93164e12 −0.627074
\(597\) 3.87713e12 1.24918
\(598\) 0 0
\(599\) −3.37603e12 −1.07148 −0.535742 0.844382i \(-0.679968\pi\)
−0.535742 + 0.844382i \(0.679968\pi\)
\(600\) −7.64359e11 −0.240778
\(601\) 2.63880e12 0.825034 0.412517 0.910950i \(-0.364650\pi\)
0.412517 + 0.910950i \(0.364650\pi\)
\(602\) −5.12871e11 −0.159156
\(603\) 5.64745e11 0.173950
\(604\) 1.10415e12 0.337568
\(605\) −6.91346e11 −0.209795
\(606\) 2.64007e12 0.795221
\(607\) −5.15939e12 −1.54259 −0.771293 0.636481i \(-0.780390\pi\)
−0.771293 + 0.636481i \(0.780390\pi\)
\(608\) 3.30700e11 0.0981449
\(609\) 5.70353e11 0.168022
\(610\) −4.62105e11 −0.135132
\(611\) 0 0
\(612\) −2.95007e11 −0.0850063
\(613\) 5.37354e11 0.153705 0.0768525 0.997042i \(-0.475513\pi\)
0.0768525 + 0.997042i \(0.475513\pi\)
\(614\) 2.91163e12 0.826759
\(615\) −2.05214e11 −0.0578455
\(616\) −2.18943e11 −0.0612658
\(617\) −4.63358e12 −1.28716 −0.643582 0.765378i \(-0.722552\pi\)
−0.643582 + 0.765378i \(0.722552\pi\)
\(618\) 1.98648e12 0.547818
\(619\) −3.06267e12 −0.838480 −0.419240 0.907876i \(-0.637703\pi\)
−0.419240 + 0.907876i \(0.637703\pi\)
\(620\) −2.91631e11 −0.0792632
\(621\) 4.79496e11 0.129382
\(622\) 1.43727e12 0.385019
\(623\) 3.02827e10 0.00805374
\(624\) 0 0
\(625\) −4.73661e10 −0.0124167
\(626\) −8.82230e10 −0.0229613
\(627\) 2.76244e12 0.713821
\(628\) −1.08328e12 −0.277922
\(629\) 1.06670e12 0.271715
\(630\) 6.16608e10 0.0155947
\(631\) 5.46928e10 0.0137340 0.00686702 0.999976i \(-0.497814\pi\)
0.00686702 + 0.999976i \(0.497814\pi\)
\(632\) −2.01565e12 −0.502560
\(633\) 7.25654e12 1.79644
\(634\) −3.11689e11 −0.0766160
\(635\) −1.04365e12 −0.254727
\(636\) −6.63585e11 −0.160820
\(637\) 0 0
\(638\) 3.45014e12 0.824410
\(639\) 1.80151e12 0.427449
\(640\) 2.33539e11 0.0550236
\(641\) −4.78068e12 −1.11848 −0.559240 0.829006i \(-0.688907\pi\)
−0.559240 + 0.829006i \(0.688907\pi\)
\(642\) −4.92744e12 −1.14476
\(643\) −4.10484e12 −0.946994 −0.473497 0.880795i \(-0.657008\pi\)
−0.473497 + 0.880795i \(0.657008\pi\)
\(644\) 4.98401e10 0.0114181
\(645\) 4.56977e12 1.03962
\(646\) −1.24972e12 −0.282337
\(647\) −5.49263e12 −1.23228 −0.616142 0.787635i \(-0.711305\pi\)
−0.616142 + 0.787635i \(0.711305\pi\)
\(648\) 1.87333e12 0.417375
\(649\) −6.30178e12 −1.39432
\(650\) 0 0
\(651\) −1.94463e11 −0.0424348
\(652\) −2.12083e11 −0.0459612
\(653\) 4.15994e12 0.895320 0.447660 0.894204i \(-0.352258\pi\)
0.447660 + 0.894204i \(0.352258\pi\)
\(654\) 3.35624e12 0.717386
\(655\) −2.42395e12 −0.514561
