Properties

Label 338.8.a.c.1.1
Level $338$
Weight $8$
Character 338.1
Self dual yes
Analytic conductor $105.586$
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] = [338,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(105.586138614\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} -39.0000 q^{3} +64.0000 q^{4} -385.000 q^{5} -312.000 q^{6} +293.000 q^{7} +512.000 q^{8} -666.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -39.0000 q^{3} +64.0000 q^{4} -385.000 q^{5} -312.000 q^{6} +293.000 q^{7} +512.000 q^{8} -666.000 q^{9} -3080.00 q^{10} +5402.00 q^{11} -2496.00 q^{12} +2344.00 q^{14} +15015.0 q^{15} +4096.00 q^{16} -21011.0 q^{17} -5328.00 q^{18} +27326.0 q^{19} -24640.0 q^{20} -11427.0 q^{21} +43216.0 q^{22} -63072.0 q^{23} -19968.0 q^{24} +70100.0 q^{25} +111267. q^{27} +18752.0 q^{28} +122238. q^{29} +120120. q^{30} +208396. q^{31} +32768.0 q^{32} -210678. q^{33} -168088. q^{34} -112805. q^{35} -42624.0 q^{36} +442379. q^{37} +218608. q^{38} -197120. q^{40} -58000.0 q^{41} -91416.0 q^{42} -202025. q^{43} +345728. q^{44} +256410. q^{45} -504576. q^{46} -588511. q^{47} -159744. q^{48} -737694. q^{49} +560800. q^{50} +819429. q^{51} +1.68434e6 q^{53} +890136. q^{54} -2.07977e6 q^{55} +150016. q^{56} -1.06571e6 q^{57} +977904. q^{58} +442630. q^{59} +960960. q^{60} -1.08361e6 q^{61} +1.66717e6 q^{62} -195138. q^{63} +262144. q^{64} -1.68542e6 q^{66} -3.44349e6 q^{67} -1.34470e6 q^{68} +2.45981e6 q^{69} -902440. q^{70} -2.08470e6 q^{71} -340992. q^{72} -5.93789e6 q^{73} +3.53903e6 q^{74} -2.73390e6 q^{75} +1.74886e6 q^{76} +1.58279e6 q^{77} -6.60926e6 q^{79} -1.57696e6 q^{80} -2.88287e6 q^{81} -464000. q^{82} +142740. q^{83} -731328. q^{84} +8.08924e6 q^{85} -1.61620e6 q^{86} -4.76728e6 q^{87} +2.76582e6 q^{88} +6.98529e6 q^{89} +2.05128e6 q^{90} -4.03661e6 q^{92} -8.12744e6 q^{93} -4.70809e6 q^{94} -1.05205e7 q^{95} -1.27795e6 q^{96} +200762. q^{97} -5.90155e6 q^{98} -3.59773e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −39.0000 −0.833950 −0.416975 0.908918i \(-0.636910\pi\)
−0.416975 + 0.908918i \(0.636910\pi\)
\(4\) 64.0000 0.500000
\(5\) −385.000 −1.37742 −0.688709 0.725038i \(-0.741822\pi\)
−0.688709 + 0.725038i \(0.741822\pi\)
\(6\) −312.000 −0.589692
\(7\) 293.000 0.322868 0.161434 0.986884i \(-0.448388\pi\)
0.161434 + 0.986884i \(0.448388\pi\)
\(8\) 512.000 0.353553
\(9\) −666.000 −0.304527
\(10\) −3080.00 −0.973982
\(11\) 5402.00 1.22371 0.611857 0.790968i \(-0.290423\pi\)
0.611857 + 0.790968i \(0.290423\pi\)
\(12\) −2496.00 −0.416975
\(13\) 0 0
\(14\) 2344.00 0.228302
\(15\) 15015.0 1.14870
\(16\) 4096.00 0.250000
\(17\) −21011.0 −1.03723 −0.518616 0.855008i \(-0.673552\pi\)
−0.518616 + 0.855008i \(0.673552\pi\)
\(18\) −5328.00 −0.215333
\(19\) 27326.0 0.913984 0.456992 0.889471i \(-0.348927\pi\)
0.456992 + 0.889471i \(0.348927\pi\)
\(20\) −24640.0 −0.688709
\(21\) −11427.0 −0.269256
\(22\) 43216.0 0.865297
\(23\) −63072.0 −1.08091 −0.540455 0.841373i \(-0.681748\pi\)
−0.540455 + 0.841373i \(0.681748\pi\)
\(24\) −19968.0 −0.294846
\(25\) 70100.0 0.897280
\(26\) 0 0
\(27\) 111267. 1.08791
\(28\) 18752.0 0.161434
\(29\) 122238. 0.930708 0.465354 0.885125i \(-0.345927\pi\)
0.465354 + 0.885125i \(0.345927\pi\)
\(30\) 120120. 0.812252
\(31\) 208396. 1.25639 0.628194 0.778057i \(-0.283795\pi\)
0.628194 + 0.778057i \(0.283795\pi\)
\(32\) 32768.0 0.176777
\(33\) −210678. −1.02052
\(34\) −168088. −0.733433
\(35\) −112805. −0.444724
\(36\) −42624.0 −0.152263
\(37\) 442379. 1.43578 0.717891 0.696156i \(-0.245108\pi\)
0.717891 + 0.696156i \(0.245108\pi\)
\(38\) 218608. 0.646284
\(39\) 0 0
\(40\) −197120. −0.486991
\(41\) −58000.0 −0.131427 −0.0657135 0.997839i \(-0.520932\pi\)
−0.0657135 + 0.997839i \(0.520932\pi\)
\(42\) −91416.0 −0.190392
\(43\) −202025. −0.387494 −0.193747 0.981051i \(-0.562064\pi\)
−0.193747 + 0.981051i \(0.562064\pi\)
\(44\) 345728. 0.611857
\(45\) 256410. 0.419461
\(46\) −504576. −0.764318
\(47\) −588511. −0.826822 −0.413411 0.910545i \(-0.635663\pi\)
−0.413411 + 0.910545i \(0.635663\pi\)
\(48\) −159744. −0.208488
\(49\) −737694. −0.895757
\(50\) 560800. 0.634473
\(51\) 819429. 0.864999
\(52\) 0 0
\(53\) 1.68434e6 1.55404 0.777022 0.629474i \(-0.216729\pi\)
0.777022 + 0.629474i \(0.216729\pi\)
\(54\) 890136. 0.769269
\(55\) −2.07977e6 −1.68557
\(56\) 150016. 0.114151
\(57\) −1.06571e6 −0.762217
\(58\) 977904. 0.658110
\(59\) 442630. 0.280581 0.140291 0.990110i \(-0.455196\pi\)
0.140291 + 0.990110i \(0.455196\pi\)
\(60\) 960960. 0.574349
\(61\) −1.08361e6 −0.611248 −0.305624 0.952152i \(-0.598865\pi\)
−0.305624 + 0.952152i \(0.598865\pi\)
\(62\) 1.66717e6 0.888400
\(63\) −195138. −0.0983218
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) −1.68542e6 −0.721615
\(67\) −3.44349e6 −1.39874 −0.699369 0.714761i \(-0.746536\pi\)
−0.699369 + 0.714761i \(0.746536\pi\)
\(68\) −1.34470e6 −0.518616
\(69\) 2.45981e6 0.901425
\(70\) −902440. −0.314467
\(71\) −2.08470e6 −0.691258 −0.345629 0.938371i \(-0.612334\pi\)
−0.345629 + 0.938371i \(0.612334\pi\)
\(72\) −340992. −0.107666
\(73\) −5.93789e6 −1.78650 −0.893248 0.449564i \(-0.851579\pi\)
−0.893248 + 0.449564i \(0.851579\pi\)
\(74\) 3.53903e6 1.01525
\(75\) −2.73390e6 −0.748287
