Properties

Label 234.8.a.d.1.1
Level $234$
Weight $8$
Character 234.1
Self dual yes
Analytic conductor $73.098$
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] = [234,8,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(73.0980959633\)
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 \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +64.0000 q^{4} -385.000 q^{5} -293.000 q^{7} +512.000 q^{8} +O(q^{10})\) \(q+8.00000 q^{2} +64.0000 q^{4} -385.000 q^{5} -293.000 q^{7} +512.000 q^{8} -3080.00 q^{10} +5402.00 q^{11} +2197.00 q^{13} -2344.00 q^{14} +4096.00 q^{16} +21011.0 q^{17} -27326.0 q^{19} -24640.0 q^{20} +43216.0 q^{22} +63072.0 q^{23} +70100.0 q^{25} +17576.0 q^{26} -18752.0 q^{28} -122238. q^{29} -208396. q^{31} +32768.0 q^{32} +168088. q^{34} +112805. q^{35} -442379. q^{37} -218608. q^{38} -197120. q^{40} -58000.0 q^{41} -202025. q^{43} +345728. q^{44} +504576. q^{46} -588511. q^{47} -737694. q^{49} +560800. q^{50} +140608. q^{52} -1.68434e6 q^{53} -2.07977e6 q^{55} -150016. q^{56} -977904. q^{58} +442630. q^{59} -1.08361e6 q^{61} -1.66717e6 q^{62} +262144. q^{64} -845845. q^{65} +3.44349e6 q^{67} +1.34470e6 q^{68} +902440. q^{70} -2.08470e6 q^{71} +5.93789e6 q^{73} -3.53903e6 q^{74} -1.74886e6 q^{76} -1.58279e6 q^{77} -6.60926e6 q^{79} -1.57696e6 q^{80} -464000. q^{82} +142740. q^{83} -8.08924e6 q^{85} -1.61620e6 q^{86} +2.76582e6 q^{88} +6.98529e6 q^{89} -643721. q^{91} +4.03661e6 q^{92} -4.70809e6 q^{94} +1.05205e7 q^{95} -200762. q^{97} -5.90155e6 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) −385.000 −1.37742 −0.688709 0.725038i \(-0.741822\pi\)
−0.688709 + 0.725038i \(0.741822\pi\)
\(6\) 0 0
\(7\) −293.000 −0.322868 −0.161434 0.986884i \(-0.551612\pi\)
−0.161434 + 0.986884i \(0.551612\pi\)
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) −3080.00 −0.973982
\(11\) 5402.00 1.22371 0.611857 0.790968i \(-0.290423\pi\)
0.611857 + 0.790968i \(0.290423\pi\)
\(12\) 0 0
\(13\) 2197.00 0.277350
\(14\) −2344.00 −0.228302
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 21011.0 1.03723 0.518616 0.855008i \(-0.326448\pi\)
0.518616 + 0.855008i \(0.326448\pi\)
\(18\) 0 0
\(19\) −27326.0 −0.913984 −0.456992 0.889471i \(-0.651073\pi\)
−0.456992 + 0.889471i \(0.651073\pi\)
\(20\) −24640.0 −0.688709
\(21\) 0 0
\(22\) 43216.0 0.865297
\(23\) 63072.0 1.08091 0.540455 0.841373i \(-0.318252\pi\)
0.540455 + 0.841373i \(0.318252\pi\)
\(24\) 0 0
\(25\) 70100.0 0.897280
\(26\) 17576.0 0.196116
\(27\) 0 0
\(28\) −18752.0 −0.161434
\(29\) −122238. −0.930708 −0.465354 0.885125i \(-0.654073\pi\)
−0.465354 + 0.885125i \(0.654073\pi\)
\(30\) 0 0
\(31\) −208396. −1.25639 −0.628194 0.778057i \(-0.716205\pi\)
−0.628194 + 0.778057i \(0.716205\pi\)
\(32\) 32768.0 0.176777
\(33\) 0 0
\(34\) 168088. 0.733433
\(35\) 112805. 0.444724
\(36\) 0 0
\(37\) −442379. −1.43578 −0.717891 0.696156i \(-0.754892\pi\)
−0.717891 + 0.696156i \(0.754892\pi\)
\(38\) −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 \(0.520932\pi\)
\(42\) 0 0
\(43\) −202025. −0.387494 −0.193747 0.981051i \(-0.562064\pi\)
−0.193747 + 0.981051i \(0.562064\pi\)
\(44\) 345728. 0.611857
\(45\) 0 0
\(46\) 504576. 0.764318
\(47\) −588511. −0.826822 −0.413411 0.910545i \(-0.635663\pi\)
−0.413411 + 0.910545i \(0.635663\pi\)
\(48\) 0 0
\(49\) −737694. −0.895757
\(50\) 560800. 0.634473
\(51\) 0 0
\(52\) 140608. 0.138675
\(53\) −1.68434e6 −1.55404 −0.777022 0.629474i \(-0.783271\pi\)
−0.777022 + 0.629474i \(0.783271\pi\)
\(54\) 0 0
\(55\) −2.07977e6 −1.68557
\(56\) −150016. −0.114151
\(57\) 0 0
\(58\) −977904. −0.658110
\(59\) 442630. 0.280581 0.140291 0.990110i \(-0.455196\pi\)
0.140291 + 0.990110i \(0.455196\pi\)
\(60\) 0 0
\(61\) −1.08361e6 −0.611248 −0.305624 0.952152i \(-0.598865\pi\)
−0.305624 + 0.952152i \(0.598865\pi\)
\(62\) −1.66717e6 −0.888400
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −845845. −0.382027
\(66\) 0 0
\(67\) 3.44349e6 1.39874 0.699369 0.714761i \(-0.253464\pi\)
0.699369 + 0.714761i \(0.253464\pi\)
\(68\) 1.34470e6 0.518616
\(69\) 0 0
\(70\) 902440. 0.314467
\(71\) −2.08470e6 −0.691258 −0.345629 0.938371i \(-0.612334\pi\)
−0.345629 + 0.938371i \(0.612334\pi\)
\(72\) 0 0
\(73\) 5.93789e6 1.78650 0.893248 0.449564i \(-0.148421\pi\)
0.893248 + 0.449564i \(0.148421\pi\)
\(74\) −3.53903e6 −1.01525
\(75\) 0 0
\(76\) −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
\(81\) 0 0
\(82\) −464000. −0.0929329