\(656\) −9.90932e10 −0.0208918
\(657\) −1.18763e12 −0.248678
\(658\) −1.61171e11 −0.0335174
\(659\) 2.15295e12 0.444681 0.222341 0.974969i \(-0.428630\pi\)
0.222341 + 0.974969i \(0.428630\pi\)
\(660\) 1.95082e12 0.400194
\(661\) −8.63978e12 −1.76034 −0.880169 0.474660i \(-0.842571\pi\)
−0.880169 + 0.474660i \(0.842571\pi\)
\(662\) −1.70882e12 −0.345807
\(663\) 0 0
\(664\) −1.87359e12 −0.374041
\(665\) 2.61210e11 0.0517956
\(666\) 3.20653e11 0.0631541
\(667\) −7.85387e11 −0.153645
\(668\) 7.30880e11 0.142021
\(669\) 7.27502e12 1.40416
\(670\) 1.68950e12 0.323908
\(671\) −1.86396e12 −0.354964
\(672\) 1.55726e11 0.0294578
\(673\) −2.90788e12 −0.546398 −0.273199 0.961957i \(-0.588082\pi\)
−0.273199 + 0.961957i \(0.588082\pi\)
\(674\) 2.81242e12 0.524942
\(675\) −2.80476e12 −0.520031
\(676\) 0 0
\(677\) 4.26822e12 0.780904 0.390452 0.920623i \(-0.372319\pi\)
0.390452 + 0.920623i \(0.372319\pi\)
\(678\) 6.75618e12 1.22791
\(679\) 6.41203e11 0.115766
\(680\) −8.82548e11 −0.158288
\(681\) 4.14753e12 0.738971
\(682\) −1.17633e12 −0.208209
\(683\) 7.69165e11 0.135247 0.0676233 0.997711i \(-0.478458\pi\)
0.0676233 + 0.997711i \(0.478458\pi\)
\(684\) −3.75671e11 −0.0656228
\(685\) 2.50763e12 0.435166
\(686\) 1.21553e12 0.209559
\(687\) 6.23855e12 1.06851
\(688\) 2.20664e12 0.375477
\(689\) 0 0
\(690\) −4.44085e11 −0.0745838
\(691\) −1.38648e12 −0.231347 −0.115673 0.993287i \(-0.536903\pi\)
−0.115673 + 0.993287i \(0.536903\pi\)
\(692\) −4.52327e12 −0.749851
\(693\) 2.48716e11 0.0409642
\(694\) −1.33284e12 −0.218103
\(695\) −1.87608e12 −0.305014
\(696\) −2.45396e12 −0.396392
\(697\) 3.74475e11 0.0601002
\(698\) 5.14189e12 0.819923
\(699\) −1.01921e12 −0.161479
\(700\) −2.91534e11 −0.0458932
\(701\) −5.51186e12 −0.862119 −0.431059 0.902324i \(-0.641860\pi\)
−0.431059 + 0.902324i \(0.641860\pi\)
\(702\) 0 0
\(703\) 1.35837e12 0.209758
\(704\) 9.42007e11 0.144536
\(705\) 1.43606e12 0.218939
\(706\) 6.52302e12 0.988160
\(707\) 1.00695e12 0.151572
\(708\) 4.48222e12 0.670415
\(709\) −2.43152e12 −0.361385 −0.180693 0.983540i \(-0.557834\pi\)
−0.180693 + 0.983540i \(0.557834\pi\)
\(710\) 5.38944e12 0.795942
\(711\) 2.28975e12 0.336028
\(712\) −1.30292e11 −0.0190001
\(713\) 2.67779e11 0.0388038
\(714\) −5.88492e11 −0.0847420
\(715\) 0 0
\(716\) −1.50203e11 −0.0213585
\(717\) −8.84166e12 −1.24939
\(718\) −8.96126e12 −1.25837
\(719\) −2.70890e12 −0.378018 −0.189009 0.981975i \(-0.560528\pi\)
−0.189009 + 0.981975i \(0.560528\pi\)
\(720\) −2.65297e11 −0.0367905
\(721\) 7.57664e11 0.104416
