Properties

Label 26.8.a.a.1.1
Level $26$
Weight $8$
Character 26.1
Self dual yes
Analytic conductor $8.122$
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] = [26,8,Mod(1,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, 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("26.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 26.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: \(8.12201066259\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 26.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -39.0000 q^{3} +64.0000 q^{4} +385.000 q^{5} +312.000 q^{6} -293.000 q^{7} -512.000 q^{8} -666.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -39.0000 q^{3} +64.0000 q^{4} +385.000 q^{5} +312.000 q^{6} -293.000 q^{7} -512.000 q^{8} -666.000 q^{9} -3080.00 q^{10} -5402.00 q^{11} -2496.00 q^{12} +2197.00 q^{13} +2344.00 q^{14} -15015.0 q^{15} +4096.00 q^{16} -21011.0 q^{17} +5328.00 q^{18} -27326.0 q^{19} +24640.0 q^{20} +11427.0 q^{21} +43216.0 q^{22} -63072.0 q^{23} +19968.0 q^{24} +70100.0 q^{25} -17576.0 q^{26} +111267. q^{27} -18752.0 q^{28} +122238. q^{29} +120120. q^{30} -208396. q^{31} -32768.0 q^{32} +210678. q^{33} +168088. q^{34} -112805. q^{35} -42624.0 q^{36} -442379. q^{37} +218608. q^{38} -85683.0 q^{39} -197120. q^{40} +58000.0 q^{41} -91416.0 q^{42} -202025. q^{43} -345728. q^{44} -256410. q^{45} +504576. q^{46} +588511. q^{47} -159744. q^{48} -737694. q^{49} -560800. q^{50} +819429. q^{51} +140608. q^{52} +1.68434e6 q^{53} -890136. q^{54} -2.07977e6 q^{55} +150016. q^{56} +1.06571e6 q^{57} -977904. q^{58} -442630. q^{59} -960960. q^{60} -1.08361e6 q^{61} +1.66717e6 q^{62} +195138. q^{63} +262144. q^{64} +845845. q^{65} -1.68542e6 q^{66} +3.44349e6 q^{67} -1.34470e6 q^{68} +2.45981e6 q^{69} +902440. q^{70} +2.08470e6 q^{71} +340992. q^{72} +5.93789e6 q^{73} +3.53903e6 q^{74} -2.73390e6 q^{75} -1.74886e6 q^{76} +1.58279e6 q^{77} +685464. q^{78} -6.60926e6 q^{79} +1.57696e6 q^{80} -2.88287e6 q^{81} -464000. q^{82} -142740. q^{83} +731328. q^{84} -8.08924e6 q^{85} +1.61620e6 q^{86} -4.76728e6 q^{87} +2.76582e6 q^{88} -6.98529e6 q^{89} +2.05128e6 q^{90} -643721. q^{91} -4.03661e6 q^{92} +8.12744e6 q^{93} -4.70809e6 q^{94} -1.05205e7 q^{95} +1.27795e6 q^{96} -200762. q^{97} +5.90155e6 q^{98} +3.59773e6 q^{99} +O(q^{100})\)

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 \(-0.258178\pi\)
0.688709 + 0.725038i \(0.258178\pi\)
\(6\) 312.000 0.589692
\(7\) −293.000 −0.322868 −0.161434 0.986884i \(-0.551612\pi\)
−0.161434 + 0.986884i \(0.551612\pi\)
\(8\) −512.000 −0.353553
\(9\) −666.000 −0.304527
\(10\) −3080.00 −0.973982
\(11\) −5402.00 −1.22371 −0.611857 0.790968i \(-0.709577\pi\)
−0.611857 + 0.790968i \(0.709577\pi\)
\(12\) −2496.00 −0.416975
\(13\) 2197.00 0.277350
\(14\) 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 \(-0.651073\pi\)
−0.456992 + 0.889471i \(0.651073\pi\)
\(20\) 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
\(26\) −17576.0 −0.196116
\(27\) 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 \(-0.716205\pi\)
−0.628194 + 0.778057i \(0.716205\pi\)
\(32\) −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 \(-0.754892\pi\)
−0.717891 + 0.696156i \(0.754892\pi\)
\(38\) 218608. 0.646284
\(39\) −85683.0 −0.231296
\(40\) −197120. −0.486991
\(41\) 58000.0 0.131427 0.0657135 0.997839i \(-0.479068\pi\)
0.0657135 + 0.997839i \(0.479068\pi\)
\(42\) −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 \(-0.364337\pi\)
0.413411 + 0.910545i \(0.364337\pi\)
\(48\) −159744. −0.208488
\(49\) −737694. −0.895757
\(50\) −560800. −0.634473