\(83\) 142740. 0.0274014 0.0137007 0.999906i \(-0.495639\pi\)
0.0137007 + 0.999906i \(0.495639\pi\)
\(84\) 0 0
\(85\) −8.08924e6 −1.42870
\(86\) −1.61620e6 −0.274000
\(87\) 0 0
\(88\) 2.76582e6 0.432648
\(89\) 6.98529e6 1.05031 0.525157 0.851005i \(-0.324007\pi\)
0.525157 + 0.851005i \(0.324007\pi\)
\(90\) 0 0
\(91\) −643721. −0.0895474
\(92\) 4.03661e6 0.540455
\(93\) 0 0
\(94\) −4.70809e6 −0.584652
\(95\) 1.05205e7 1.25894
\(96\) 0 0
\(97\) −200762. −0.0223347 −0.0111674 0.999938i \(-0.503555\pi\)
−0.0111674 + 0.999938i \(0.503555\pi\)
\(98\) −5.90155e6 −0.633395
\(99\) 0 0
\(100\) 4.48640e6 0.448640
\(101\) 5.42144e6 0.523588 0.261794 0.965124i \(-0.415686\pi\)
0.261794 + 0.965124i \(0.415686\pi\)
\(102\) 0 0
\(103\) −1.71897e7 −1.55002 −0.775011 0.631948i \(-0.782255\pi\)
−0.775011 + 0.631948i \(0.782255\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 0 0
\(106\) −1.34747e7 −1.09887
\(107\) −1.23582e7 −0.975242 −0.487621 0.873055i \(-0.662135\pi\)
−0.487621 + 0.873055i \(0.662135\pi\)
\(108\) 0 0
\(109\) 1.70569e7 1.26156 0.630778 0.775964i \(-0.282736\pi\)
0.630778 + 0.775964i \(0.282736\pi\)
\(110\) −1.66382e7 −1.19188
\(111\) 0 0
\(112\) −1.20013e6 −0.0807169
\(113\) −2.11250e7 −1.37728 −0.688639 0.725104i \(-0.741792\pi\)
−0.688639 + 0.725104i \(0.741792\pi\)
\(114\) 0 0
\(115\) −2.42827e7 −1.48886
\(116\) −7.82323e6 −0.465354
\(117\) 0 0
\(118\) 3.54104e6 0.198401
\(119\) −6.15622e6 −0.334888
\(120\) 0 0
\(121\) 9.69443e6 0.497478
\(122\) −8.66886e6 −0.432218
\(123\) 0 0
\(124\) −1.33373e7 −0.628194
\(125\) 3.08962e6 0.141488
\(126\) 0 0
\(127\) −3.24008e7 −1.40360 −0.701800 0.712374i \(-0.747620\pi\)
−0.701800 + 0.712374i \(0.747620\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 0 0
\(130\) −6.76676e6 −0.270134
\(131\) 2.64669e7 1.02862 0.514308 0.857605i \(-0.328049\pi\)
0.514308 + 0.857605i \(0.328049\pi\)
\(132\) 0 0
\(133\) 8.00652e6 0.295096
\(134\) 2.75479e7 0.989057
\(135\) 0 0
\(136\) 1.07576e7 0.366717
\(137\) −5.36201e7 −1.78158 −0.890791 0.454413i \(-0.849849\pi\)
−0.890791 + 0.454413i \(0.849849\pi\)
\(138\) 0 0
\(139\) 7.58784e6 0.239644 0.119822 0.992795i \(-0.461768\pi\)
0.119822 + 0.992795i \(0.461768\pi\)
\(140\) 7.21952e6 0.222362
\(141\) 0 0
\(142\) −1.66776e7 −0.488793
\(143\) 1.18682e7 0.339397
\(144\) 0 0
\(145\) 4.70616e7 1.28197
\(146\) 4.75031e7 1.26324
\(147\) 0 0
\(148\) −2.83123e7 −0.717891
\(149\) 5.70297e7 1.41237 0.706187 0.708026i \(-0.250414\pi\)
0.706187 + 0.708026i \(0.250414\pi\)
\(150\) 0 0
\(151\) −2.00648e7 −0.474259 −0.237130 0.971478i \(-0.576207\pi\)
−0.237130 + 0.971478i \(0.576207\pi\)
\(152\) −1.39909e7 −0.323142
\(153\) 0 0
\(154\) −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
\(159\) 0 0
\(160\) −1.26157e7 −0.243495
\(161\) −1.84801e7 −0.348991
\(162\) 0 0
\(163\) −3.13938e7 −0.567789 −0.283895 0.958855i \(-0.591627\pi\)
−0.283895 + 0.958855i \(0.591627\pi\)
\(164\) −3.71200e6 −0.0657135
\(165\) 0 0
\(166\) 1.14192e6 0.0193757
\(167\) −9.22170e7 −1.53216 −0.766079 0.642747i \(-0.777795\pi\)
−0.766079 + 0.642747i \(0.777795\pi\)
\(168\) 0 0
\(169\) 4.82681e6 0.0769231
\(170\) −6.47139e7 −1.01024
\(171\) 0 0
\(172\) −1.29296e7 −0.193747
\(173\) 6.57015e7 0.964748 0.482374 0.875965i \(-0.339775\pi\)
0.482374 + 0.875965i \(0.339775\pi\)
\(174\) 0 0
\(175\) −2.05393e7 −0.289703
\(176\) 2.21266e7 0.305929
\(177\) 0 0
\(178\) 5.58823e7 0.742684
\(179\) 3.20402e6 0.0417551 0.0208776 0.999782i \(-0.493354\pi\)
0.0208776 + 0.999782i \(0.493354\pi\)
\(180\) 0 0
\(181\) −4.45759e7 −0.558760 −0.279380 0.960181i \(-0.590129\pi\)
−0.279380 + 0.960181i \(0.590129\pi\)
\(182\) −5.14977e6 −0.0633195
\(183\) 0 0
\(184\) 3.22929e7 0.382159
\(185\) 1.70316e8 1.97767
\(186\) 0 0
\(187\) 1.13501e8 1.26927
\(188\) −3.76647e7 −0.413411
\(189\) 0 0
\(190\) 8.41641e7 0.890203
\(191\) −1.86394e8 −1.93559 −0.967797 0.251733i \(-0.919000\pi\)
−0.967797 + 0.251733i \(0.919000\pi\)
\(192\) 0 0
\(193\) −1.52927e8 −1.53120 −0.765602 0.643314i \(-0.777559\pi\)
−0.765602 + 0.643314i \(0.777559\pi\)
\(194\) −1.60610e6 −0.0157930
\(195\) 0 0
\(196\) −4.72124e7 −0.447878
\(197\) −9.51837e7 −0.887015 −0.443507 0.896271i \(-0.646266\pi\)
−0.443507 + 0.896271i \(0.646266\pi\)
\(198\) 0 0
\(199\) 1.78585e8 1.60642 0.803212 0.595693i \(-0.203122\pi\)
0.803212 + 0.595693i \(0.203122\pi\)
\(200\) 3.58912e7 0.317236
\(201\) 0 0
\(202\) 4.33715e7 0.370232
\(203\) 3.58157e7 0.300495
\(204\) 0 0
\(205\) 2.23300e7 0.181030