\(722\) 3.57157e12 0.489150
\(723\) 7.19927e11 0.0979864
\(724\) −1.39109e12 −0.188162
\(725\) 4.59404e12 0.617553
\(726\) 1.98345e12 0.264976
\(727\) 5.26647e11 0.0699221 0.0349611 0.999389i \(-0.488869\pi\)
0.0349611 + 0.999389i \(0.488869\pi\)
\(728\) 0 0
\(729\) 5.07138e12 0.665047
\(730\) −3.55294e12 −0.463058
\(731\) −8.33893e12 −1.08015
\(732\) 1.32576e12 0.170674
\(733\) 2.78009e12 0.355706 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(734\) −6.01690e12 −0.765140
\(735\) −5.35379e12 −0.676656
\(736\) −2.14438e11 −0.0269371
\(737\) 6.81481e12 0.850844
\(738\) 1.12569e11 0.0139689
\(739\) 2.36558e12 0.291768 0.145884 0.989302i \(-0.453397\pi\)
0.145884 + 0.989302i \(0.453397\pi\)
\(740\) 9.59272e11 0.117598
\(741\) 0 0
\(742\) −2.53098e11 −0.0306529
\(743\) −1.31398e13 −1.58175 −0.790876 0.611977i \(-0.790375\pi\)
−0.790876 + 0.611977i \(0.790375\pi\)
\(744\) 8.36680e11 0.100111
\(745\) 6.56457e12 0.780733
\(746\) 1.29959e12 0.153632
\(747\) 2.12838e12 0.250095
\(748\) −3.55987e12 −0.415792
\(749\) −1.87938e12 −0.218195
\(750\) 6.83888e12 0.789240
\(751\) −7.29436e12 −0.836773 −0.418387 0.908269i \(-0.637404\pi\)
−0.418387 + 0.908269i \(0.637404\pi\)
\(752\) 6.93441e11 0.0790732
\(753\) −1.06110e13 −1.20276
\(754\) 0 0
\(755\) −3.75238e12 −0.420287
\(756\) 5.71427e11 0.0636227
\(757\) −1.63020e13 −1.80430 −0.902150 0.431423i \(-0.858012\pi\)
−0.902150 + 0.431423i \(0.858012\pi\)
\(758\) 3.24367e12 0.356882
\(759\) −1.79127e12 −0.195917
\(760\) −1.12386e12 −0.122195
\(761\) 9.68945e12 1.04729 0.523646 0.851936i \(-0.324571\pi\)
0.523646 + 0.851936i \(0.324571\pi\)
\(762\) 2.99420e12 0.321724
\(763\) 1.28010e12 0.136737
\(764\) 8.27629e12 0.878851
\(765\) 1.00256e12 0.105836
\(766\) −7.62906e11 −0.0800648
\(767\) 0 0
\(768\) −6.70015e11 −0.0694959
\(769\) −1.21329e13 −1.25111 −0.625554 0.780180i \(-0.715127\pi\)
−0.625554 + 0.780180i \(0.715127\pi\)
\(770\) 7.44064e11 0.0762785
\(771\) 1.46009e13 1.48811
\(772\) 3.31262e12 0.335655
\(773\) 1.68647e13 1.69891 0.849454 0.527663i \(-0.176932\pi\)
0.849454 + 0.527663i \(0.176932\pi\)
\(774\) −2.50671e12 −0.251055
\(775\) −1.56635e12 −0.155966
\(776\) −2.75879e12 −0.273112
\(777\) 6.39653e11 0.0629578
\(778\) 3.03671e12 0.297164
\(779\) 4.76868e11 0.0463959
\(780\) 0 0
\(781\) 2.17390e13 2.09079
\(782\) 8.10366e11 0.0774909
\(783\) −9.00463e12 −0.856126
\(784\) −2.58522e12 −0.244385
\(785\) 3.68147e12 0.346025
\(786\) 6.95422e12 0.649901
\(787\) 6.00116e12 0.557634 0.278817 0.960344i \(-0.410058\pi\)