\(76\) 1.74886e6 0.456992
\(77\) 1.58279e6 0.395098
\(78\) 0 0
\(79\) −6.60926e6 −1.50820 −0.754098 0.656762i \(-0.771926\pi\)
−0.754098 + 0.656762i \(0.771926\pi\)
\(80\) −1.57696e6 −0.344354
\(81\) −2.88287e6 −0.602737
\(82\) −464000. −0.0929329
\(83\) 142740. 0.0274014 0.0137007 0.999906i \(-0.495639\pi\)
0.0137007 + 0.999906i \(0.495639\pi\)
\(84\) −731328. −0.134628
\(85\) 8.08924e6 1.42870
\(86\) −1.61620e6 −0.274000
\(87\) −4.76728e6 −0.776164
\(88\) 2.76582e6 0.432648
\(89\) 6.98529e6 1.05031 0.525157 0.851005i \(-0.324007\pi\)
0.525157 + 0.851005i \(0.324007\pi\)
\(90\) 2.05128e6 0.296603
\(91\) 0 0
\(92\) −4.03661e6 −0.540455
\(93\) −8.12744e6 −1.04776
\(94\) −4.70809e6 −0.584652
\(95\) −1.05205e7 −1.25894
\(96\) −1.27795e6 −0.147423
\(97\) 200762. 0.0223347 0.0111674 0.999938i \(-0.496445\pi\)
0.0111674 + 0.999938i \(0.496445\pi\)
\(98\) −5.90155e6 −0.633395
\(99\) −3.59773e6 −0.372654
\(100\) 4.48640e6 0.448640
\(101\) −5.42144e6 −0.523588 −0.261794 0.965124i \(-0.584314\pi\)
−0.261794 + 0.965124i \(0.584314\pi\)
\(102\) 6.55543e6 0.611647
\(103\) −1.71897e7 −1.55002 −0.775011 0.631948i \(-0.782255\pi\)
−0.775011 + 0.631948i \(0.782255\pi\)
\(104\) 0 0
\(105\) 4.39940e6 0.370877
\(106\) 1.34747e7 1.09887
\(107\) 1.23582e7 0.975242 0.487621 0.873055i \(-0.337865\pi\)
0.487621 + 0.873055i \(0.337865\pi\)
\(108\) 7.12109e6 0.543955
\(109\) −1.70569e7 −1.26156 −0.630778 0.775964i \(-0.717264\pi\)
−0.630778 + 0.775964i \(0.717264\pi\)
\(110\) −1.66382e7 −1.19188
\(111\) −1.72528e7 −1.19737
\(112\) 1.20013e6 0.0807169
\(113\) 2.11250e7 1.37728 0.688639 0.725104i \(-0.258208\pi\)
0.688639 + 0.725104i \(0.258208\pi\)
\(114\) −8.52571e6 −0.538969
\(115\) 2.42827e7 1.48886
\(116\) 7.82323e6 0.465354
\(117\) 0 0
\(118\) 3.54104e6 0.198401
\(119\) −6.15622e6 −0.334888
\(120\) 7.68768e6 0.406126
\(121\) 9.69443e6 0.497478
\(122\) −8.66886e6 −0.432218
\(123\) 2.26200e6 0.109604
\(124\) 1.33373e7 0.628194
\(125\) 3.08962e6 0.141488
\(126\) −1.56110e6 −0.0695240
\(127\) −3.24008e7 −1.40360 −0.701800 0.712374i \(-0.747620\pi\)
−0.701800 + 0.712374i \(0.747620\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 7.87898e6 0.323151
\(130\) 0 0
\(131\) −2.64669e7 −1.02862 −0.514308 0.857605i \(-0.671951\pi\)
−0.514308 + 0.857605i \(0.671951\pi\)
\(132\) −1.34834e7 −0.510259
\(133\) 8.00652e6 0.295096
\(134\) −2.75479e7 −0.989057
\(135\) −4.28378e7 −1.49851
\(136\) −1.07576e7 −0.366717
\(137\) −5.36201e7 −1.78158 −0.890791 0.454413i \(-0.849849\pi\)
−0.890791 + 0.454413i \(0.849849\pi\)
\(138\) 1.96785e7 0.637403
\(139\) 7.58784e6 0.239644 0.119822 0.992795i \(-0.461768\pi\)
0.119822 + 0.992795i \(0.461768\pi\)
\(140\) −7.21952e6 −0.222362
\(141\) 2.29519e7 0.689529
\(142\) −1.66776e7 −0.488793
\(143\) 0 0
\(144\) −2.72794e6 −0.0761317
\(145\) −4.70616e7 −1.28197
\(146\) −4.75031e7 −1.26324
\(147\) 2.87701e7 0.747016
\(148\) 2.83123e7 0.717891
\(149\) 5.70297e7 1.41237 0.706187 0.708026i \(-0.250414\pi\)
0.706187 + 0.708026i \(0.250414\pi\)
\(150\) −2.18712e7 −0.529119
\(151\) 2.00648e7 0.474259 0.237130 0.971478i \(-0.423793\pi\)
0.237130 + 0.971478i \(0.423793\pi\)
\(152\) 1.39909e7 0.323142
\(153\) 1.39933e7 0.315865
\(154\) 1.26623e7 0.279376
\(155\) −8.02325e7 −1.73057
\(156\) 0 0
\(157\) −3.15314e7 −0.650272 −0.325136 0.945667i \(-0.605410\pi\)
−0.325136 + 0.945667i \(0.605410\pi\)
\(158\) −5.28740e7 −1.06646
\(159\) −6.56891e7 −1.29600
\(160\) −1.26157e7 −0.243495
\(161\) −1.84801e7 −0.348991
\(162\) −2.30630e7 −0.426199
\(163\) 3.13938e7 0.567789 0.283895 0.958855i \(-0.408373\pi\)
0.283895 + 0.958855i \(0.408373\pi\)
\(164\) −3.71200e6 −0.0657135
\(165\) 8.11110e7 1.40568
\(166\) 1.14192e6 0.0193757
\(167\) −9.22170e7 −1.53216 −0.766079 0.642747i \(-0.777795\pi\)
−0.766079 + 0.642747i \(0.777795\pi\)
\(168\) −5.85062e6 −0.0951962
\(169\) 0 0
\(170\) 6.47139e7 1.01024
\(171\) −1.81991e7 −0.278332
\(172\) −1.29296e7 −0.193747
\(173\) −6.57015e7 −0.964748 −0.482374 0.875965i \(-0.660225\pi\)
−0.482374 + 0.875965i \(0.660225\pi\)
\(174\) −3.81383e7 −0.548831
\(175\) 2.05393e7 0.289703
\(176\) 2.21266e7 0.305929
\(177\) −1.72626e7 −0.233991
\(178\) 5.58823e7 0.742684
\(179\) −3.20402e6 −0.0417551 −0.0208776 0.999782i \(-0.506646\pi\)
−0.0208776 + 0.999782i \(0.506646\pi\)
\(180\) 1.64102e7 0.209730
\(181\) −4.45759e7 −0.558760 −0.279380 0.960181i \(-0.590129\pi\)
−0.279380 + 0.960181i \(0.590129\pi\)
\(182\) 0 0
\(183\) 4.22607e7 0.509751
\(184\) −3.22929e7 −0.382159
\(185\) −1.70316e8 −1.97767
\(186\) −6.50196e7 −0.740881
\(187\) −1.13501e8 −1.26927
\(188\) −3.76647e7 −0.413411
\(189\) 3.26012e7 0.351251
\(190\) −8.41641e7 −0.890203
\(191\) 1.86394e8 1.93559 0.967797 0.251733i \(-0.0810004\pi\)
0.967797 + 0.251733i \(0.0810004\pi\)
\(192\) −1.02236e7 −0.104244
\(193\) 1.52927e8 1.53120 0.765602 0.643314i \(-0.222441\pi\)
0.765602 + 0.643314i \(0.222441\pi\)
\(194\) 1.60610e6 0.0157930
\(195\) 0 0
\(196\) −4.72124e7 −0.447878
\(197\) −9.51837e7 −0.887015 −0.443507 0.896271i \(-0.646266\pi\)
−0.443507 + 0.896271i \(0.646266\pi\)
\(198\) −2.87819e7 −0.263506
\(199\) 1.78585e8 1.60642 0.803212 0.595693i \(-0.203122\pi\)
0.803212 + 0.595693i \(0.203122\pi\)
\(200\) 3.58912e7 0.317236