\(206\) −1.37517e8 −1.09603
\(207\) 0 0
\(208\) 8.99891e6 0.0693375
\(209\) −1.47615e8 −1.11846
\(210\) 0 0
\(211\) −1.33235e8 −0.976406 −0.488203 0.872730i \(-0.662348\pi\)
−0.488203 + 0.872730i \(0.662348\pi\)
\(212\) −1.07798e8 −0.777022
\(213\) 0 0
\(214\) −9.88657e7 −0.689600
\(215\) 7.77796e7 0.533742
\(216\) 0 0
\(217\) 6.10600e7 0.405647
\(218\) 1.36455e8 0.892054
\(219\) 0 0
\(220\) −1.33105e8 −0.842783
\(221\) 4.61612e7 0.287676
\(222\) 0 0
\(223\) 1.19394e8 0.720969 0.360484 0.932765i \(-0.382611\pi\)
0.360484 + 0.932765i \(0.382611\pi\)
\(224\) −9.60102e6 −0.0570755
\(225\) 0 0
\(226\) −1.69000e8 −0.973883
\(227\) −1.13656e7 −0.0644911 −0.0322456 0.999480i \(-0.510266\pi\)
−0.0322456 + 0.999480i \(0.510266\pi\)
\(228\) 0 0
\(229\) −1.46559e7 −0.0806470 −0.0403235 0.999187i \(-0.512839\pi\)
−0.0403235 + 0.999187i \(0.512839\pi\)
\(230\) −1.94262e8 −1.05279
\(231\) 0 0
\(232\) −6.25859e7 −0.329055
\(233\) 2.46924e8 1.27885 0.639423 0.768855i \(-0.279173\pi\)
0.639423 + 0.768855i \(0.279173\pi\)
\(234\) 0 0
\(235\) 2.26577e8 1.13888
\(236\) 2.83283e7 0.140291
\(237\) 0 0
\(238\) −4.92498e7 −0.236802
\(239\) 1.61239e7 0.0763971 0.0381985 0.999270i \(-0.487838\pi\)
0.0381985 + 0.999270i \(0.487838\pi\)
\(240\) 0 0
\(241\) 1.14256e8 0.525798 0.262899 0.964823i \(-0.415321\pi\)
0.262899 + 0.964823i \(0.415321\pi\)
\(242\) 7.75555e7 0.351770
\(243\) 0 0
\(244\) −6.93509e7 −0.305624
\(245\) 2.84012e8 1.23383
\(246\) 0 0
\(247\) −6.00352e7 −0.253493
\(248\) −1.06699e8 −0.444200
\(249\) 0 0
\(250\) 2.47170e7 0.100047
\(251\) −2.22704e8 −0.888935 −0.444467 0.895795i \(-0.646607\pi\)
−0.444467 + 0.895795i \(0.646607\pi\)
\(252\) 0 0
\(253\) 3.40715e8 1.32272
\(254\) −2.59207e8 −0.992494
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) −2.82302e8 −1.03741 −0.518703 0.854955i \(-0.673585\pi\)
−0.518703 + 0.854955i \(0.673585\pi\)
\(258\) 0 0
\(259\) 1.29617e8 0.463567
\(260\) −5.41341e7 −0.191013
\(261\) 0 0
\(262\) 2.11735e8 0.727342
\(263\) 2.36490e8 0.801619 0.400809 0.916162i \(-0.368729\pi\)
0.400809 + 0.916162i \(0.368729\pi\)
\(264\) 0 0
\(265\) 6.48469e8 2.14057
\(266\) 6.40521e7 0.208664
\(267\) 0 0
\(268\) 2.20383e8 0.699369
\(269\) 4.82172e8 1.51032 0.755160 0.655541i \(-0.227559\pi\)
0.755160 + 0.655541i \(0.227559\pi\)
\(270\) 0 0
\(271\) 4.66372e8 1.42344 0.711721 0.702462i \(-0.247916\pi\)
0.711721 + 0.702462i \(0.247916\pi\)
\(272\) 8.60611e7 0.259308
\(273\) 0 0
\(274\) −4.28961e8 −1.25977
\(275\) 3.78680e8 1.09801
\(276\) 0 0
\(277\) −1.88709e8 −0.533475 −0.266738 0.963769i \(-0.585946\pi\)
−0.266738 + 0.963769i \(0.585946\pi\)
\(278\) 6.07027e7 0.169454
\(279\) 0 0
\(280\) 5.77562e7 0.157234
\(281\) 7.15402e8 1.92344 0.961718 0.274040i \(-0.0883600\pi\)
0.961718 + 0.274040i \(0.0883600\pi\)
\(282\) 0 0
\(283\) −4.04602e8 −1.06115 −0.530573 0.847639i \(-0.678023\pi\)
−0.530573 + 0.847639i \(0.678023\pi\)
\(284\) −1.33421e8 −0.345629
\(285\) 0 0
\(286\) 9.49456e7 0.239990
\(287\) 1.69940e7 0.0424335
\(288\) 0 0
\(289\) 3.11234e7 0.0758482
\(290\) 3.76493e8 0.906492
\(291\) 0 0
\(292\) 3.80025e8 0.893248
\(293\) 8.11321e8 1.88433 0.942163 0.335156i \(-0.108789\pi\)
0.942163 + 0.335156i \(0.108789\pi\)
\(294\) 0 0
\(295\) −1.70413e8 −0.386478
\(296\) −2.26498e8 −0.507626
\(297\) 0 0
\(298\) 4.56238e8 0.998699
\(299\) 1.38569e8 0.299790
\(300\) 0 0
\(301\) 5.91933e7 0.125109
\(302\) −1.60519e8 −0.335352
\(303\) 0 0
\(304\) −1.11927e8 −0.228496
\(305\) 4.17189e8 0.841945
\(306\) 0 0
\(307\) 4.60958e8 0.909237 0.454618 0.890686i \(-0.349776\pi\)
0.454618 + 0.890686i \(0.349776\pi\)
\(308\) −1.01298e8 −0.197549
\(309\) 0 0
\(310\) 6.41860e8 1.22370
\(311\) 2.87718e8 0.542383 0.271192 0.962525i \(-0.412582\pi\)
0.271192 + 0.962525i \(0.412582\pi\)
\(312\) 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
\(315\) 0 0
\(316\) −4.22992e8 −0.754098
\(317\) −4.92761e8 −0.868818 −0.434409 0.900716i \(-0.643043\pi\)
−0.434409 + 0.900716i \(0.643043\pi\)
\(318\) 0 0
\(319\) −6.60330e8 −1.13892
\(320\) −1.00925e8 −0.172177
\(321\) 0 0
\(322\) −1.47841e8 −0.246774
\(323\) −5.74147e8 −0.948012
\(324\) 0 0
\(325\) 1.54010e8 0.248861
\(326\) −2.51150e8 −0.401488
\(327\) 0 0
\(328\) −2.96960e7 −0.0464665
\(329\) 1.72434e8 0.266954
\(330\) 0 0
\(331\) −4.83358e8 −0.732607 −0.366304 0.930495i \(-0.619377\pi\)
−0.366304 + 0.930495i \(0.619377\pi\)
\(332\) 9.13536e6 0.0137007
\(333\) 0 0
\(334\) −7.37736e8 −1.08340