0.278817 + 0.960344i \(0.410058\pi\)
\(788\) −2.25559e12 −0.208397
\(789\) 1.46703e13 1.34769
\(790\) 6.85006e12 0.625709
\(791\) 2.57688e12 0.234045
\(792\) −1.07011e12 −0.0966416
\(793\) 0 0
\(794\) 6.40380e12 0.571801
\(795\) 2.25515e12 0.200228
\(796\) −6.36247e12 −0.561716
\(797\) −1.43155e13 −1.25674 −0.628368 0.777916i \(-0.716277\pi\)
−0.628368 + 0.777916i \(0.716277\pi\)
\(798\) −7.49403e11 −0.0654188
\(799\) −2.62053e12 −0.227472
\(800\) 1.25433e12 0.108270
\(801\) 1.48010e11 0.0127041
\(802\) 1.40190e13 1.19655
\(803\) −1.43312e13 −1.21636
\(804\) −4.84712e12 −0.409102
\(805\) −1.69378e11 −0.0142160
\(806\) 0 0
\(807\) −1.91450e13 −1.58900
\(808\) −4.33242e12 −0.357585
\(809\) −1.09893e13 −0.901992 −0.450996 0.892526i \(-0.648931\pi\)
−0.450996 + 0.892526i \(0.648931\pi\)
\(810\) −6.36638e12 −0.519649
\(811\) −2.15444e13 −1.74880 −0.874399 0.485207i \(-0.838744\pi\)
−0.874399 + 0.485207i \(0.838744\pi\)
\(812\) −9.35964e11 −0.0755539
\(813\) −2.56242e13 −2.05704
\(814\) 3.86934e12 0.308907
\(815\) 7.20750e11 0.0572236
\(816\) 2.53200e12 0.199921
\(817\) −1.06190e13 −0.833846
\(818\) 9.15680e12 0.715079
\(819\) 0 0
\(820\) 3.36762e11 0.0260112
\(821\) −5.71748e12 −0.439198 −0.219599 0.975590i \(-0.570475\pi\)
−0.219599 + 0.975590i \(0.570475\pi\)
\(822\) −7.19430e12 −0.549623
\(823\) −1.00524e13 −0.763787 −0.381893 0.924206i \(-0.624728\pi\)
−0.381893 + 0.924206i \(0.624728\pi\)
\(824\) −3.25987e12 −0.246336
\(825\) 1.04778e13 0.787461
\(826\) 1.70957e12 0.127784
\(827\) −2.29581e13 −1.70672 −0.853359 0.521324i \(-0.825438\pi\)
−0.853359 + 0.521324i \(0.825438\pi\)
\(828\) 2.43599e11 0.0180110
\(829\) −1.57277e13 −1.15657 −0.578283 0.815836i \(-0.696277\pi\)
−0.578283 + 0.815836i \(0.696277\pi\)
\(830\) 6.36729e12 0.465697
\(831\) −1.01421e13 −0.737773
\(832\) 0 0
\(833\) 9.76960e12 0.703031
\(834\) 5.38240e12 0.385238
\(835\) −2.48385e12 −0.176822
\(836\) −4.53324e12 −0.320982
\(837\) 3.07014e12 0.216219
\(838\) −6.66708e12 −0.467022
\(839\) 8.52168e12 0.593740 0.296870 0.954918i \(-0.404057\pi\)
0.296870 + 0.954918i \(0.404057\pi\)
\(840\) −5.29225e11 −0.0366762
\(841\) 2.41910e11 0.0166752
\(842\) −1.90469e13 −1.30593
\(843\) 8.12705e12 0.554254
\(844\) −1.19082e13 −0.807801
\(845\) 0 0
\(846\) −7.87740e11 −0.0528708
\(847\) 7.56507e11 0.0505054
\(848\) 1.08896e12 0.0723154
\(849\) 1.41491e13 0.934638
\(850\) −4.74015e12 −0.311463
\(851\) −8.80815e11 −0.0575707
\(852\) −1.54621e13 −1.00529