\(201\) 1.34296e8 1.16648
\(202\) −4.33715e7 −0.370232
\(203\) 3.58157e7 0.300495
\(204\) 5.24435e7 0.432500
\(205\) 2.23300e7 0.181030
\(206\) −1.37517e8 −1.09603
\(207\) 4.20060e7 0.329166
\(208\) 0 0
\(209\) 1.47615e8 1.11846
\(210\) 3.51952e7 0.262250
\(211\) −1.33235e8 −0.976406 −0.488203 0.872730i \(-0.662348\pi\)
−0.488203 + 0.872730i \(0.662348\pi\)
\(212\) 1.07798e8 0.777022
\(213\) 8.13035e7 0.576475
\(214\) 9.88657e7 0.689600
\(215\) 7.77796e7 0.533742
\(216\) 5.69687e7 0.384634
\(217\) 6.10600e7 0.405647
\(218\) −1.36455e8 −0.892054
\(219\) 2.31578e8 1.48985
\(220\) −1.33105e8 −0.842783
\(221\) 0 0
\(222\) −1.38022e8 −0.846669
\(223\) −1.19394e8 −0.720969 −0.360484 0.932765i \(-0.617389\pi\)
−0.360484 + 0.932765i \(0.617389\pi\)
\(224\) 9.60102e6 0.0570755
\(225\) −4.66866e7 −0.273246
\(226\) 1.69000e8 0.973883
\(227\) −1.13656e7 −0.0644911 −0.0322456 0.999480i \(-0.510266\pi\)
−0.0322456 + 0.999480i \(0.510266\pi\)
\(228\) −6.82057e7 −0.381109
\(229\) 1.46559e7 0.0806470 0.0403235 0.999187i \(-0.487161\pi\)
0.0403235 + 0.999187i \(0.487161\pi\)
\(230\) 1.94262e8 1.05279
\(231\) −6.17287e7 −0.329492
\(232\) 6.25859e7 0.329055
\(233\) −2.46924e8 −1.27885 −0.639423 0.768855i \(-0.720827\pi\)
−0.639423 + 0.768855i \(0.720827\pi\)
\(234\) 0 0
\(235\) 2.26577e8 1.13888
\(236\) 2.83283e7 0.140291
\(237\) 2.57761e8 1.25776
\(238\) −4.92498e7 −0.236802
\(239\) 1.61239e7 0.0763971 0.0381985 0.999270i \(-0.487838\pi\)
0.0381985 + 0.999270i \(0.487838\pi\)
\(240\) 6.15014e7 0.287175
\(241\) −1.14256e8 −0.525798 −0.262899 0.964823i \(-0.584679\pi\)
−0.262899 + 0.964823i \(0.584679\pi\)
\(242\) 7.75555e7 0.351770
\(243\) −1.30909e8 −0.585258
\(244\) −6.93509e7 −0.305624
\(245\) 2.84012e8 1.23383
\(246\) 1.80960e7 0.0775014
\(247\) 0 0
\(248\) 1.06699e8 0.444200
\(249\) −5.56686e6 −0.0228514
\(250\) 2.47170e7 0.100047
\(251\) 2.22704e8 0.888935 0.444467 0.895795i \(-0.353393\pi\)
0.444467 + 0.895795i \(0.353393\pi\)
\(252\) −1.24888e7 −0.0491609
\(253\) −3.40715e8 −1.32272
\(254\) −2.59207e8 −0.992494
\(255\) −3.15480e8 −1.19147
\(256\) 1.67772e7 0.0625000
\(257\) 2.82302e8 1.03741 0.518703 0.854955i \(-0.326415\pi\)
0.518703 + 0.854955i \(0.326415\pi\)
\(258\) 6.30318e7 0.228502
\(259\) 1.29617e8 0.463567
\(260\) 0 0
\(261\) −8.14105e7 −0.283425
\(262\) −2.11735e8 −0.727342
\(263\) −2.36490e8 −0.801619 −0.400809 0.916162i \(-0.631271\pi\)
−0.400809 + 0.916162i \(0.631271\pi\)
\(264\) −1.07867e8 −0.360807
\(265\) −6.48469e8 −2.14057
\(266\) 6.40521e7 0.208664
\(267\) −2.72426e8 −0.875910
\(268\) −2.20383e8 −0.699369
\(269\) −4.82172e8 −1.51032 −0.755160 0.655541i \(-0.772441\pi\)
−0.755160 + 0.655541i \(0.772441\pi\)
\(270\) −3.42702e8 −1.05960
\(271\) −4.66372e8 −1.42344 −0.711721 0.702462i \(-0.752084\pi\)
−0.711721 + 0.702462i \(0.752084\pi\)
\(272\) −8.60611e7 −0.259308
\(273\) 0 0
\(274\) −4.28961e8 −1.25977
\(275\) 3.78680e8 1.09801
\(276\) 1.57428e8 0.450712
\(277\) −1.88709e8 −0.533475 −0.266738 0.963769i \(-0.585946\pi\)
−0.266738 + 0.963769i \(0.585946\pi\)
\(278\) 6.07027e7 0.169454
\(279\) −1.38792e8 −0.382603
\(280\) −5.77562e7 −0.157234
\(281\) 7.15402e8 1.92344 0.961718 0.274040i \(-0.0883600\pi\)
0.961718 + 0.274040i \(0.0883600\pi\)
\(282\) 1.83615e8 0.487570
\(283\) −4.04602e8 −1.06115 −0.530573 0.847639i \(-0.678023\pi\)
−0.530573 + 0.847639i \(0.678023\pi\)
\(284\) −1.33421e8 −0.345629
\(285\) 4.10300e8 1.04989
\(286\) 0 0
\(287\) −1.69940e7 −0.0424335
\(288\) −2.18235e7 −0.0538332
\(289\) 3.11234e7 0.0758482
\(290\) −3.76493e8 −0.906492
\(291\) −7.82972e6 −0.0186260
\(292\) −3.80025e8 −0.893248
\(293\) 8.11321e8 1.88433 0.942163 0.335156i \(-0.108789\pi\)
0.942163 + 0.335156i \(0.108789\pi\)
\(294\) 2.30161e8 0.528220
\(295\) −1.70413e8 −0.386478
\(296\) 2.26498e8 0.507626
\(297\) 6.01064e8 1.33129
\(298\) 4.56238e8 0.998699
\(299\) 0 0
\(300\) −1.74970e8 −0.374144
\(301\) −5.91933e7 −0.125109
\(302\) 1.60519e8 0.335352
\(303\) 2.11436e8 0.436646
\(304\) 1.11927e8 0.228496
\(305\) 4.17189e8 0.841945
\(306\) 1.11947e8 0.223350
\(307\) −4.60958e8 −0.909237 −0.454618 0.890686i \(-0.650224\pi\)
−0.454618 + 0.890686i \(0.650224\pi\)
\(308\) 1.01298e8 0.197549
\(309\) 6.70398e8 1.29264
\(310\) −6.41860e8 −1.22370
\(311\) −2.87718e8 −0.542383 −0.271192 0.962525i \(-0.587418\pi\)
−0.271192 + 0.962525i \(0.587418\pi\)
\(312\) 0 0
\(313\) −9.56179e8 −1.76252 −0.881260 0.472632i \(-0.843304\pi\)
−0.881260 + 0.472632i \(0.843304\pi\)
\(314\) −2.52252e8 −0.459812
\(315\) 7.51281e7 0.135430
\(316\) −4.22992e8 −0.754098
\(317\) −4.92761e8 −0.868818 −0.434409 0.900716i \(-0.643043\pi\)
−0.434409 + 0.900716i \(0.643043\pi\)
\(318\) −5.25513e8 −0.916407
\(319\) 6.60330e8 1.13892
\(320\) −1.00925e8 −0.172177
\(321\) −4.81970e8 −0.813303
\(322\) −1.47841e8 −0.246774
\(323\) −5.74147e8 −0.948012
\(324\) −1.84504e8 −0.301368
\(325\) 0 0
\(326\) 2.51150e8 0.401488
\(327\) 6.65218e8 1.05207
\(328\) −2.96960e7 −0.0464665
\(329\) −1.72434e8 −0.266954
\(330\) 6.48888e8 0.993965
\(331\) 4.83358e8 0.732607 0.366304 0.930495i \(-0.380623\pi\)
0.366304 + 0.930495i \(0.380623\pi\)
\(332\) 9.13536e6 0.0137007
\(333\) −2.94624e8 −0.437234
\(334\) −7.37736e8 −1.08340