\(335\) −1.32574e9 −1.92665
\(336\) 0 0
\(337\) 1.30823e9 1.86200 0.930998 0.365025i \(-0.118940\pi\)
0.930998 + 0.365025i \(0.118940\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 0 0
\(340\) −5.17711e8 −0.714350
\(341\) −1.12576e9 −1.53746
\(342\) 0 0
\(343\) 4.57442e8 0.612078
\(344\) −1.03437e8 −0.137000
\(345\) 0 0
\(346\) 5.25612e8 0.682180
\(347\) 8.94842e8 1.14972 0.574861 0.818251i \(-0.305056\pi\)
0.574861 + 0.818251i \(0.305056\pi\)
\(348\) 0 0
\(349\) −5.41626e8 −0.682041 −0.341020 0.940056i \(-0.610772\pi\)
−0.341020 + 0.940056i \(0.610772\pi\)
\(350\) −1.64314e8 −0.204851
\(351\) 0 0
\(352\) 1.77013e8 0.216324
\(353\) −2.25334e8 −0.272656 −0.136328 0.990664i \(-0.543530\pi\)
−0.136328 + 0.990664i \(0.543530\pi\)
\(354\) 0 0
\(355\) 8.02611e8 0.952152
\(356\) 4.47058e8 0.525157
\(357\) 0 0
\(358\) 2.56322e7 0.0295253
\(359\) 4.38763e8 0.500495 0.250247 0.968182i \(-0.419488\pi\)
0.250247 + 0.968182i \(0.419488\pi\)
\(360\) 0 0
\(361\) −1.47161e8 −0.164634
\(362\) −3.56607e8 −0.395103
\(363\) 0 0
\(364\) −4.11981e7 −0.0447737
\(365\) −2.28609e9 −2.46075
\(366\) 0 0
\(367\) 8.08568e8 0.853857 0.426929 0.904285i \(-0.359596\pi\)
0.426929 + 0.904285i \(0.359596\pi\)
\(368\) 2.58343e8 0.270227
\(369\) 0 0
\(370\) 1.36253e9 1.39843
\(371\) 4.93510e8 0.501750
\(372\) 0 0
\(373\) −1.17884e9 −1.17618 −0.588092 0.808794i \(-0.700121\pi\)
−0.588092 + 0.808794i \(0.700121\pi\)
\(374\) 9.08011e8 0.897513
\(375\) 0 0
\(376\) −3.01318e8 −0.292326
\(377\) −2.68557e8 −0.258132
\(378\) 0 0
\(379\) −1.79168e9 −1.69053 −0.845266 0.534345i \(-0.820558\pi\)
−0.845266 + 0.534345i \(0.820558\pi\)
\(380\) 6.73313e8 0.629469
\(381\) 0 0
\(382\) −1.49115e9 −1.36867
\(383\) 1.19775e9 1.08936 0.544680 0.838644i \(-0.316651\pi\)
0.544680 + 0.838644i \(0.316651\pi\)
\(384\) 0 0
\(385\) 6.09373e8 0.544215
\(386\) −1.22341e9 −1.08273
\(387\) 0 0
\(388\) −1.28488e7 −0.0111674
\(389\) 1.43672e8 0.123751 0.0618754 0.998084i \(-0.480292\pi\)
0.0618754 + 0.998084i \(0.480292\pi\)
\(390\) 0 0
\(391\) 1.32521e9 1.12115
\(392\) −3.77699e8 −0.316698
\(393\) 0 0
\(394\) −7.61470e8 −0.627214
\(395\) 2.54456e9 2.07742
\(396\) 0 0
\(397\) −6.17334e8 −0.495169 −0.247584 0.968866i \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(398\) 1.42868e9 1.13591
\(399\) 0 0
\(400\) 2.87130e8 0.224320
\(401\) −1.13305e9 −0.877491 −0.438746 0.898611i \(-0.644577\pi\)
−0.438746 + 0.898611i \(0.644577\pi\)
\(402\) 0 0
\(403\) −4.57846e8 −0.348459
\(404\) 3.46972e8 0.261794
\(405\) 0 0
\(406\) 2.86526e8 0.212482
\(407\) −2.38973e9 −1.75699
\(408\) 0 0
\(409\) −1.04283e9 −0.753670 −0.376835 0.926280i \(-0.622988\pi\)
−0.376835 + 0.926280i \(0.622988\pi\)
\(410\) 1.78640e8 0.128007
\(411\) 0 0
\(412\) −1.10014e9 −0.775011
\(413\) −1.29691e8 −0.0905906
\(414\) 0 0
\(415\) −5.49549e7 −0.0377431
\(416\) 7.19913e7 0.0490290
\(417\) 0 0
\(418\) −1.18092e9 −0.790867
\(419\) −7.09302e8 −0.471066 −0.235533 0.971866i \(-0.575684\pi\)
−0.235533 + 0.971866i \(0.575684\pi\)
\(420\) 0 0
\(421\) −1.19877e9 −0.782974 −0.391487 0.920184i \(-0.628039\pi\)
−0.391487 + 0.920184i \(0.628039\pi\)
\(422\) −1.06588e9 −0.690424
\(423\) 0 0
\(424\) −8.62380e8 −0.549437
\(425\) 1.47287e9 0.930687
\(426\) 0 0
\(427\) 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 \(0.592666\pi\)
\(432\) 0 0
\(433\) −3.81628e8 −0.225908 −0.112954 0.993600i \(-0.536031\pi\)
−0.112954 + 0.993600i \(0.536031\pi\)
\(434\) 4.88480e8 0.286836
\(435\) 0 0
\(436\) 1.09164e9 0.630778
\(437\) −1.72351e9 −0.987933
\(438\) 0 0
\(439\) 1.11683e8 0.0630031 0.0315015 0.999504i \(-0.489971\pi\)
0.0315015 + 0.999504i \(0.489971\pi\)
\(440\) −1.06484e9 −0.595938
\(441\) 0 0
\(442\) 3.69289e8 0.203418
\(443\) −1.45991e9 −0.797837 −0.398919 0.916986i \(-0.630614\pi\)
−0.398919 + 0.916986i \(0.630614\pi\)
\(444\) 0 0
\(445\) −2.68934e9 −1.44672
\(446\) 9.55154e8 0.509802
\(447\) 0 0
\(448\) −7.68082e7 −0.0403585
\(449\) 6.34009e8 0.330547 0.165273 0.986248i \(-0.447149\pi\)
0.165273 + 0.986248i \(0.447149\pi\)
\(450\) 0 0
\(451\) −3.13316e8 −0.160829
\(452\) −1.35200e9 −0.688639
\(453\) 0 0
\(454\) −9.09244e7 −0.0456021
\(455\) 2.47833e8 0.123344
\(456\) 0 0
\(457\) 6.04376e8 0.296211 0.148105 0.988972i \(-0.452683\pi\)
0.148105 + 0.988972i \(0.452683\pi\)
\(458\) −1.17247e8 −0.0570261
\(459\) 0 0
\(460\) −1.55409e9 −0.744432
\(461\) −2.20565e9 −1.04853 −0.524267 0.851554i \(-0.675661\pi\)
−0.524267 + 0.851554i \(0.675661\pi\)