\(853\) 1.83160e12 0.118457 0.0592285 0.998244i \(-0.481136\pi\)
0.0592285 + 0.998244i \(0.481136\pi\)
\(854\) 5.05660e11 0.0325311
\(855\) 1.27669e12 0.0817032
\(856\) 8.08606e12 0.514760
\(857\) −6.20072e11 −0.0392671 −0.0196335 0.999807i \(-0.506250\pi\)
−0.0196335 + 0.999807i \(0.506250\pi\)
\(858\) 0 0
\(859\) −5.71581e12 −0.358186 −0.179093 0.983832i \(-0.557316\pi\)
−0.179093 + 0.983832i \(0.557316\pi\)
\(860\) −7.49912e12 −0.467484
\(861\) 2.24556e11 0.0139255
\(862\) −6.98328e12 −0.430801
\(863\) −2.02596e13 −1.24332 −0.621658 0.783289i \(-0.713540\pi\)
−0.621658 + 0.783289i \(0.713540\pi\)
\(864\) −2.45858e12 −0.150097
\(865\) 1.53720e13 0.933596
\(866\) −7.17488e12 −0.433495
\(867\) 8.93122e12 0.536815
\(868\) 3.19118e11 0.0190815
\(869\) 2.76305e13 1.64362
\(870\) 8.33961e12 0.493525
\(871\) 0 0
\(872\) −5.50768e12 −0.322585
\(873\) 3.13394e12 0.182611
\(874\) 1.03194e12 0.0598211
\(875\) 2.60842e12 0.150432
\(876\) 1.01933e13 0.584851
\(877\) 9.14573e10 0.00522059 0.00261030 0.999997i \(-0.499169\pi\)
0.00261030 + 0.999997i \(0.499169\pi\)
\(878\) 1.05552e13 0.599436
\(879\) 1.13189e13 0.639519
\(880\) −3.20135e12 −0.179954
\(881\) 1.73150e13 0.968347 0.484174 0.874972i \(-0.339120\pi\)
0.484174 + 0.874972i \(0.339120\pi\)
\(882\) 2.93677e12 0.163404
\(883\) 1.38781e13 0.768259 0.384130 0.923279i \(-0.374502\pi\)
0.384130 + 0.923279i \(0.374502\pi\)
\(884\) 0 0
\(885\) −1.52325e13 −0.834695
\(886\) −1.51761e13 −0.827387
\(887\) 1.76586e13 0.957853 0.478926 0.877855i \(-0.341026\pi\)
0.478926 + 0.877855i \(0.341026\pi\)
\(888\) −2.75212e12 −0.148528
\(889\) 1.14202e12 0.0613219
\(890\) 4.42788e11 0.0236560
\(891\) −2.56796e13 −1.36502
\(892\) −1.19385e13 −0.631404
\(893\) −3.33706e12 −0.175603
\(894\) −1.88335e13 −0.986081
\(895\) 5.10457e11 0.0265923
\(896\) −2.55551e11 −0.0132462
\(897\) 0 0
\(898\) −9.78821e12 −0.502296
\(899\) −5.02872e12 −0.256767
\(900\) −1.42490e12 −0.0723927
\(901\) −4.11520e12 −0.208032
\(902\) 1.35837e12 0.0683264
\(903\) −5.00049e12 −0.250275
\(904\) −1.10871e13 −0.552152
\(905\) 4.72754e12 0.234270
\(906\) 1.07655e13 0.530830
\(907\) 1.23924e13 0.608025 0.304013 0.952668i \(-0.401673\pi\)
0.304013 + 0.952668i \(0.401673\pi\)
\(908\) −6.80620e12 −0.332291
\(909\) 4.92157e12 0.239092
\(910\) 0 0
\(911\) −1.38104e13 −0.664313 −0.332157 0.943224i \(-0.607776\pi\)
−0.332157 + 0.943224i \(0.607776\pi\)
\(912\) 3.22432e12 0.154334
\(913\) 2.56832e13 1.22329
\(914\) −6.06452e12 −0.287434
\(915\) −4.50553e12 −0.212496
\(916\) −1.02376e13 −0.480474