\(335\) 1.32574e9 1.92665
\(336\) −4.68050e7 −0.0673139
\(337\) 1.30823e9 1.86200 0.930998 0.365025i \(-0.118940\pi\)
0.930998 + 0.365025i \(0.118940\pi\)
\(338\) 0 0
\(339\) −8.23874e8 −1.14858
\(340\) 5.17711e8 0.714350
\(341\) 1.12576e9 1.53746
\(342\) −1.45593e8 −0.196811
\(343\) −4.57442e8 −0.612078
\(344\) −1.03437e8 −0.137000
\(345\) −9.47026e8 −1.24164
\(346\) −5.25612e8 −0.682180
\(347\) −8.94842e8 −1.14972 −0.574861 0.818251i \(-0.694944\pi\)
−0.574861 + 0.818251i \(0.694944\pi\)
\(348\) −3.05106e8 −0.388082
\(349\) 5.41626e8 0.682041 0.341020 0.940056i \(-0.389228\pi\)
0.341020 + 0.940056i \(0.389228\pi\)
\(350\) 1.64314e8 0.204851
\(351\) 0 0
\(352\) 1.77013e8 0.216324
\(353\) −2.25334e8 −0.272656 −0.136328 0.990664i \(-0.543530\pi\)
−0.136328 + 0.990664i \(0.543530\pi\)
\(354\) −1.38101e8 −0.165457
\(355\) 8.02611e8 0.952152
\(356\) 4.47058e8 0.525157
\(357\) 2.40093e8 0.279280
\(358\) −2.56322e7 −0.0295253
\(359\) 4.38763e8 0.500495 0.250247 0.968182i \(-0.419488\pi\)
0.250247 + 0.968182i \(0.419488\pi\)
\(360\) 1.31282e8 0.148302
\(361\) −1.47161e8 −0.164634
\(362\) −3.56607e8 −0.395103
\(363\) −3.78083e8 −0.414872
\(364\) 0 0
\(365\) 2.28609e9 2.46075
\(366\) 3.38086e8 0.360448
\(367\) 8.08568e8 0.853857 0.426929 0.904285i \(-0.359596\pi\)
0.426929 + 0.904285i \(0.359596\pi\)
\(368\) −2.58343e8 −0.270227
\(369\) 3.86280e7 0.0400230
\(370\) −1.36253e9 −1.39843
\(371\) 4.93510e8 0.501750
\(372\) −5.20156e8 −0.523882
\(373\) −1.17884e9 −1.17618 −0.588092 0.808794i \(-0.700121\pi\)
−0.588092 + 0.808794i \(0.700121\pi\)
\(374\) −9.08011e8 −0.897513
\(375\) −1.20495e8 −0.117994
\(376\) −3.01318e8 −0.292326
\(377\) 0 0
\(378\) 2.60810e8 0.248372
\(379\) 1.79168e9 1.69053 0.845266 0.534345i \(-0.179442\pi\)
0.845266 + 0.534345i \(0.179442\pi\)
\(380\) −6.73313e8 −0.629469
\(381\) 1.26363e9 1.17053
\(382\) 1.49115e9 1.36867
\(383\) 1.19775e9 1.08936 0.544680 0.838644i \(-0.316651\pi\)
0.544680 + 0.838644i \(0.316651\pi\)
\(384\) −8.17889e7 −0.0737115
\(385\) −6.09373e8 −0.544215
\(386\) 1.22341e9 1.08273
\(387\) 1.34549e8 0.118002
\(388\) 1.28488e7 0.0111674
\(389\) −1.43672e8 −0.123751 −0.0618754 0.998084i \(-0.519708\pi\)
−0.0618754 + 0.998084i \(0.519708\pi\)
\(390\) 0 0
\(391\) 1.32521e9 1.12115
\(392\) −3.77699e8 −0.316698
\(393\) 1.03221e9 0.857815
\(394\) −7.61470e8 −0.627214
\(395\) 2.54456e9 2.07742
\(396\) −2.30255e8 −0.186327
\(397\) 6.17334e8 0.495169 0.247584 0.968866i \(-0.420363\pi\)
0.247584 + 0.968866i \(0.420363\pi\)
\(398\) 1.42868e9 1.13591
\(399\) −3.12254e8 −0.246095
\(400\) 2.87130e8 0.224320
\(401\) −1.13305e9 −0.877491 −0.438746 0.898611i \(-0.644577\pi\)
−0.438746 + 0.898611i \(0.644577\pi\)
\(402\) 1.07437e9 0.824825
\(403\) 0 0
\(404\) −3.46972e8 −0.261794
\(405\) 1.10991e9 0.830220
\(406\) 2.86526e8 0.212482
\(407\) 2.38973e9 1.75699
\(408\) 4.19548e8 0.305823
\(409\) 1.04283e9 0.753670 0.376835 0.926280i \(-0.377012\pi\)
0.376835 + 0.926280i \(0.377012\pi\)
\(410\) 1.78640e8 0.128007
\(411\) 2.09119e9 1.48575
\(412\) −1.10014e9 −0.775011
\(413\) 1.29691e8 0.0905906
\(414\) 3.36048e8 0.232755
\(415\) −5.49549e7 −0.0377431
\(416\) 0 0
\(417\) −2.95926e8 −0.199851
\(418\) 1.18092e9 0.790867
\(419\) 7.09302e8 0.471066 0.235533 0.971866i \(-0.424316\pi\)
0.235533 + 0.971866i \(0.424316\pi\)
\(420\) 2.81561e8 0.185439
\(421\) 1.19877e9 0.782974 0.391487 0.920184i \(-0.371961\pi\)
0.391487 + 0.920184i \(0.371961\pi\)
\(422\) −1.06588e9 −0.690424
\(423\) 3.91948e8 0.251789
\(424\) 8.62380e8 0.549437
\(425\) −1.47287e9 −0.930687
\(426\) 6.50428e8 0.407629
\(427\) −3.17497e8 −0.197352
\(428\) 7.90926e8 0.487621
\(429\) 0 0
\(430\) 6.22237e8 0.377412
\(431\) −9.54153e8 −0.574047 −0.287024 0.957924i \(-0.592666\pi\)
−0.287024 + 0.957924i \(0.592666\pi\)
\(432\) 4.55750e8 0.271978
\(433\) −3.81628e8 −0.225908 −0.112954 0.993600i \(-0.536031\pi\)
−0.112954 + 0.993600i \(0.536031\pi\)
\(434\) 4.88480e8 0.286836
\(435\) 1.83540e9 1.06910
\(436\) −1.09164e9 −0.630778
\(437\) −1.72351e9 −0.987933
\(438\) 1.85262e9 1.05348
\(439\) 1.11683e8 0.0630031 0.0315015 0.999504i \(-0.489971\pi\)
0.0315015 + 0.999504i \(0.489971\pi\)
\(440\) −1.06484e9 −0.595938
\(441\) 4.91304e8 0.272782
\(442\) 0 0
\(443\) 1.45991e9 0.797837 0.398919 0.916986i \(-0.369386\pi\)
0.398919 + 0.916986i \(0.369386\pi\)
\(444\) −1.10418e9 −0.598685
\(445\) −2.68934e9 −1.44672
\(446\) −9.55154e8 −0.509802
\(447\) −2.22416e9 −1.17785
\(448\) 7.68082e7 0.0403585
\(449\) 6.34009e8 0.330547 0.165273 0.986248i \(-0.447149\pi\)
0.165273 + 0.986248i \(0.447149\pi\)
\(450\) −3.73493e8 −0.193214
\(451\) −3.13316e8 −0.160829
\(452\) 1.35200e9 0.688639
\(453\) −7.82528e8 −0.395509
\(454\) −9.09244e7 −0.0456021
\(455\) 0 0
\(456\) −5.45646e8 −0.269484
\(457\) −6.04376e8 −0.296211 −0.148105 0.988972i \(-0.547317\pi\)
−0.148105 + 0.988972i \(0.547317\pi\)
\(458\) 1.17247e8 0.0570261
\(459\) −2.33783e9 −1.12841
\(460\) 1.55409e9 0.744432
\(461\) −2.20565e9 −1.04853 −0.524267 0.851554i \(-0.675661\pi\)
−0.524267 + 0.851554i \(0.675661\pi\)
\(462\) −4.93829e8 −0.232986
\(463\) −1.04925e9 −0.491299 −0.245650 0.969359i \(-0.579001\pi\)
−0.245650 + 0.969359i \(0.579001\pi\)
\(464\) 5.00687e8 0.232677