\(51\) 819429. 0.864999
\(52\) 140608. 0.138675
\(53\) 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 \(-0.544804\pi\)
−0.140291 + 0.990110i \(0.544804\pi\)
\(60\) −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
\(65\) 845845. 0.382027
\(66\) −1.68542e6 −0.721615
\(67\) 3.44349e6 1.39874 0.699369 0.714761i \(-0.253464\pi\)
0.699369 + 0.714761i \(0.253464\pi\)
\(68\) −1.34470e6 −0.518616
\(69\) 2.45981e6 0.901425
\(70\) 902440. 0.314467
\(71\) 2.08470e6 0.691258 0.345629 0.938371i \(-0.387666\pi\)
0.345629 + 0.938371i \(0.387666\pi\)
\(72\) 340992. 0.107666
\(73\) 5.93789e6 1.78650 0.893248 0.449564i \(-0.148421\pi\)
0.893248 + 0.449564i \(0.148421\pi\)
\(74\) 3.53903e6 1.01525
\(75\) −2.73390e6 −0.748287
\(76\) −1.74886e6 −0.456992
\(77\) 1.58279e6 0.395098
\(78\) 685464. 0.163551
\(79\) −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 \(-0.504361\pi\)
−0.0137007 + 0.999906i \(0.504361\pi\)
\(84\) 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 \(-0.675993\pi\)
−0.525157 + 0.851005i \(0.675993\pi\)
\(90\) 2.05128e6 0.296603
\(91\) −643721. −0.0895474
\(92\) −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 \(-0.503555\pi\)
−0.0111674 + 0.999938i \(0.503555\pi\)
\(98\) 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 \(0.782255\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 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 \(-0.282736\pi\)
0.630778 + 0.775964i \(0.282736\pi\)
\(110\) 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
\(117\) −1.46320e6 −0.0844605
\(118\) 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
\(130\) −6.76676e6 −0.270134
\(131\) −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 \(-0.150151\pi\)
0.890791 + 0.454413i \(0.150151\pi\)
\(138\) −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
\(143\) −1.18682e7 −0.339397
\(144\) −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 \(-0.749586\pi\)
−0.706187 + 0.708026i \(0.749586\pi\)
\(150\) 2.18712e7 0.529119
\(151\) −2.00648e7 −0.474259 −0.237130 0.971478i \(-0.576207\pi\)
−0.237130 + 0.971478i \(0.576207\pi\)
\(152\) 1.39909e7 0.323142
\(153\) 1.39933e7 0.315865
\(154\) −1.26623e7 −0.279376
\(155\) −8.02325e7 −1.73057
\(156\) −5.48371e6 −0.115648
\(157\) −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 \(-0.591627\pi\)
−0.283895 + 0.958855i \(0.591627\pi\)
\(164\) 3.71200e6 0.0657135
\(165\) 8.11110e7 1.40568
\(166\) 1.14192e6 0.0193757
\(167\) 9.22170e7 1.53216 0.766079 0.642747i \(-0.222205\pi\)
0.766079 + 0.642747i \(0.222205\pi\)
\(168\) −5.85062e6 −0.0951962
\(169\) 4.82681e6 0.0769231
\(170\) 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 \(0.590129\pi\)
\(182\) 5.14977e6 0.0633195
\(183\) 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 \(-0.777559\pi\)
−0.765602 + 0.643314i \(0.777559\pi\)
\(194\) 1.60610e6 0.0157930
\(195\) −3.29880e7 −0.318592
\(196\) −4.72124e7 −0.447878
\(197\) 9.51837e7 0.887015 0.443507 0.896271i \(-0.353734\pi\)
0.443507 + 0.896271i \(0.353734\pi\)
\(198\) −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
\(208\) 8.99891e6 0.0693375
\(209\) 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
\(221\) −4.61612e7 −0.287676
\(222\) −1.38022e8 −0.846669
\(223\) 1.19394e8 0.720969 0.360484 0.932765i \(-0.382611\pi\)
0.360484 + 0.932765i \(0.382611\pi\)
\(224\) 9.60102e6 0.0570755
\(225\) −4.66866e7 −0.273246
\(226\) −1.69000e8 −0.973883
\(227\) 1.13656e7 0.0644911 0.0322456 0.999480i \(-0.489734\pi\)
0.0322456 + 0.999480i \(0.489734\pi\)
\(228\) 6.82057e7 0.381109
\(229\) −1.46559e7 −0.0806470 −0.0403235 0.999187i \(-0.512839\pi\)