\(462\) 0 0
\(463\) 1.04925e9 0.491299 0.245650 0.969359i \(-0.420999\pi\)
0.245650 + 0.969359i \(0.420999\pi\)
\(464\) −5.00687e8 −0.232677
\(465\) 0 0
\(466\) 1.97539e9 0.904281
\(467\) 2.01461e9 0.915337 0.457668 0.889123i \(-0.348685\pi\)
0.457668 + 0.889123i \(0.348685\pi\)
\(468\) 0 0
\(469\) −1.00894e9 −0.451607
\(470\) 1.81261e9 0.805309
\(471\) 0 0
\(472\) 2.26627e8 0.0992005
\(473\) −1.09134e9 −0.474183
\(474\) 0 0
\(475\) −1.91555e9 −0.820099
\(476\) −3.93998e8 −0.167444
\(477\) 0 0
\(478\) 1.28991e8 0.0540209
\(479\) −3.67842e9 −1.52928 −0.764639 0.644458i \(-0.777083\pi\)
−0.764639 + 0.644458i \(0.777083\pi\)
\(480\) 0 0
\(481\) −9.71907e8 −0.398214
\(482\) 9.14048e8 0.371796
\(483\) 0 0
\(484\) 6.20444e8 0.248739
\(485\) 7.72934e7 0.0307642
\(486\) 0 0
\(487\) −1.91497e8 −0.0751294 −0.0375647 0.999294i \(-0.511960\pi\)
−0.0375647 + 0.999294i \(0.511960\pi\)
\(488\) −5.54807e8 −0.216109
\(489\) 0 0
\(490\) 2.27210e9 0.872450
\(491\) 3.22321e8 0.122886 0.0614431 0.998111i \(-0.480430\pi\)
0.0614431 + 0.998111i \(0.480430\pi\)
\(492\) 0 0
\(493\) −2.56834e9 −0.965359
\(494\) −4.80282e8 −0.179247
\(495\) 0 0
\(496\) −8.53590e8 −0.314097
\(497\) 6.10819e8 0.223185
\(498\) 0 0
\(499\) 3.86695e9 1.39321 0.696604 0.717455i \(-0.254693\pi\)
0.696604 + 0.717455i \(0.254693\pi\)
\(500\) 1.97736e8 0.0707442
\(501\) 0 0
\(502\) −1.78163e9 −0.628572
\(503\) 3.43814e8 0.120458 0.0602290 0.998185i \(-0.480817\pi\)
0.0602290 + 0.998185i \(0.480817\pi\)
\(504\) 0 0
\(505\) −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 \(0.615688\pi\)
\(510\) 0 0
\(511\) −1.73980e9 −0.576802
\(512\) 1.34218e8 0.0441942
\(513\) 0 0
\(514\) −2.25842e9 −0.733556
\(515\) 6.61803e9 2.13503
\(516\) 0 0
\(517\) −3.17914e9 −1.01179
\(518\) 1.03694e9 0.327792
\(519\) 0 0
\(520\) −4.33073e8 −0.135067
\(521\) 1.40622e9 0.435634 0.217817 0.975990i \(-0.430106\pi\)
0.217817 + 0.975990i \(0.430106\pi\)
\(522\) 0 0
\(523\) 2.18120e9 0.666712 0.333356 0.942801i \(-0.391819\pi\)
0.333356 + 0.942801i \(0.391819\pi\)
\(524\) 1.69388e9 0.514308
\(525\) 0 0
\(526\) 1.89192e9 0.566830
\(527\) −4.37861e9 −1.30316
\(528\) 0 0
\(529\) 5.73252e8 0.168365
\(530\) 5.18775e9 1.51361
\(531\) 0 0
\(532\) 5.12417e8 0.147548
\(533\) −1.27426e8 −0.0364513
\(534\) 0 0
\(535\) 4.75791e9 1.34332
\(536\) 1.76306e9 0.494529
\(537\) 0 0
\(538\) 3.85737e9 1.06796
\(539\) −3.98502e9 −1.09615
\(540\) 0 0
\(541\) 2.54634e8 0.0691395 0.0345698 0.999402i \(-0.488994\pi\)
0.0345698 + 0.999402i \(0.488994\pi\)
\(542\) 3.73097e9 1.00653
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 3.02944e9 0.776414
\(551\) 3.34028e9 0.850652
\(552\) 0 0
\(553\) 1.93651e9 0.486948
\(554\) −1.50967e9 −0.377224
\(555\) 0 0
\(556\) 4.85622e8 0.119822
\(557\) 7.71518e9 1.89170 0.945852 0.324599i \(-0.105229\pi\)
0.945852 + 0.324599i \(0.105229\pi\)
\(558\) 0 0
\(559\) −4.43849e8 −0.107472
\(560\) 4.62049e8 0.111181
\(561\) 0 0
\(562\) 5.72321e9 1.36008
\(563\) 8.12996e7 0.0192003 0.00960017 0.999954i \(-0.496944\pi\)
0.00960017 + 0.999954i \(0.496944\pi\)
\(564\) 0 0
\(565\) 8.13312e9 1.89709
\(566\) −3.23681e9 −0.750343
\(567\) 0 0
\(568\) −1.06737e9 −0.244397
\(569\) 5.08814e9 1.15789 0.578944 0.815367i \(-0.303465\pi\)
0.578944 + 0.815367i \(0.303465\pi\)
\(570\) 0 0
\(571\) −5.61762e9 −1.26277 −0.631387 0.775468i \(-0.717514\pi\)
−0.631387 + 0.775468i \(0.717514\pi\)
\(572\) 7.59564e8 0.169699
\(573\) 0 0
\(574\) 1.35952e8 0.0300050
\(575\) 4.42135e9 0.969878
\(576\) 0 0
\(577\) 4.12728e9 0.894435 0.447218 0.894425i \(-0.352415\pi\)
0.447218 + 0.894425i \(0.352415\pi\)
\(578\) 2.48988e8 0.0536328
\(579\) 0 0
\(580\) 3.01194e9 0.640987
\(581\) −4.18228e7 −0.00884702
\(582\) 0 0
\(583\) −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 \(0.561020\pi\)
\(588\) 0 0
\(589\) 5.69463e9 1.14832
\(590\) −1.36330e9 −0.273281
\(591\) 0 0
\(592\) −1.81198e9 −0.358945
\(593\) −3.31544e9 −0.652905 −0.326453 0.945214i \(-0.605853\pi\)
−0.326453 + 0.945214i \(0.605853\pi\)
\(594\) 0 0
\(595\) 2.37015e9 0.461281
\(596\) 3.64990e9 0.706187
\(597\) 0 0
\(598\) 1.10855e9 0.211984
\(599\) −1.93367e9 −0.367610 −0.183805 0.982963i \(-0.558842\pi\)
−0.183805 + 0.982963i \(0.558842\pi\)
\(600\) 0 0
\(601\) −5.88820e9 −1.10643 −0.553213 0.833040i \(-0.686598\pi\)
−0.553213 + 0.833040i \(0.686598\pi\)
\(602\) 4.73547e8 0.0884657
\(603\) 0 0
\(604\) −1.28415e9 −0.237130
\(605\) −3.73236e9 −0.685235