\(465\) 3.12907e9 1.44321
\(466\) −1.97539e9 −0.904281
\(467\) −2.01461e9 −0.915337 −0.457668 0.889123i \(-0.651315\pi\)
−0.457668 + 0.889123i \(0.651315\pi\)
\(468\) 0 0
\(469\) −1.00894e9 −0.451607
\(470\) 1.81261e9 0.805309
\(471\) 1.22973e9 0.542295
\(472\) 2.26627e8 0.0992005
\(473\) −1.09134e9 −0.474183
\(474\) 2.06209e9 0.889371
\(475\) 1.91555e9 0.820099
\(476\) −3.93998e8 −0.167444
\(477\) −1.12177e9 −0.473248
\(478\) 1.28991e8 0.0540209
\(479\) −3.67842e9 −1.52928 −0.764639 0.644458i \(-0.777083\pi\)
−0.764639 + 0.644458i \(0.777083\pi\)
\(480\) 4.92012e8 0.203063
\(481\) 0 0
\(482\) −9.14048e8 −0.371796
\(483\) 7.20724e8 0.291041
\(484\) 6.20444e8 0.248739
\(485\) −7.72934e7 −0.0307642
\(486\) −1.04727e9 −0.413840
\(487\) 1.91497e8 0.0751294 0.0375647 0.999294i \(-0.488040\pi\)
0.0375647 + 0.999294i \(0.488040\pi\)
\(488\) −5.54807e8 −0.216109
\(489\) −1.22436e9 −0.473508
\(490\) 2.27210e9 0.872450
\(491\) −3.22321e8 −0.122886 −0.0614431 0.998111i \(-0.519570\pi\)
−0.0614431 + 0.998111i \(0.519570\pi\)
\(492\) 1.44768e8 0.0548018
\(493\) −2.56834e9 −0.965359
\(494\) 0 0
\(495\) 1.38513e9 0.513300
\(496\) 8.53590e8 0.314097
\(497\) −6.10819e8 −0.223185
\(498\) −4.45349e7 −0.0161584
\(499\) −3.86695e9 −1.39321 −0.696604 0.717455i \(-0.745307\pi\)
−0.696604 + 0.717455i \(0.745307\pi\)
\(500\) 1.97736e8 0.0707442
\(501\) 3.59646e9 1.27774
\(502\) 1.78163e9 0.628572
\(503\) −3.43814e8 −0.120458 −0.0602290 0.998185i \(-0.519183\pi\)
−0.0602290 + 0.998185i \(0.519183\pi\)
\(504\) −9.99107e7 −0.0347620
\(505\) 2.08725e9 0.721199
\(506\) −2.72572e9 −0.935307
\(507\) 0 0
\(508\) −2.07365e9 −0.701800
\(509\) −2.11533e9 −0.710993 −0.355497 0.934678i \(-0.615688\pi\)
−0.355497 + 0.934678i \(0.615688\pi\)
\(510\) −2.52384e9 −0.842493
\(511\) −1.73980e9 −0.576802
\(512\) 1.34218e8 0.0441942
\(513\) 3.04048e9 0.994333
\(514\) 2.25842e9 0.733556
\(515\) 6.61803e9 2.13503
\(516\) 5.04254e8 0.161576
\(517\) −3.17914e9 −1.01179
\(518\) 1.03694e9 0.327792
\(519\) 2.56236e9 0.804552
\(520\) 0 0
\(521\) −1.40622e9 −0.435634 −0.217817 0.975990i \(-0.569894\pi\)
−0.217817 + 0.975990i \(0.569894\pi\)
\(522\) −6.51284e8 −0.200412
\(523\) 2.18120e9 0.666712 0.333356 0.942801i \(-0.391819\pi\)
0.333356 + 0.942801i \(0.391819\pi\)
\(524\) −1.69388e9 −0.514308
\(525\) −8.01033e8 −0.241598
\(526\) −1.89192e9 −0.566830
\(527\) −4.37861e9 −1.30316
\(528\) −8.62937e8 −0.255129
\(529\) 5.73252e8 0.168365
\(530\) −5.18775e9 −1.51361
\(531\) −2.94792e8 −0.0854445
\(532\) 5.12417e8 0.147548
\(533\) 0 0
\(534\) −2.17941e9 −0.619362
\(535\) −4.75791e9 −1.34332
\(536\) −1.76306e9 −0.494529
\(537\) 1.24957e8 0.0348217
\(538\) −3.85737e9 −1.06796
\(539\) −3.98502e9 −1.09615
\(540\) −2.74162e9 −0.749254
\(541\) −2.54634e8 −0.0691395 −0.0345698 0.999402i \(-0.511006\pi\)
−0.0345698 + 0.999402i \(0.511006\pi\)
\(542\) −3.73097e9 −1.00653
\(543\) 1.73846e9 0.465978
\(544\) −6.88488e8 −0.183358
\(545\) 6.56689e9 1.73769
\(546\) 0 0
\(547\) 2.15158e9 0.562085 0.281043 0.959695i \(-0.409320\pi\)
0.281043 + 0.959695i \(0.409320\pi\)
\(548\) −3.43169e9 −0.890791
\(549\) 7.21683e8 0.186142
\(550\) 3.02944e9 0.776414
\(551\) 3.34028e9 0.850652
\(552\) 1.25942e9 0.318702
\(553\) −1.93651e9 −0.486948
\(554\) −1.50967e9 −0.377224
\(555\) 6.64232e9 1.64928
\(556\) 4.85622e8 0.119822
\(557\) 7.71518e9 1.89170 0.945852 0.324599i \(-0.105229\pi\)
0.945852 + 0.324599i \(0.105229\pi\)
\(558\) −1.11033e9 −0.270542
\(559\) 0 0
\(560\) −4.62049e8 −0.111181
\(561\) 4.42656e9 1.05851
\(562\) 5.72321e9 1.36008
\(563\) −8.12996e7 −0.0192003 −0.00960017 0.999954i \(-0.503056\pi\)
−0.00960017 + 0.999954i \(0.503056\pi\)
\(564\) 1.46892e9 0.344764
\(565\) −8.13312e9 −1.89709
\(566\) −3.23681e9 −0.750343
\(567\) −8.44681e8 −0.194604
\(568\) −1.06737e9 −0.244397
\(569\) −5.08814e9 −1.15789 −0.578944 0.815367i \(-0.696535\pi\)
−0.578944 + 0.815367i \(0.696535\pi\)
\(570\) 3.28240e9 0.742385
\(571\) −5.61762e9 −1.26277 −0.631387 0.775468i \(-0.717514\pi\)
−0.631387 + 0.775468i \(0.717514\pi\)
\(572\) 0 0
\(573\) −7.26935e9 −1.61419
\(574\) −1.35952e8 −0.0300050
\(575\) −4.42135e9 −0.969878
\(576\) −1.74588e8 −0.0380658
\(577\) −4.12728e9 −0.894435 −0.447218 0.894425i \(-0.647585\pi\)
−0.447218 + 0.894425i \(0.647585\pi\)
\(578\) 2.48988e8 0.0536328
\(579\) −5.96415e9 −1.27695
\(580\) −3.01194e9 −0.640987
\(581\) 4.18228e7 0.00884702
\(582\) −6.26377e7 −0.0131706
\(583\) 9.09878e9 1.90171
\(584\) −3.04020e9 −0.631622
\(585\) 0 0
\(586\) 6.49057e9 1.33242
\(587\) −1.86734e9 −0.381056 −0.190528 0.981682i \(-0.561020\pi\)
−0.190528 + 0.981682i \(0.561020\pi\)
\(588\) 1.84128e9 0.373508
\(589\) 5.69463e9 1.14832
\(590\) −1.36330e9 −0.273281
\(591\) 3.71216e9 0.739726
\(592\) 1.81198e9 0.358945
\(593\) −3.31544e9 −0.652905 −0.326453 0.945214i \(-0.605853\pi\)
−0.326453 + 0.945214i \(0.605853\pi\)
\(594\) 4.80851e9 0.941366
\(595\) 2.37015e9 0.461281
\(596\) 3.64990e9 0.706187
\(597\) −6.96483e9 −1.33968
\(598\) 0 0
\(599\) 1.93367e9 0.367610 0.183805 0.982963i \(-0.441158\pi\)
0.183805 + 0.982963i \(0.441158\pi\)
\(600\) −1.39976e9 −0.264559
\(601\) −5.88820e9 −1.10643 −0.553213 0.833040i \(-0.686598\pi\)
−0.553213 + 0.833040i \(0.686598\pi\)