−0.0403235 + 0.999187i \(0.512839\pi\)
\(230\) 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 \(0.720827\pi\)
\(234\) 1.17056e7 0.0597226
\(235\) 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 \(-0.512162\pi\)
−0.0381985 + 0.999270i \(0.512162\pi\)
\(240\) −6.15014e7 −0.287175
\(241\) 1.14256e8 0.525798 0.262899 0.964823i \(-0.415321\pi\)
0.262899 + 0.964823i \(0.415321\pi\)
\(242\) −7.75555e7 −0.351770
\(243\) −1.30909e8 −0.585258
\(244\) −6.93509e7 −0.305624
\(245\) −2.84012e8 −1.23383
\(246\) 1.80960e7 0.0775014
\(247\) −6.00352e7 −0.253493
\(248\) 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
\(260\) 5.41341e7 0.191013
\(261\) −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 \(-0.247916\pi\)
0.711721 + 0.702462i \(0.247916\pi\)
\(272\) −8.60611e7 −0.259308
\(273\) 2.51051e7 0.0746781
\(274\) −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 \(-0.911640\pi\)
−0.961718 + 0.274040i \(0.911640\pi\)
\(282\) 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
\(286\) 9.49456e7 0.239990
\(287\) −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 \(-0.891211\pi\)
−0.942163 + 0.335156i \(0.891211\pi\)
\(294\) −2.30161e8 −0.528220
\(295\) −1.70413e8 −0.386478
\(296\) 2.26498e8 0.507626
\(297\) −6.01064e8 −1.33129
\(298\) 4.56238e8 0.998699
\(299\) −1.38569e8 −0.299790
\(300\) −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 \(-0.349776\pi\)
0.454618 + 0.890686i \(0.349776\pi\)
\(308\) 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 \(0.587418\pi\)
\(312\) 4.38697e7 0.0817756
\(313\) −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 \(-0.356957\pi\)
0.434409 + 0.900716i \(0.356957\pi\)
\(318\) 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
\(325\) 1.54010e8 0.248861
\(326\) 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 \(-0.619377\pi\)
−0.366304 + 0.930495i \(0.619377\pi\)
\(332\) −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 \(0.118940\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) −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 \(-0.610772\pi\)
−0.341020 + 0.940056i \(0.610772\pi\)
\(350\) 1.64314e8 0.204851
\(351\) 2.44454e8 0.301732
\(352\) 1.77013e8 0.216324
\(353\) 2.25334e8 0.272656 0.136328 0.990664i \(-0.456470\pi\)
0.136328 + 0.990664i \(0.456470\pi\)
\(354\) −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 \(-0.580512\pi\)
−0.250247 + 0.968182i \(0.580512\pi\)
\(360\) 1.31282e8 0.148302
\(361\) −1.47161e8 −0.164634
\(362\) 3.56607e8 0.395103
\(363\) −3.78083e8 −0.414872
\(364\) −4.11981e7 −0.0447737
\(365\) 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
\(377\) 2.68557e8 0.258132
\(378\) 2.60810e8 0.248372
\(379\) −1.79168e9 −1.69053 −0.845266 0.534345i \(-0.820558\pi\)
−0.845266 + 0.534345i \(0.820558\pi\)
\(380\) −6.73313e8 −0.629469
\(381\) 1.26363e9 1.17053
\(382\) −1.49115e9 −1.36867
\(383\) −1.19775e9 −1.08936 −0.544680 0.838644i \(-0.683349\pi\)
−0.544680 + 0.838644i \(0.683349\pi\)
\(384\) 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 \(0.519708\pi\)
\(390\) 2.63904e8 0.225278
\(391\) 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 \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(398\) −1.42868e9 −1.13591
\(399\) −3.12254e8 −0.246095
\(400\) 2.87130e8 0.224320
\(401\) 1.13305e9 0.877491 0.438746 0.898611i \(-0.355423\pi\)
0.438746 + 0.898611i \(0.355423\pi\)
\(402\) 1.07437e9 0.824825
\(403\) −4.57846e8 −0.348459
\(404\) −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 \(-0.622988\pi\)
−0.376835 + 0.926280i \(0.622988\pi\)