\(917\) 2.65241e12 0.123874
\(918\) 9.29101e12 0.431788
\(919\) 8.56606e11 0.0396152 0.0198076 0.999804i \(-0.493695\pi\)
0.0198076 + 0.999804i \(0.493695\pi\)
\(920\) 7.28754e11 0.0335379
\(921\) 2.83884e13 1.30009
\(922\) −1.42410e13 −0.649010
\(923\) 0 0
\(924\) −2.13469e12 −0.0963412
\(925\) 5.15223e12 0.231397
\(926\) 2.12563e13 0.950034
\(927\) 3.70316e12 0.164708
\(928\) 4.02700e12 0.178244
\(929\) −1.29169e13 −0.568968 −0.284484 0.958681i \(-0.591822\pi\)
−0.284484 + 0.958681i \(0.591822\pi\)
\(930\) −2.84341e12 −0.124642
\(931\) 1.24409e13 0.542723
\(932\) 1.67254e12 0.0726116
\(933\) 1.40134e13 0.605447
\(934\) −2.65020e13 −1.13951
\(935\) 1.20980e13 0.517679
\(936\) 0 0
\(937\) −3.67867e12 −0.155906 −0.0779530 0.996957i \(-0.524838\pi\)
−0.0779530 + 0.996957i \(0.524838\pi\)
\(938\) −1.84874e12 −0.0779765
\(939\) −8.60174e11 −0.0361070
\(940\) −2.35662e12 −0.0984495
\(941\) −4.45145e13 −1.85075 −0.925376 0.379051i \(-0.876251\pi\)
−0.925376 + 0.379051i \(0.876251\pi\)
\(942\) −1.05620e13 −0.437036
\(943\) −3.09219e11 −0.0127339
\(944\) −7.35544e12 −0.301463
\(945\) −1.94196e12 −0.0792130
\(946\) −3.02486e13 −1.22799
\(947\) 1.99543e13 0.806236 0.403118 0.915148i \(-0.367926\pi\)
0.403118 + 0.915148i \(0.367926\pi\)
\(948\) −1.96526e13 −0.790282
\(949\) 0 0
\(950\) −6.03625e12 −0.240442
\(951\) −3.03897e12 −0.120480
\(952\) 9.65731e11 0.0381057
\(953\) 7.13202e12 0.280088 0.140044 0.990145i \(-0.455276\pi\)
0.140044 + 0.990145i \(0.455276\pi\)
\(954\) −1.23704e12 −0.0483523
\(955\) −2.81264e13 −1.09421
\(956\) 1.45094e13 0.561809
\(957\) 3.36388e13 1.29639
\(958\) −1.72757e13 −0.662660
\(959\) −2.74398e12 −0.104760
\(960\) 2.27700e12 0.0865253
\(961\) −2.47251e13 −0.935152
\(962\) 0 0
\(963\) −9.18565e12 −0.344185
\(964\) −1.18142e12 −0.0440613
\(965\) −1.12577e13 −0.417905
\(966\) 4.85941e11 0.0179550
\(967\) −2.84176e12 −0.104512 −0.0522562 0.998634i \(-0.516641\pi\)
−0.0522562 + 0.998634i \(0.516641\pi\)
\(968\) −3.25489e12 −0.119151
\(969\) −1.21848e13 −0.443978
\(970\) 9.37557e12 0.340036
\(971\) −3.90309e13 −1.40903 −0.704517 0.709687i \(-0.748836\pi\)
−0.704517 + 0.709687i \(0.748836\pi\)
\(972\) 6.45045e12 0.231788
\(973\) 2.05290e12 0.0734278
\(974\) 2.56878e13 0.914558
\(975\) 0 0
\(976\) −2.17561e12 −0.0767463
\(977\) 2.17556e13 0.763915 0.381958 0.924180i \(-0.375250\pi\)
0.381958 + 0.924180i \(0.375250\pi\)
\(978\) −2.06781e12 −0.0722745
\(979\) 1.78604e12 0.0621397
\(980\) 8.78570e12 0.304270
\(981\) 6.25664e12 0.215690