\(606\) 0 0
\(607\) 7.94197e9 1.44135 0.720673 0.693276i \(-0.243833\pi\)
0.720673 + 0.693276i \(0.243833\pi\)
\(608\) −8.95418e8 −0.161571
\(609\) 0 0
\(610\) 3.33751e9 0.595345
\(611\) −1.29296e9 −0.229319
\(612\) 0 0
\(613\) −2.36146e8 −0.0414065 −0.0207033 0.999786i \(-0.506591\pi\)
−0.0207033 + 0.999786i \(0.506591\pi\)
\(614\) 3.68766e9 0.642927
\(615\) 0 0
\(616\) −8.10386e8 −0.139688
\(617\) −1.27029e9 −0.217723 −0.108862 0.994057i \(-0.534721\pi\)
−0.108862 + 0.994057i \(0.534721\pi\)
\(618\) 0 0
\(619\) 1.63555e9 0.277170 0.138585 0.990351i \(-0.455745\pi\)
0.138585 + 0.990351i \(0.455745\pi\)
\(620\) 5.13488e9 0.865285
\(621\) 0 0
\(622\) 2.30175e9 0.383523
\(623\) −2.04669e9 −0.339112
\(624\) 0 0
\(625\) −6.66607e9 −1.09217
\(626\) −7.64943e9 −1.24629
\(627\) 0 0
\(628\) −2.01801e9 −0.325136
\(629\) −9.29483e9 −1.48924
\(630\) 0 0
\(631\) 1.68242e9 0.266582 0.133291 0.991077i \(-0.457445\pi\)
0.133291 + 0.991077i \(0.457445\pi\)
\(632\) −3.38394e9 −0.533228
\(633\) 0 0
\(634\) −3.94209e9 −0.614347
\(635\) 1.24743e10 1.93334
\(636\) 0 0
\(637\) −1.62071e9 −0.248438
\(638\) −5.28264e9 −0.805338
\(639\) 0 0
\(640\) −8.07404e8 −0.121748
\(641\) 1.70575e9 0.255807 0.127903 0.991787i \(-0.459175\pi\)
0.127903 + 0.991787i \(0.459175\pi\)
\(642\) 0 0
\(643\) −1.45635e9 −0.216036 −0.108018 0.994149i \(-0.534450\pi\)
−0.108018 + 0.994149i \(0.534450\pi\)
\(644\) −1.18273e9 −0.174495
\(645\) 0 0
\(646\) −4.59317e9 −0.670346
\(647\) 3.56464e9 0.517430 0.258715 0.965954i \(-0.416701\pi\)
0.258715 + 0.965954i \(0.416701\pi\)
\(648\) 0 0
\(649\) 2.39109e9 0.343352
\(650\) 1.23208e9 0.175971
\(651\) 0 0
\(652\) −2.00920e9 −0.283895
\(653\) 5.86806e9 0.824705 0.412352 0.911024i \(-0.364707\pi\)
0.412352 + 0.911024i \(0.364707\pi\)
\(654\) 0 0
\(655\) −1.01898e10 −1.41683
\(656\) −2.37568e8 −0.0328567
\(657\) 0 0
\(658\) 1.37947e9 0.188765
\(659\) 2.73239e9 0.371915 0.185958 0.982558i \(-0.440461\pi\)
0.185958 + 0.982558i \(0.440461\pi\)
\(660\) 0 0
\(661\) 8.50066e9 1.14485 0.572424 0.819958i \(-0.306003\pi\)
0.572424 + 0.819958i \(0.306003\pi\)
\(662\) −3.86687e9 −0.518032
\(663\) 0 0
\(664\) 7.30829e7 0.00968785
\(665\) −3.08251e9 −0.406470
\(666\) 0 0
\(667\) −7.70980e9 −1.00601
\(668\) −5.90189e9 −0.766079
\(669\) 0 0
\(670\) −1.06059e10 −1.36235
\(671\) −5.85365e9 −0.747994
\(672\) 0 0
\(673\) −3.85727e7 −0.00487784 −0.00243892 0.999997i \(-0.500776\pi\)
−0.00243892 + 0.999997i \(0.500776\pi\)
\(674\) 1.04658e10 1.31663
\(675\) 0 0
\(676\) 3.08916e8 0.0384615
\(677\) 7.34428e9 0.909681 0.454840 0.890573i \(-0.349696\pi\)
0.454840 + 0.890573i \(0.349696\pi\)
\(678\) 0 0
\(679\) 5.88233e7 0.00721116
\(680\) −4.14169e9 −0.505122
\(681\) 0 0
\(682\) −9.00604e9 −1.08715
\(683\) 7.49577e9 0.900210 0.450105 0.892976i \(-0.351387\pi\)
0.450105 + 0.892976i \(0.351387\pi\)
\(684\) 0 0
\(685\) 2.06438e10 2.45398
\(686\) 3.65954e9 0.432805
\(687\) 0 0
\(688\) −8.27494e8 −0.0968736
\(689\) −3.70049e9 −0.431014
\(690\) 0 0
\(691\) 1.66382e10 1.91838 0.959188 0.282769i \(-0.0912530\pi\)
0.959188 + 0.282769i \(0.0912530\pi\)
\(692\) 4.20490e9 0.482374
\(693\) 0 0
\(694\) 7.15873e9 0.812976
\(695\) −2.92132e9 −0.330090
\(696\) 0 0
\(697\) −1.21864e9 −0.136320
\(698\) −4.33300e9 −0.482275
\(699\) 0 0
\(700\) −1.31452e9 −0.144851
\(701\) −3.26804e9 −0.358323 −0.179161 0.983820i \(-0.557338\pi\)
−0.179161 + 0.983820i \(0.557338\pi\)
\(702\) 0 0
\(703\) 1.20884e10 1.31228
\(704\) 1.41610e9 0.152964
\(705\) 0 0
\(706\) −1.80267e9 −0.192797
\(707\) −1.58848e9 −0.169050
\(708\) 0 0
\(709\) 4.48613e9 0.472727 0.236363 0.971665i \(-0.424044\pi\)
0.236363 + 0.971665i \(0.424044\pi\)
\(710\) 6.42089e9 0.673273
\(711\) 0 0
\(712\) 3.57647e9 0.371342
\(713\) −1.31440e10 −1.35804
\(714\) 0 0
\(715\) −4.56925e9 −0.467492
\(716\) 2.05057e8 0.0208776
\(717\) 0 0
\(718\) 3.51011e9 0.353903
\(719\) −5.42385e9 −0.544198 −0.272099 0.962269i \(-0.587718\pi\)
−0.272099 + 0.962269i \(0.587718\pi\)
\(720\) 0 0
\(721\) 5.03658e9 0.500452
\(722\) −1.17729e9 −0.116414
\(723\) 0 0
\(724\) −2.85286e9 −0.279380
\(725\) −8.56888e9 −0.835105
\(726\) 0 0
\(727\) −1.50827e10 −1.45582 −0.727911 0.685672i \(-0.759508\pi\)
−0.727911 + 0.685672i \(0.759508\pi\)
\(728\) −3.29585e8 −0.0316598
\(729\) 0 0
\(730\) −1.82887e10 −1.74001
\(731\) −4.24475e9 −0.401921
\(732\) 0 0
\(733\) −6.75596e9 −0.633612 −0.316806 0.948490i \(-0.602610\pi\)
−0.316806 + 0.948490i \(0.602610\pi\)