\(410\) −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
\(416\) −7.19913e7 −0.0490290
\(417\) −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 \(-0.628039\pi\)
−0.391487 + 0.920184i \(0.628039\pi\)
\(422\) 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
\(429\) 4.62860e8 0.283041
\(430\) 6.22237e8 0.377412
\(431\) 9.54153e8 0.574047 0.287024 0.957924i \(-0.407334\pi\)
0.287024 + 0.957924i \(0.407334\pi\)
\(432\) 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
\(442\) 3.69289e8 0.203418
\(443\) 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 \(-0.552851\pi\)
−0.165273 + 0.986248i \(0.552851\pi\)
\(450\) 3.73493e8 0.193214
\(451\) −3.13316e8 −0.160829
\(452\) 1.35200e9 0.688639
\(453\) 7.82528e8 0.395509
\(454\) −9.09244e7 −0.0456021
\(455\) −2.47833e8 −0.123344
\(456\) −5.45646e8 −0.269484
\(457\) 6.04376e8 0.296211 0.148105 0.988972i \(-0.452683\pi\)
0.148105 + 0.988972i \(0.452683\pi\)
\(458\) 1.17247e8 0.0570261
\(459\) −2.33783e9 −1.12841
\(460\) −1.55409e9 −0.744432
\(461\) 2.20565e9 1.04853 0.524267 0.851554i \(-0.324339\pi\)
0.524267 + 0.851554i \(0.324339\pi\)
\(462\) 4.93829e8 0.232986
\(463\) 1.04925e9 0.491299 0.245650 0.969359i \(-0.420999\pi\)
0.245650 + 0.969359i \(0.420999\pi\)
\(464\) 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 \(0.651315\pi\)
\(468\) −9.36449e7 −0.0422303
\(469\) −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 \(-0.222917\pi\)
0.764639 + 0.644458i \(0.222917\pi\)
\(480\) 4.92012e8 0.203063
\(481\) −9.71907e8 −0.398214
\(482\) −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 \(-0.511960\pi\)
−0.0375647 + 0.999294i \(0.511960\pi\)
\(488\) 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
\(494\) 4.80282e8 0.179247
\(495\) 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 \(-0.254693\pi\)
0.696604 + 0.717455i \(0.254693\pi\)
\(500\) −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
\(507\) −1.88246e8 −0.0641500
\(508\) −2.07365e9 −0.701800
\(509\) 2.11533e9 0.710993 0.355497 0.934678i \(-0.384312\pi\)
0.355497 + 0.934678i \(0.384312\pi\)
\(510\) −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
\(520\) −4.33073e8 −0.135067
\(521\) −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
\(533\) 1.27426e8 0.0364513
\(534\) −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 \(-0.488994\pi\)
0.0345698 + 0.999402i \(0.488994\pi\)
\(542\) −3.73097e9 −1.00653
\(543\) 1.73846e9 0.465978
\(544\) 6.88488e8 0.183358
\(545\) 6.56689e9 1.73769
\(546\) −2.00841e8 −0.0528054
\(547\) 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 \(-0.894771\pi\)
−0.945852 + 0.324599i \(0.894771\pi\)
\(558\) −1.11033e9 −0.270542
\(559\) −4.43849e8 −0.107472
\(560\) −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 \(0.717514\pi\)
\(572\) −7.59564e8 −0.169699
\(573\) −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 \(-0.352415\pi\)
0.447218 + 0.894425i \(0.352415\pi\)
\(578\) −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
\(585\) −5.63333e8 −0.116337
\(586\) 6.49057e9 1.33242
\(587\) 1.86734e9 0.381056 0.190528 0.981682i \(-0.438980\pi\)
0.190528 + 0.981682i \(0.438980\pi\)
\(588\) 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 \(-0.394147\pi\)
0.326453 + 0.945214i \(0.394147\pi\)
\(594\) 4.80851e9 0.941366
\(595\) 2.37015e9 0.461281
\(596\) −3.64990e9 −0.706187
\(597\) −6.96483e9 −1.33968
\(598\) 1.10855e9 0.211984
\(599\) 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
\(611\) 1.29296e9 0.229319
\(612\) 8.95573e8 0.157932
\(613\) −2.36146e8 −0.0414065 −0.0207033 0.999786i \(-0.506591\pi\)