\(982\) −1.26981e13 −0.435749
\(983\) −4.64998e13 −1.58840 −0.794201 0.607655i \(-0.792110\pi\)
−0.794201 + 0.607655i \(0.792110\pi\)
\(984\) −9.66159e11 −0.0328527
\(985\) 7.66548e12 0.259464
\(986\) −1.52181e13 −0.512761
\(987\) −1.57142e12 −0.0527065
\(988\) 0 0
\(989\) 6.88577e12 0.228860
\(990\) 3.63669e12 0.120323
\(991\) −1.55209e13 −0.511194 −0.255597 0.966783i \(-0.582272\pi\)
−0.255597 + 0.966783i \(0.582272\pi\)
\(992\) −1.37301e12 −0.0450166
\(993\) −1.66610e13 −0.543786
\(994\) −5.89741e12 −0.191612
\(995\) 2.16225e13 0.699361
\(996\) −1.82675e13 −0.588184
\(997\) −4.26334e13 −1.36654 −0.683268 0.730167i \(-0.739442\pi\)
−0.683268 + 0.730167i \(0.739442\pi\)
\(998\) −3.15121e13 −1.00552
\(999\) −1.00987e13 −0.320791
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 338.10.a.a.1.1 1
13.12 even 2 2.10.a.a.1.1 1
39.38 odd 2 18.10.a.a.1.1 1
52.51 odd 2 16.10.a.d.1.1 1
65.12 odd 4 50.10.b.a.49.2 2
65.38 odd 4 50.10.b.a.49.1 2
65.64 even 2 50.10.a.c.1.1 1
91.12 odd 6 98.10.c.b.67.1 2
91.25 even 6 98.10.c.c.79.1 2
91.38 odd 6 98.10.c.b.79.1 2
91.51 even 6 98.10.c.c.67.1 2
91.90 odd 2 98.10.a.c.1.1 1
104.51 odd 2 64.10.a.b.1.1 1
104.77 even 2 64.10.a.h.1.1 1
117.25 even 6 162.10.c.b.109.1 2
117.38 odd 6 162.10.c.i.109.1 2
117.77 odd 6 162.10.c.i.55.1 2
117.103 even 6 162.10.c.b.55.1 2
143.142 odd 2 242.10.a.a.1.1 1
156.155 even 2 144.10.a.d.1.1 1
208.51 odd 4 256.10.b.e.129.2 2
208.77 even 4 256.10.b.g.129.1 2
208.155 odd 4 256.10.b.e.129.1 2
208.181 even 4 256.10.b.g.129.2 2
260.103 even 4 400.10.c.d.49.2 2
260.207 even 4 400.10.c.d.49.1 2
260.259 odd 2 400.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.10.a.a.1.1 1 13.12 even 2
16.10.a.d.1.1 1 52.51 odd 2
18.10.a.a.1.1 1 39.38 odd 2
50.10.a.c.1.1 1 65.64 even 2
50.10.b.a.49.1 2 65.38 odd 4
50.10.b.a.49.2 2 65.12 odd 4
64.10.a.b.1.1 1 104.51 odd 2
64.10.a.h.1.1 1 104.77 even 2
98.10.a.c.1.1 1 91.90 odd 2
98.10.c.b.67.1 2 91.12 odd 6
98.10.c.b.79.1 2 91.38 odd 6
98.10.c.c.67.1 2 91.51 even 6
98.10.c.c.79.1 2 91.25 even 6
144.10.a.d.1.1 1 156.155 even 2
162.10.c.b.55.1 2 117.103 even 6
162.10.c.b.109.1 2 117.25 even 6
162.10.c.i.55.1 2 117.77 odd 6
162.10.c.i.109.1 2 117.38 odd 6
242.10.a.a.1.1 1 143.142 odd 2
256.10.b.e.129.1 2 208.155 odd 4
256.10.b.e.129.2 2 208.51 odd 4
256.10.b.g.129.1 2 208.77 even 4
256.10.b.g.129.2 2 208.181 even 4
338.10.a.a.1.1 1 1.1 even 1 trivial
400.10.a.b.1.1 1 260.259 odd 2
400.10.c.d.49.1 2 260.207 even 4
400.10.c.d.49.2 2 260.103 even 4