\(734\) 6.46854e9 0.603768
\(735\) 0 0
\(736\) 2.06674e9 0.191080
\(737\) 1.86017e10 1.71166
\(738\) 0 0
\(739\) 1.08154e10 0.985797 0.492899 0.870087i \(-0.335937\pi\)
0.492899 + 0.870087i \(0.335937\pi\)
\(740\) 1.09002e10 0.988836
\(741\) 0 0
\(742\) 3.94808e9 0.354791
\(743\) 3.71897e9 0.332630 0.166315 0.986073i \(-0.446813\pi\)
0.166315 + 0.986073i \(0.446813\pi\)
\(744\) 0 0
\(745\) −2.19565e10 −1.94543
\(746\) −9.43075e9 −0.831687
\(747\) 0 0
\(748\) 7.26409e9 0.634637
\(749\) 3.62096e9 0.314874
\(750\) 0 0
\(751\) −2.15786e10 −1.85902 −0.929510 0.368797i \(-0.879770\pi\)
−0.929510 + 0.368797i \(0.879770\pi\)
\(752\) −2.41054e9 −0.206706
\(753\) 0 0
\(754\) −2.14846e9 −0.182527
\(755\) 7.72496e9 0.653253
\(756\) 0 0
\(757\) 7.42446e9 0.622056 0.311028 0.950401i \(-0.399327\pi\)
0.311028 + 0.950401i \(0.399327\pi\)
\(758\) −1.43335e10 −1.19539
\(759\) 0 0
\(760\) 5.38650e9 0.445102
\(761\) −8.57002e9 −0.704913 −0.352457 0.935828i \(-0.614654\pi\)
−0.352457 + 0.935828i \(0.614654\pi\)
\(762\) 0 0
\(763\) −4.99766e9 −0.407315
\(764\) −1.19292e10 −0.967797
\(765\) 0 0
\(766\) 9.58202e9 0.770294
\(767\) 9.72458e8 0.0778193
\(768\) 0 0
\(769\) 7.81741e9 0.619899 0.309949 0.950753i \(-0.399688\pi\)
0.309949 + 0.950753i \(0.399688\pi\)
\(770\) 4.87498e9 0.384818
\(771\) 0 0
\(772\) −9.78732e9 −0.765602
\(773\) −1.30864e10 −1.01904 −0.509522 0.860457i \(-0.670178\pi\)
−0.509522 + 0.860457i \(0.670178\pi\)
\(774\) 0 0
\(775\) −1.46086e10 −1.12733
\(776\) −1.02790e8 −0.00789651
\(777\) 0 0
\(778\) 1.14937e9 0.0875051
\(779\) 1.58491e9 0.120122
\(780\) 0 0
\(781\) −1.12616e10 −0.845903
\(782\) 1.06016e10 0.792775
\(783\) 0 0
\(784\) −3.02159e9 −0.223939
\(785\) 1.21396e10 0.895696
\(786\) 0 0
\(787\) 5.35561e9 0.391649 0.195825 0.980639i \(-0.437262\pi\)
0.195825 + 0.980639i \(0.437262\pi\)
\(788\) −6.09176e9 −0.443507
\(789\) 0 0
\(790\) 2.03565e10 1.46895
\(791\) 6.18962e9 0.444679
\(792\) 0 0
\(793\) −2.38069e9 −0.169530
\(794\) −4.93867e9 −0.350137
\(795\) 0 0
\(796\) 1.14295e10 0.803212
\(797\) −1.21863e10 −0.852641 −0.426320 0.904572i \(-0.640190\pi\)
−0.426320 + 0.904572i \(0.640190\pi\)
\(798\) 0 0
\(799\) −1.23652e10 −0.857606
\(800\) 2.29704e9 0.158618
\(801\) 0 0
\(802\) −9.06438e9 −0.620480
\(803\) 3.20765e10 2.18616
\(804\) 0 0
\(805\) 7.11484e9 0.480706
\(806\) −3.66277e9 −0.246398
\(807\) 0 0
\(808\) 2.77578e9 0.185116
\(809\) −1.32472e10 −0.879636 −0.439818 0.898087i \(-0.644957\pi\)
−0.439818 + 0.898087i \(0.644957\pi\)
\(810\) 0 0
\(811\) 1.45473e10 0.957658 0.478829 0.877908i \(-0.341061\pi\)
0.478829 + 0.877908i \(0.341061\pi\)
\(812\) 2.29221e9 0.150248
\(813\) 0 0
\(814\) −1.91179e10 −1.24238
\(815\) 1.20866e10 0.782083
\(816\) 0 0
\(817\) 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 \(0.565901\pi\)
\(822\) 0 0
\(823\) −6.77944e9 −0.423930 −0.211965 0.977277i \(-0.567986\pi\)
−0.211965 + 0.977277i \(0.567986\pi\)
\(824\) −8.80112e9 −0.548015
\(825\) 0 0
\(826\) −1.03752e9 −0.0640572
\(827\) 7.96808e9 0.489874 0.244937 0.969539i \(-0.421233\pi\)
0.244937 + 0.969539i \(0.421233\pi\)
\(828\) 0 0
\(829\) 3.74439e9 0.228265 0.114133 0.993466i \(-0.463591\pi\)
0.114133 + 0.993466i \(0.463591\pi\)
\(830\) −4.39639e8 −0.0266884
\(831\) 0 0
\(832\) 5.75930e8 0.0346688
\(833\) −1.54997e10 −0.929106
\(834\) 0 0
\(835\) 3.55035e10 2.11042
\(836\) −9.44736e9 −0.559228
\(837\) 0 0
\(838\) −5.67442e9 −0.333094
\(839\) 8.06205e9 0.471280 0.235640 0.971840i \(-0.424281\pi\)
0.235640 + 0.971840i \(0.424281\pi\)
\(840\) 0 0
\(841\) −2.30775e9 −0.133783
\(842\) −9.59013e9 −0.553646
\(843\) 0 0
\(844\) −8.52706e9 −0.488203
\(845\) −1.85832e9 −0.105955
\(846\) 0 0
\(847\) −2.84047e9 −0.160619
\(848\) −6.89904e9 −0.388511
\(849\) 0 0
\(850\) 1.17830e10 0.658095
\(851\) −2.79017e10 −1.55195
\(852\) 0 0
\(853\) −3.31854e9 −0.183074 −0.0915368 0.995802i \(-0.529178\pi\)
−0.0915368 + 0.995802i \(0.529178\pi\)
\(854\) 2.53998e9 0.139549
\(855\) 0 0
\(856\) −6.32740e9 −0.344800
\(857\) 1.81939e10 0.987398 0.493699 0.869633i \(-0.335644\pi\)
0.493699 + 0.869633i \(0.335644\pi\)
\(858\) 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
\(861\) 0 0
\(862\) −7.63322e9 −0.405913
\(863\) 2.77943e10 1.47203 0.736017 0.676963i \(-0.236704\pi\)
0.736017 + 0.676963i \(0.236704\pi\)
\(864\) 0 0
\(865\) −2.52951e10 −1.32886
\(866\) −3.05302e9 −0.159741
\(867\) 0 0
\(868\) 3.90784e9 0.202823
\(869\) −3.57032e10 −1.84560
\(870\) 0 0