−0.0207033 + 0.999786i \(0.506591\pi\)
\(614\) −3.68766e9 −0.642927
\(615\) −8.70870e8 −0.150970
\(616\) −8.10386e8 −0.139688
\(617\) 1.27029e9 0.217723 0.108862 0.994057i \(-0.465279\pi\)
0.108862 + 0.994057i \(0.465279\pi\)
\(618\) −5.36318e9 −0.914035
\(619\) 1.63555e9 0.277170 0.138585 0.990351i \(-0.455745\pi\)
0.138585 + 0.990351i \(0.455745\pi\)
\(620\) −5.13488e9 −0.865285
\(621\) −7.01783e9 −1.17593
\(622\) 2.30175e9 0.383523
\(623\) 2.04669e9 0.339112
\(624\) −3.50958e8 −0.0578241
\(625\) −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 \(-0.457445\pi\)
0.133291 + 0.991077i \(0.457445\pi\)
\(632\) 3.38394e9 0.533228
\(633\) 5.19618e9 0.814275
\(634\) −3.94209e9 −0.614347
\(635\) −1.24743e10 −1.93334
\(636\) −4.20410e9 −0.647998
\(637\) −1.62071e9 −0.248438
\(638\) 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 \(-0.534450\pi\)
−0.108018 + 0.994149i \(0.534450\pi\)
\(644\) 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
\(650\) −1.23208e9 −0.175971
\(651\) −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 \(-0.306003\pi\)
0.572424 + 0.819958i \(0.306003\pi\)
\(662\) 3.86687e9 0.518032
\(663\) 1.80029e9 0.239908
\(664\) 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
\(676\) 3.08916e8 0.0384615
\(677\) −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 \(-0.648613\pi\)
−0.450105 + 0.892976i \(0.648613\pi\)
\(684\) 1.16474e9 0.139166
\(685\) 2.06438e10 2.45398
\(686\) −3.65954e9 −0.432805
\(687\) 5.71580e8 0.0672556
\(688\) −8.27494e8 −0.0968736
\(689\) 3.70049e9 0.431014
\(690\) −7.57621e9 −0.877971
\(691\) 1.66382e10 1.91838 0.959188 0.282769i \(-0.0912530\pi\)
0.959188 + 0.282769i \(0.0912530\pi\)
\(692\) −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 \(0.442662\pi\)
\(702\) −1.95563e9 −0.213357
\(703\) 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 \(-0.424044\pi\)
0.236363 + 0.971665i \(0.424044\pi\)
\(710\) −6.42089e9 −0.673273
\(711\) 4.40176e9 0.459286
\(712\) 3.57647e9 0.371342
\(713\) 1.31440e10 1.35804
\(714\) 1.92074e9 0.197481
\(715\) −4.56925e9 −0.467492
\(716\) −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 \(0.759508\pi\)
\(728\) 3.29585e8 0.0316598
\(729\) 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 \(-0.602610\pi\)
−0.316806 + 0.948490i \(0.602610\pi\)
\(734\) −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 \(-0.335937\pi\)
0.492899 + 0.870087i \(0.335937\pi\)
\(740\) −1.09002e10 −0.988836
\(741\) 2.34137e9 0.211401
\(742\) 3.94808e9 0.354791
\(743\) −3.71897e9 −0.332630 −0.166315 0.986073i \(-0.553187\pi\)
−0.166315 + 0.986073i \(0.553187\pi\)
\(744\) −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
\(754\) −2.14846e9 −0.182527
\(755\) −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 \(-0.385346\pi\)
0.352457 + 0.935828i \(0.385346\pi\)
\(762\) −1.01091e10 −0.827691
\(763\) −4.99766e9 −0.407315
\(764\) 1.19292e10 0.967797
\(765\) 5.38743e9 0.435078
\(766\) 9.58202e9 0.770294
\(767\) −9.72458e8 −0.0778193
\(768\) −6.54311e8 −0.0521219
\(769\) 7.81741e9 0.619899 0.309949 0.950753i \(-0.399688\pi\)
0.309949 + 0.950753i \(0.399688\pi\)
\(770\) −4.87498e9 −0.384818
\(771\) −1.10098e10 −0.865144
\(772\) −9.78732e9 −0.765602
\(773\) 1.30864e10 1.01904 0.509522 0.860457i \(-0.329822\pi\)
0.509522 + 0.860457i \(0.329822\pi\)
\(774\) −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
\(780\) −2.11123e9 −0.159296
\(781\) −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 \(-0.437262\pi\)
0.195825 + 0.980639i \(0.437262\pi\)
\(788\) 6.09176e9 0.443507
\(789\) 9.22311e9 0.668510