\(602\) −4.73547e8 −0.0884657
\(603\) 2.29336e9 0.425953
\(604\) 1.28415e9 0.237130
\(605\) −3.73236e9 −0.685235
\(606\) 1.69149e9 0.308756
\(607\) 7.94197e9 1.44135 0.720673 0.693276i \(-0.243833\pi\)
0.720673 + 0.693276i \(0.243833\pi\)
\(608\) 8.95418e8 0.161571
\(609\) −1.39681e9 −0.250598
\(610\) 3.33751e9 0.595345
\(611\) 0 0
\(612\) 8.95573e8 0.157932
\(613\) 2.36146e8 0.0414065 0.0207033 0.999786i \(-0.493409\pi\)
0.0207033 + 0.999786i \(0.493409\pi\)
\(614\) −3.68766e9 −0.642927
\(615\) −8.70870e8 −0.150970
\(616\) 8.10386e8 0.139688
\(617\) −1.27029e9 −0.217723 −0.108862 0.994057i \(-0.534721\pi\)
−0.108862 + 0.994057i \(0.534721\pi\)
\(618\) 5.36318e9 0.914035
\(619\) −1.63555e9 −0.277170 −0.138585 0.990351i \(-0.544255\pi\)
−0.138585 + 0.990351i \(0.544255\pi\)
\(620\) −5.13488e9 −0.865285
\(621\) −7.01783e9 −1.17593
\(622\) −2.30175e9 −0.383523
\(623\) 2.04669e9 0.339112
\(624\) 0 0
\(625\) −6.66607e9 −1.09217
\(626\) −7.64943e9 −1.24629
\(627\) −5.75699e9 −0.932736
\(628\) −2.01801e9 −0.325136
\(629\) −9.29483e9 −1.48924
\(630\) 6.01025e8 0.0957636
\(631\) −1.68242e9 −0.266582 −0.133291 0.991077i \(-0.542555\pi\)
−0.133291 + 0.991077i \(0.542555\pi\)
\(632\) −3.38394e9 −0.533228
\(633\) 5.19618e9 0.814275
\(634\) −3.94209e9 −0.614347
\(635\) 1.24743e10 1.93334
\(636\) −4.20410e9 −0.647998
\(637\) 0 0
\(638\) 5.28264e9 0.805338
\(639\) 1.38841e9 0.210507
\(640\) −8.07404e8 −0.121748
\(641\) −1.70575e9 −0.255807 −0.127903 0.991787i \(-0.540825\pi\)
−0.127903 + 0.991787i \(0.540825\pi\)
\(642\) −3.85576e9 −0.575092
\(643\) 1.45635e9 0.216036 0.108018 0.994149i \(-0.465550\pi\)
0.108018 + 0.994149i \(0.465550\pi\)
\(644\) −1.18273e9 −0.174495
\(645\) −3.03341e9 −0.445114
\(646\) −4.59317e9 −0.670346
\(647\) −3.56464e9 −0.517430 −0.258715 0.965954i \(-0.583299\pi\)
−0.258715 + 0.965954i \(0.583299\pi\)
\(648\) −1.47603e9 −0.213100
\(649\) 2.39109e9 0.343352
\(650\) 0 0
\(651\) −2.38134e9 −0.338289
\(652\) 2.00920e9 0.283895
\(653\) −5.86806e9 −0.824705 −0.412352 0.911024i \(-0.635293\pi\)
−0.412352 + 0.911024i \(0.635293\pi\)
\(654\) 5.32174e9 0.743929
\(655\) 1.01898e10 1.41683
\(656\) −2.37568e8 −0.0328567
\(657\) 3.95463e9 0.544036
\(658\) −1.37947e9 −0.188765
\(659\) −2.73239e9 −0.371915 −0.185958 0.982558i \(-0.559539\pi\)
−0.185958 + 0.982558i \(0.559539\pi\)
\(660\) 5.19111e9 0.702839
\(661\) −8.50066e9 −1.14485 −0.572424 0.819958i \(-0.693997\pi\)
−0.572424 + 0.819958i \(0.693997\pi\)
\(662\) 3.86687e9 0.518032
\(663\) 0 0
\(664\) 7.30829e7 0.00968785
\(665\) −3.08251e9 −0.406470
\(666\) −2.35700e9 −0.309171
\(667\) −7.70980e9 −1.00601
\(668\) −5.90189e9 −0.766079
\(669\) 4.65638e9 0.601252
\(670\) 1.06059e10 1.36235
\(671\) −5.85365e9 −0.747994
\(672\) −3.74440e8 −0.0475981
\(673\) −3.85727e7 −0.00487784 −0.00243892 0.999997i \(-0.500776\pi\)
−0.00243892 + 0.999997i \(0.500776\pi\)
\(674\) 1.04658e10 1.31663
\(675\) 7.79982e9 0.976160
\(676\) 0 0
\(677\) −7.34428e9 −0.909681 −0.454840 0.890573i \(-0.650304\pi\)
−0.454840 + 0.890573i \(0.650304\pi\)
\(678\) −6.59100e9 −0.812170
\(679\) 5.88233e7 0.00721116
\(680\) 4.14169e9 0.505122
\(681\) 4.43257e8 0.0537824
\(682\) 9.00604e9 1.08715
\(683\) 7.49577e9 0.900210 0.450105 0.892976i \(-0.351387\pi\)
0.450105 + 0.892976i \(0.351387\pi\)
\(684\) −1.16474e9 −0.139166
\(685\) 2.06438e10 2.45398
\(686\) −3.65954e9 −0.432805
\(687\) −5.71580e8 −0.0672556
\(688\) −8.27494e8 −0.0968736
\(689\) 0 0
\(690\) −7.57621e9 −0.877971
\(691\) −1.66382e10 −1.91838 −0.959188 0.282769i \(-0.908747\pi\)
−0.959188 + 0.282769i \(0.908747\pi\)
\(692\) −4.20490e9 −0.482374
\(693\) −1.05414e9 −0.120318
\(694\) −7.15873e9 −0.812976
\(695\) −2.92132e9 −0.330090
\(696\) −2.44085e9 −0.274415
\(697\) 1.21864e9 0.136320
\(698\) 4.33300e9 0.482275
\(699\) 9.63005e9 1.06649
\(700\) 1.31452e9 0.144851
\(701\) 3.26804e9 0.358323 0.179161 0.983820i \(-0.442662\pi\)
0.179161 + 0.983820i \(0.442662\pi\)
\(702\) 0 0
\(703\) 1.20884e10 1.31228
\(704\) 1.41610e9 0.152964
\(705\) −8.83649e9 −0.949769
\(706\) −1.80267e9 −0.192797
\(707\) −1.58848e9 −0.169050
\(708\) −1.10480e9 −0.116995
\(709\) −4.48613e9 −0.472727 −0.236363 0.971665i \(-0.575956\pi\)
−0.236363 + 0.971665i \(0.575956\pi\)
\(710\) 6.42089e9 0.673273
\(711\) 4.40176e9 0.459286
\(712\) 3.57647e9 0.371342
\(713\) −1.31440e10 −1.35804
\(714\) 1.92074e9 0.197481
\(715\) 0 0
\(716\) −2.05057e8 −0.0208776
\(717\) −6.28831e8 −0.0637114
\(718\) 3.51011e9 0.353903
\(719\) 5.42385e9 0.544198 0.272099 0.962269i \(-0.412282\pi\)
0.272099 + 0.962269i \(0.412282\pi\)
\(720\) 1.05026e9 0.104865
\(721\) −5.03658e9 −0.500452
\(722\) −1.17729e9 −0.116414
\(723\) 4.45598e9 0.438490
\(724\) −2.85286e9 −0.279380
\(725\) 8.56888e9 0.835105
\(726\) −3.02466e9 −0.293359
\(727\) −1.50827e10 −1.45582 −0.727911 0.685672i \(-0.759508\pi\)
−0.727911 + 0.685672i \(0.759508\pi\)
\(728\) 0 0
\(729\) 1.14103e10 1.09081
\(730\) 1.82887e10 1.74001
\(731\) 4.24475e9 0.401921
\(732\) 2.70469e9 0.254875
\(733\) 6.75596e9 0.633612 0.316806 0.948490i \(-0.397390\pi\)
0.316806 + 0.948490i \(0.397390\pi\)
\(734\) 6.46854e9 0.603768
\(735\) −1.10765e10 −1.02895
\(736\) −2.06674e9 −0.191080
\(737\) −1.86017e10 −1.71166