\(871\) 7.56534e9 0.387940
\(872\) 8.73311e9 0.446027
\(873\) 0 0
\(874\) −1.37880e10 −0.698574
\(875\) −9.05260e8 −0.0456820
\(876\) 0 0
\(877\) −3.40401e10 −1.70409 −0.852043 0.523471i \(-0.824637\pi\)
−0.852043 + 0.523471i \(0.824637\pi\)
\(878\) 8.93465e8 0.0445499
\(879\) 0 0
\(880\) −8.51874e9 −0.421392
\(881\) −3.24476e10 −1.59870 −0.799350 0.600866i \(-0.794823\pi\)
−0.799350 + 0.600866i \(0.794823\pi\)
\(882\) 0 0
\(883\) 1.15866e10 0.566362 0.283181 0.959066i \(-0.408610\pi\)
0.283181 + 0.959066i \(0.408610\pi\)
\(884\) 2.95431e9 0.143838
\(885\) 0 0
\(886\) −1.16793e10 −0.564156
\(887\) 2.86160e10 1.37682 0.688408 0.725323i \(-0.258310\pi\)
0.688408 + 0.725323i \(0.258310\pi\)
\(888\) 0 0
\(889\) 9.49344e9 0.453177
\(890\) −2.15147e10 −1.02299
\(891\) 0 0
\(892\) 7.64123e9 0.360484
\(893\) 1.60817e10 0.755702
\(894\) 0 0
\(895\) −1.23355e9 −0.0575142
\(896\) −6.14466e8 −0.0285377
\(897\) 0 0
\(898\) 5.07207e9 0.233732
\(899\) 2.54739e10 1.16933
\(900\) 0 0
\(901\) −3.53896e10 −1.61190
\(902\) −2.50653e9 −0.113723
\(903\) 0 0
\(904\) −1.08160e10 −0.486942
\(905\) 1.71617e10 0.769646
\(906\) 0 0
\(907\) 2.28278e10 1.01587 0.507936 0.861395i \(-0.330409\pi\)
0.507936 + 0.861395i \(0.330409\pi\)
\(908\) −7.27395e8 −0.0322456
\(909\) 0 0
\(910\) 1.98266e9 0.0872175
\(911\) 1.76175e10 0.772024 0.386012 0.922494i \(-0.373852\pi\)
0.386012 + 0.922494i \(0.373852\pi\)
\(912\) 0 0
\(913\) 7.71081e8 0.0335315
\(914\) 4.83501e9 0.209453
\(915\) 0 0
\(916\) −9.37978e8 −0.0403235
\(917\) −7.75480e9 −0.332107
\(918\) 0 0
\(919\) −4.93202e9 −0.209614 −0.104807 0.994493i \(-0.533423\pi\)
−0.104807 + 0.994493i \(0.533423\pi\)
\(920\) −1.24328e10 −0.526393
\(921\) 0 0
\(922\) −1.76452e10 −0.741425
\(923\) −4.58010e9 −0.191721
\(924\) 0 0
\(925\) −3.10108e10 −1.28830
\(926\) 8.39401e9 0.347401
\(927\) 0 0
\(928\) −4.00549e9 −0.164527
\(929\) 2.68352e10 1.09812 0.549061 0.835783i \(-0.314986\pi\)
0.549061 + 0.835783i \(0.314986\pi\)
\(930\) 0 0
\(931\) 2.01582e10 0.818707
\(932\) 1.58032e10 0.639423
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 1.45009e10 0.569440
\(941\) 3.07099e10 1.20147 0.600737 0.799447i \(-0.294874\pi\)
0.600737 + 0.799447i \(0.294874\pi\)
\(942\) 0 0
\(943\) −3.65818e9 −0.142061
\(944\) 1.81301e9 0.0701453
\(945\) 0 0
\(946\) −8.73071e9 −0.335298
\(947\) 1.03377e9 0.0395548 0.0197774 0.999804i \(-0.493704\pi\)
0.0197774 + 0.999804i \(0.493704\pi\)
\(948\) 0 0
\(949\) 1.30455e10 0.495485
\(950\) −1.53244e10 −0.579898
\(951\) 0 0
\(952\) −3.15199e9 −0.118401
\(953\) 1.78629e9 0.0668540 0.0334270 0.999441i \(-0.489358\pi\)
0.0334270 + 0.999441i \(0.489358\pi\)
\(954\) 0 0
\(955\) 7.17615e10 2.66612
\(956\) 1.03193e9 0.0381985
\(957\) 0 0
\(958\) −2.94273e10 −1.08136
\(959\) 1.57107e10 0.575215
\(960\) 0 0
\(961\) 1.59163e10 0.578508
\(962\) −7.77525e9 −0.281580
\(963\) 0 0
\(964\) 7.31238e9 0.262899
\(965\) 5.88768e10 2.10911
\(966\) 0 0
\(967\) −1.53418e10 −0.545613 −0.272806 0.962069i \(-0.587952\pi\)
−0.272806 + 0.962069i \(0.587952\pi\)
\(968\) 4.96355e9 0.175885
\(969\) 0 0
\(970\) 6.18347e8 0.0217536
\(971\) 5.22182e8 0.0183043 0.00915217 0.999958i \(-0.497087\pi\)
0.00915217 + 0.999958i \(0.497087\pi\)
\(972\) 0 0
\(973\) −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 \(0.929769\pi\)
\(978\) 0 0
\(979\) 3.77345e10 1.28528
\(980\) 1.81768e10 0.616916
\(981\) 0 0
\(982\) 2.57857e9 0.0868936
\(983\) 1.15736e10 0.388624 0.194312 0.980940i \(-0.437753\pi\)
0.194312 + 0.980940i \(0.437753\pi\)
\(984\) 0 0
\(985\) 3.66457e10 1.22179
\(986\) −2.05467e10 −0.682612
\(987\) 0 0
\(988\) −3.84225e9 −0.126747
\(989\) −1.27421e10 −0.418846
\(990\) 0 0
\(991\) 1.38509e10 0.452086 0.226043 0.974117i \(-0.427421\pi\)
0.226043 + 0.974117i \(0.427421\pi\)
\(992\) −6.82872e9 −0.222100
\(993\) 0 0
\(994\) 4.88655e9 0.157816
\(995\) −6.87554e10 −2.21272
\(996\) 0 0
\(997\) 5.45316e9 0.174267 0.0871334 0.996197i \(-0.472229\pi\)
0.0871334 + 0.996197i \(0.472229\pi\)
\(998\) 3.09356e10 0.985147
\(999\) 0 0
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 234.8.a.d.1.1 1
3.2 odd 2 26.8.a.a.1.1 1
12.11 even 2 208.8.a.c.1.1 1
39.5 even 4 338.8.b.b.337.2 2
39.8 even 4 338.8.b.b.337.1 2
39.38 odd 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 3.2 odd 2
208.8.a.c.1.1 1 12.11 even 2
234.8.a.d.1.1 1 1.1 even 1 trivial
338.8.a.c.1.1 1 39.38 odd 2
338.8.b.b.337.1 2 39.8 even 4
338.8.b.b.337.2 2 39.5 even 4