\(790\) 2.03565e10 1.46895
\(791\) −6.18962e9 −0.444679
\(792\) −1.84204e9 −0.131753
\(793\) −2.38069e9 −0.169530
\(794\) 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
\(806\) 3.66277e9 0.246398
\(807\) 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 \(-0.341061\pi\)
0.478829 + 0.877908i \(0.341061\pi\)
\(812\) −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
\(819\) 4.28718e8 0.0272696
\(820\) 1.42912e9 0.0905149
\(821\) 6.51876e9 0.411115 0.205558 0.978645i \(-0.434099\pi\)
0.205558 + 0.978645i \(0.434099\pi\)
\(822\) 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 \(-0.578767\pi\)
−0.244937 + 0.969539i \(0.578767\pi\)
\(828\) 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
\(832\) 5.75930e8 0.0346688
\(833\) 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 \(-0.575719\pi\)
−0.235640 + 0.971840i \(0.575719\pi\)
\(840\) −2.25249e9 −0.131125
\(841\) −2.30775e9 −0.133783
\(842\) 9.59013e9 0.553646
\(843\) 2.79007e10 1.60405
\(844\) −8.52706e9 −0.488203
\(845\) 1.85832e9 0.105955
\(846\) 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 \(-0.529178\pi\)
−0.0915368 + 0.995802i \(0.529178\pi\)
\(854\) −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 \(0.664356\pi\)
\(858\) −3.70288e9 −0.200140
\(859\) −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 \(-0.763296\pi\)
−0.736017 + 0.676963i \(0.763296\pi\)
\(864\) −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
\(871\) 7.56534e9 0.387940
\(872\) −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 \(-0.824637\pi\)
−0.852043 + 0.523471i \(0.824637\pi\)
\(878\) −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 \(0.408610\pi\)
\(884\) −2.95431e9 −0.143838
\(885\) 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
\(897\) 5.40420e9 0.250010
\(898\) 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
\(910\) 1.98266e9 0.0872175
\(911\) −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
\(923\) 4.58010e9 0.191721
\(924\) −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 \(-0.685014\pi\)
−0.549061 + 0.835783i \(0.685014\pi\)
\(930\) −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
\(936\) 7.49159e8 0.0298613
\(937\) −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 \(-0.705126\pi\)
−0.600737 + 0.799447i \(0.705126\pi\)
\(942\) −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 \(-0.506296\pi\)
−0.0197774 + 0.999804i \(0.506296\pi\)
\(948\) 1.64967e10 0.628880
\(949\) 1.30455e10 0.495485
\(950\) 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
\(962\) 7.77525e9 0.281580
\(963\) −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 \(-0.587952\pi\)
−0.272806 + 0.962069i \(0.587952\pi\)
\(968\) −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
\(975\) −6.00638e9 −0.207537
\(976\) −4.43846e9 −0.152812
\(977\) 5.68857e10 1.95152 0.975758 0.218851i \(-0.0702309\pi\)
0.975758 + 0.218851i \(0.0702309\pi\)
\(978\) −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 \(-0.562247\pi\)
−0.194312 + 0.980940i \(0.562247\pi\)
\(984\) 1.15814e9 0.0387507
\(985\) 3.66457e10 1.22179
\(986\) 2.05467e10 0.682612
\(987\) 6.72492e9 0.222626
\(988\) −3.84225e9 −0.126747
\(989\) 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 26.8.a.a.1.1 1
3.2 odd 2 234.8.a.d.1.1 1
4.3 odd 2 208.8.a.c.1.1 1
13.5 odd 4 338.8.b.b.337.2 2
13.8 odd 4 338.8.b.b.337.1 2
13.12 even 2 338.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.a.1.1 1 1.1 even 1 trivial
208.8.a.c.1.1 1 4.3 odd 2
234.8.a.d.1.1 1 3.2 odd 2
338.8.a.c.1.1 1 13.12 even 2
338.8.b.b.337.1 2 13.8 odd 4
338.8.b.b.337.2 2 13.5 odd 4