\(738\) 3.09024e8 0.0283006
\(739\) −1.08154e10 −0.985797 −0.492899 0.870087i \(-0.664063\pi\)
−0.492899 + 0.870087i \(0.664063\pi\)
\(740\) −1.09002e10 −0.988836
\(741\) 0 0
\(742\) 3.94808e9 0.354791
\(743\) 3.71897e9 0.332630 0.166315 0.986073i \(-0.446813\pi\)
0.166315 + 0.986073i \(0.446813\pi\)
\(744\) −4.16125e9 −0.370441
\(745\) −2.19565e10 −1.94543
\(746\) −9.43075e9 −0.831687
\(747\) −9.50648e7 −0.00834445
\(748\) −7.26409e9 −0.634637
\(749\) 3.62096e9 0.314874
\(750\) −9.63963e8 −0.0834346
\(751\) −2.15786e10 −1.85902 −0.929510 0.368797i \(-0.879770\pi\)
−0.929510 + 0.368797i \(0.879770\pi\)
\(752\) −2.41054e9 −0.206706
\(753\) −8.68546e9 −0.741328
\(754\) 0 0
\(755\) −7.72496e9 −0.653253
\(756\) 2.08648e9 0.175626
\(757\) 7.42446e9 0.622056 0.311028 0.950401i \(-0.399327\pi\)
0.311028 + 0.950401i \(0.399327\pi\)
\(758\) 1.43335e10 1.19539
\(759\) 1.32879e10 1.10309
\(760\) −5.38650e9 −0.445102
\(761\) −8.57002e9 −0.704913 −0.352457 0.935828i \(-0.614654\pi\)
−0.352457 + 0.935828i \(0.614654\pi\)
\(762\) 1.01091e10 0.827691
\(763\) −4.99766e9 −0.407315
\(764\) 1.19292e10 0.967797
\(765\) −5.38743e9 −0.435078
\(766\) 9.58202e9 0.770294
\(767\) 0 0
\(768\) −6.54311e8 −0.0521219
\(769\) −7.81741e9 −0.619899 −0.309949 0.950753i \(-0.600312\pi\)
−0.309949 + 0.950753i \(0.600312\pi\)
\(770\) −4.87498e9 −0.384818
\(771\) −1.10098e10 −0.865144
\(772\) 9.78732e9 0.765602
\(773\) −1.30864e10 −1.01904 −0.509522 0.860457i \(-0.670178\pi\)
−0.509522 + 0.860457i \(0.670178\pi\)
\(774\) 1.07639e9 0.0834403
\(775\) 1.46086e10 1.12733
\(776\) 1.02790e8 0.00789651
\(777\) −5.05506e9 −0.386592
\(778\) −1.14937e9 −0.0875051
\(779\) −1.58491e9 −0.120122
\(780\) 0 0
\(781\) −1.12616e10 −0.845903
\(782\) 1.06016e10 0.792775
\(783\) 1.36011e10 1.01253
\(784\) −3.02159e9 −0.223939
\(785\) 1.21396e10 0.895696
\(786\) 8.25767e9 0.606567
\(787\) −5.35561e9 −0.391649 −0.195825 0.980639i \(-0.562738\pi\)
−0.195825 + 0.980639i \(0.562738\pi\)
\(788\) −6.09176e9 −0.443507
\(789\) 9.22311e9 0.668510
\(790\) 2.03565e10 1.46895
\(791\) 6.18962e9 0.444679
\(792\) −1.84204e9 −0.131753
\(793\) 0 0
\(794\) 4.93867e9 0.350137
\(795\) 2.52903e10 1.78513
\(796\) 1.14295e10 0.803212
\(797\) 1.21863e10 0.852641 0.426320 0.904572i \(-0.359810\pi\)
0.426320 + 0.904572i \(0.359810\pi\)
\(798\) −2.49803e9 −0.174016
\(799\) 1.23652e10 0.857606
\(800\) 2.29704e9 0.158618
\(801\) −4.65220e9 −0.319849
\(802\) −9.06438e9 −0.620480
\(803\) −3.20765e10 −2.18616
\(804\) 8.59494e9 0.583239
\(805\) 7.11484e9 0.480706
\(806\) 0 0
\(807\) 1.88047e10 1.25953
\(808\) −2.77578e9 −0.185116
\(809\) 1.32472e10 0.879636 0.439818 0.898087i \(-0.355043\pi\)
0.439818 + 0.898087i \(0.355043\pi\)
\(810\) 8.87924e9 0.587054
\(811\) −1.45473e10 −0.957658 −0.478829 0.877908i \(-0.658939\pi\)
−0.478829 + 0.877908i \(0.658939\pi\)
\(812\) 2.29221e9 0.150248
\(813\) 1.81885e10 1.18708
\(814\) 1.91179e10 1.24238
\(815\) −1.20866e10 −0.782083
\(816\) 3.35638e9 0.216250
\(817\) −5.52054e9 −0.354164
\(818\) 8.34262e9 0.532925
\(819\) 0 0
\(820\) 1.42912e9 0.0905149
\(821\) −6.51876e9 −0.411115 −0.205558 0.978645i \(-0.565901\pi\)
−0.205558 + 0.978645i \(0.565901\pi\)
\(822\) 1.67295e10 1.05059
\(823\) −6.77944e9 −0.423930 −0.211965 0.977277i \(-0.567986\pi\)
−0.211965 + 0.977277i \(0.567986\pi\)
\(824\) −8.80112e9 −0.548015
\(825\) −1.47685e10 −0.915690
\(826\) 1.03752e9 0.0640572
\(827\) 7.96808e9 0.489874 0.244937 0.969539i \(-0.421233\pi\)
0.244937 + 0.969539i \(0.421233\pi\)
\(828\) 2.68838e9 0.164583
\(829\) 3.74439e9 0.228265 0.114133 0.993466i \(-0.463591\pi\)
0.114133 + 0.993466i \(0.463591\pi\)
\(830\) −4.39639e8 −0.0266884
\(831\) 7.35966e9 0.444892
\(832\) 0 0
\(833\) 1.54997e10 0.929106
\(834\) −2.36741e9 −0.141316
\(835\) 3.55035e10 2.11042
\(836\) 9.44736e9 0.559228
\(837\) 2.31876e10 1.36684
\(838\) 5.67442e9 0.333094
\(839\) 8.06205e9 0.471280 0.235640 0.971840i \(-0.424281\pi\)
0.235640 + 0.971840i \(0.424281\pi\)
\(840\) 2.25249e9 0.131125
\(841\) −2.30775e9 −0.133783
\(842\) 9.59013e9 0.553646
\(843\) −2.79007e10 −1.60405
\(844\) −8.52706e9 −0.488203
\(845\) 0 0
\(846\) 3.13559e9 0.178042
\(847\) 2.84047e9 0.160619
\(848\) 6.89904e9 0.388511
\(849\) 1.57795e10 0.884943
\(850\) −1.17830e10 −0.658095
\(851\) −2.79017e10 −1.55195
\(852\) 5.20342e9 0.288238
\(853\) 3.31854e9 0.183074 0.0915368 0.995802i \(-0.470822\pi\)
0.0915368 + 0.995802i \(0.470822\pi\)
\(854\) −2.53998e9 −0.139549
\(855\) 7.00666e9 0.383380
\(856\) 6.32740e9 0.344800
\(857\) −1.81939e10 −0.987398 −0.493699 0.869633i \(-0.664356\pi\)
−0.493699 + 0.869633i \(0.664356\pi\)
\(858\) 0 0
\(859\) −1.91859e10 −1.03278 −0.516388 0.856355i \(-0.672724\pi\)
−0.516388 + 0.856355i \(0.672724\pi\)
\(860\) 4.97790e9 0.266871
\(861\) 6.62766e8 0.0353874
\(862\) −7.63322e9 −0.405913
\(863\) 2.77943e10 1.47203 0.736017 0.676963i \(-0.236704\pi\)
0.736017 + 0.676963i \(0.236704\pi\)
\(864\) 3.64600e9 0.192317
\(865\) 2.52951e10 1.32886
\(866\) −3.05302e9 −0.159741
\(867\) −1.21381e9 −0.0632536
\(868\) 3.90784e9 0.202823
\(869\) −3.57032e10 −1.84560
\(870\) 1.46832e10 0.755969
\(871\) 0 0
\(872\) −8.73311e9 −0.446027
\(873\) −1.33707e8 −0.00680152
\(874\) −1.37880e10 −0.698574
\(875\) 9.05260e8 0.0456820
\(876\) 1.48210e10 0.744925
\(877\) 3.40401e10 1.70409 0.852043 0.523471i \(-0.175363\pi\)
0.852043 + 0.523471i \(0.175363\pi\)
\(878\) 8.93465e8 0.0445499
\(879\) −3.16415e10 −1.57143
\(880\) −8.51874e9 −0.421392
\(881\) 3.24476e10 1.59870 0.799350 0.600866i \(-0.205177\pi\)
0.799350 + 0.600866i \(0.205177\pi\)
\(882\) 3.93043e9 0.192886
\(883\) 1.15866e10 0.566362 0.283181 0.959066i \(-0.408610\pi\)
0.283181 + 0.959066i \(0.408610\pi\)
\(884\) 0 0
\(885\) 6.64609e9 0.322303
\(886\) 1.16793e10 0.564156
\(887\) −2.86160e10 −1.37682 −0.688408 0.725323i \(-0.741690\pi\)
−0.688408 + 0.725323i \(0.741690\pi\)
\(888\) −8.83342e9 −0.423335
\(889\) −9.49344e9 −0.453177
\(890\) −2.15147e10 −1.02299
\(891\) −1.55733e10 −0.737578
\(892\) −7.64123e9 −0.360484
\(893\) −1.60817e10 −0.755702
\(894\) −1.77933e10 −0.832865
\(895\) 1.23355e9 0.0575142
\(896\) 6.14466e8 0.0285377
\(897\) 0 0
\(898\) 5.07207e9 0.233732
\(899\) 2.54739e10 1.16933
\(900\) −2.98794e9 −0.136623
\(901\) −3.53896e10 −1.61190
\(902\) −2.50653e9 −0.113723
\(903\) 2.30854e9 0.104335
\(904\) 1.08160e10 0.486942
\(905\) 1.71617e10 0.769646
\(906\) −6.26022e9 −0.279667
\(907\) 2.28278e10 1.01587 0.507936 0.861395i \(-0.330409\pi\)
0.507936 + 0.861395i \(0.330409\pi\)
\(908\) −7.27395e8 −0.0322456
\(909\) 3.61068e9 0.159446
\(910\) 0 0
\(911\) −1.76175e10 −0.772024 −0.386012 0.922494i \(-0.626148\pi\)
−0.386012 + 0.922494i \(0.626148\pi\)
\(912\) −4.36516e9 −0.190554
\(913\) 7.71081e8 0.0335315
\(914\) −4.83501e9 −0.209453
\(915\) −1.62704e10 −0.702140
\(916\) 9.37978e8 0.0403235
\(917\) −7.75480e9 −0.332107
\(918\) −1.87026e10 −0.797910
\(919\) −4.93202e9 −0.209614 −0.104807 0.994493i \(-0.533423\pi\)
−0.104807 + 0.994493i \(0.533423\pi\)
\(920\) 1.24328e10 0.526393
\(921\) 1.79774e10 0.758258
\(922\) −1.76452e10 −0.741425
\(923\) 0 0
\(924\) −3.95063e9 −0.164746
\(925\) 3.10108e10 1.28830
\(926\) −8.39401e9 −0.347401
\(927\) 1.14483e10 0.472023
\(928\) 4.00549e9 0.164527
\(929\) 2.68352e10 1.09812 0.549061 0.835783i \(-0.314986\pi\)
0.549061 + 0.835783i \(0.314986\pi\)
\(930\) 2.50325e10 1.02050
\(931\) −2.01582e10 −0.818707
\(932\) −1.58032e10 −0.639423
\(933\) 1.12210e10 0.452321
\(934\) −1.61168e10 −0.647241
\(935\) 4.36980e10 1.74832
\(936\) 0 0
\(937\) −2.08650e10 −0.828570 −0.414285 0.910147i \(-0.635968\pi\)
−0.414285 + 0.910147i \(0.635968\pi\)
\(938\) −8.07153e9 −0.319335
\(939\) 3.72910e10 1.46985
\(940\) 1.45009e10 0.569440
\(941\) 3.07099e10 1.20147 0.600737 0.799447i \(-0.294874\pi\)
0.600737 + 0.799447i \(0.294874\pi\)
\(942\) 9.83781e9 0.383460
\(943\) 3.65818e9 0.142061
\(944\) 1.81301e9 0.0701453
\(945\) −1.25515e10 −0.483820
\(946\) −8.73071e9 −0.335298
\(947\) 1.03377e9 0.0395548 0.0197774 0.999804i \(-0.493704\pi\)
0.0197774 + 0.999804i \(0.493704\pi\)
\(948\) 1.64967e10 0.628880
\(949\) 0 0
\(950\) 1.53244e10 0.579898
\(951\) 1.92177e10 0.724552
\(952\) −3.15199e9 −0.118401
\(953\) −1.78629e9 −0.0668540 −0.0334270 0.999441i \(-0.510642\pi\)
−0.0334270 + 0.999441i \(0.510642\pi\)
\(954\) −8.97414e9 −0.334637
\(955\) −7.17615e10 −2.66612
\(956\) 1.03193e9 0.0381985
\(957\) −2.57529e10 −0.949803
\(958\) −2.94273e10 −1.08136
\(959\) −1.57107e10 −0.575215
\(960\) 3.93609e9 0.143587
\(961\) 1.59163e10 0.578508
\(962\) 0 0
\(963\) −8.23057e9 −0.296987
\(964\) −7.31238e9 −0.262899
\(965\) −5.88768e10 −2.10911
\(966\) 5.76579e9 0.205797
\(967\) 1.53418e10 0.545613 0.272806 0.962069i \(-0.412048\pi\)
0.272806 + 0.962069i \(0.412048\pi\)
\(968\) 4.96355e9 0.175885
\(969\) 2.23917e10 0.790595
\(970\) −6.18347e8 −0.0217536
\(971\) −5.22182e8 −0.0183043 −0.00915217 0.999958i \(-0.502913\pi\)
−0.00915217 + 0.999958i \(0.502913\pi\)
\(972\) −8.37817e9 −0.292629
\(973\) 2.22324e9 0.0773733
\(974\) 1.53197e9 0.0531245
\(975\) 0 0
\(976\) −4.43846e9 −0.152812
\(977\) −5.68857e10 −1.95152 −0.975758 0.218851i \(-0.929769\pi\)
−0.975758 + 0.218851i \(0.929769\pi\)
\(978\) −9.79486e9 −0.334821
\(979\) 3.77345e10 1.28528
\(980\) 1.81768e10 0.616916
\(981\) 1.13599e10 0.384177
\(982\) −2.57857e9 −0.0868936
\(983\) 1.15736e10 0.388624 0.194312 0.980940i \(-0.437753\pi\)
0.194312 + 0.980940i \(0.437753\pi\)
\(984\) 1.15814e9 0.0387507
\(985\) 3.66457e10 1.22179
\(986\) −2.05467e10 −0.682612
\(987\) 6.72492e9 0.222626
\(988\) 0 0
\(989\) 1.27421e10 0.418846
\(990\) 1.10810e10 0.362958
\(991\) 1.38509e10 0.452086 0.226043 0.974117i \(-0.427421\pi\)
0.226043 + 0.974117i \(0.427421\pi\)
\(992\) 6.82872e9 0.222100
\(993\) −1.88510e10 −0.610958
\(994\) −4.88655e9 −0.157816
\(995\) −6.87554e10 −2.21272
\(996\) −3.56279e8 −0.0114257
\(997\) 5.45316e9 0.174267 0.0871334 0.996197i \(-0.472229\pi\)
0.0871334 + 0.996197i \(0.472229\pi\)
\(998\) −3.09356e10 −0.985147
\(999\) 4.92222e10 1.56200
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 338.8.a.c.1.1 1
13.5 odd 4 338.8.b.b.337.1 2
13.8 odd 4 338.8.b.b.337.2 2
13.12 even 2 26.8.a.a.1.1 1
39.38 odd 2 234.8.a.d.1.1 1
52.51 odd 2 208.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.a.1.1 1 13.12 even 2
208.8.a.c.1.1 1 52.51 odd 2
234.8.a.d.1.1 1 39.38 odd 2
338.8.a.c.1.1 1 1.1 even 1 trivial
338.8.b.b.337.1 2 13.5 odd 4
338.8.b.b.337.2 2 13.8 odd 4