Properties

Label 384.8.a.l.1.2
Level $384$
Weight $8$
Character 384.1
Self dual yes
Analytic conductor $119.956$
Analytic rank $0$
Dimension $3$
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] = [384,8,Mod(1,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("384.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 286x - 1680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-12.1582\) of defining polynomial
Character \(\chi\) \(=\) 384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +223.082 q^{5} -1526.43 q^{7} +729.000 q^{9} +O(q^{10})\) \(q+27.0000 q^{3} +223.082 q^{5} -1526.43 q^{7} +729.000 q^{9} -1604.21 q^{11} -2959.89 q^{13} +6023.23 q^{15} -20142.8 q^{17} -7745.63 q^{19} -41213.7 q^{21} +71478.4 q^{23} -28359.2 q^{25} +19683.0 q^{27} -25181.4 q^{29} +270532. q^{31} -43313.6 q^{33} -340520. q^{35} -394033. q^{37} -79917.0 q^{39} +181640. q^{41} +555492. q^{43} +162627. q^{45} +669993. q^{47} +1.50646e6 q^{49} -543856. q^{51} +772991. q^{53} -357870. q^{55} -209132. q^{57} +1.30252e6 q^{59} +1.68711e6 q^{61} -1.11277e6 q^{63} -660299. q^{65} +1.08822e6 q^{67} +1.92992e6 q^{69} -859754. q^{71} +2.61965e6 q^{73} -765699. q^{75} +2.44871e6 q^{77} +6.33710e6 q^{79} +531441. q^{81} -8.43435e6 q^{83} -4.49351e6 q^{85} -679899. q^{87} -1.18695e7 q^{89} +4.51807e6 q^{91} +7.30435e6 q^{93} -1.72791e6 q^{95} +4.05179e6 q^{97} -1.16947e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 81 q^{3} + 308 q^{5} + 6 q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 81 q^{3} + 308 q^{5} + 6 q^{7} + 2187 q^{9} + 1468 q^{11} + 3070 q^{13} + 8316 q^{15} - 5126 q^{17} + 14168 q^{19} + 162 q^{21} + 59980 q^{23} + 31529 q^{25} + 59049 q^{27} + 71984 q^{29} + 177470 q^{31} + 39636 q^{33} - 84760 q^{35} - 184014 q^{37} + 82890 q^{39} + 657250 q^{41} - 120544 q^{43} + 224532 q^{45} + 665132 q^{47} + 1204647 q^{49} - 138402 q^{51} - 111888 q^{53} + 1854224 q^{55} + 382536 q^{57} - 1033308 q^{59} + 3584306 q^{61} + 4374 q^{63} + 1230472 q^{65} - 2344900 q^{67} + 1619460 q^{69} + 6311524 q^{71} - 2112174 q^{73} + 851283 q^{75} + 6670456 q^{77} + 2018830 q^{79} + 1594323 q^{81} - 7430484 q^{83} + 9821848 q^{85} + 1943568 q^{87} - 13300410 q^{89} + 10605052 q^{91} + 4791690 q^{93} + 303008 q^{95} + 2960414 q^{97} + 1070172 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 223.082 0.798124 0.399062 0.916924i \(-0.369336\pi\)
0.399062 + 0.916924i \(0.369336\pi\)
\(6\) 0 0
\(7\) −1526.43 −1.68203 −0.841017 0.541009i \(-0.818043\pi\)
−0.841017 + 0.541009i \(0.818043\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −1604.21 −0.363401 −0.181700 0.983354i \(-0.558160\pi\)
−0.181700 + 0.983354i \(0.558160\pi\)
\(12\) 0 0
\(13\) −2959.89 −0.373657 −0.186829 0.982393i \(-0.559821\pi\)
−0.186829 + 0.982393i \(0.559821\pi\)
\(14\) 0 0
\(15\) 6023.23 0.460797
\(16\) 0 0
\(17\) −20142.8 −0.994372 −0.497186 0.867644i \(-0.665633\pi\)
−0.497186 + 0.867644i \(0.665633\pi\)
\(18\) 0 0
\(19\) −7745.63 −0.259071 −0.129536 0.991575i \(-0.541349\pi\)
−0.129536 + 0.991575i \(0.541349\pi\)
\(20\) 0 0
\(21\) −41213.7 −0.971123
\(22\) 0 0
\(23\) 71478.4 1.22498 0.612488 0.790480i \(-0.290169\pi\)
0.612488 + 0.790480i \(0.290169\pi\)
\(24\) 0 0
\(25\) −28359.2 −0.362998
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −25181.4 −0.191729 −0.0958645 0.995394i \(-0.530562\pi\)
−0.0958645 + 0.995394i \(0.530562\pi\)
\(30\) 0 0
\(31\) 270532. 1.63099 0.815496 0.578762i \(-0.196464\pi\)
0.815496 + 0.578762i \(0.196464\pi\)
\(32\) 0 0
\(33\) −43313.6 −0.209810
\(34\) 0 0
\(35\) −340520. −1.34247
\(36\) 0 0
\(37\) −394033. −1.27887 −0.639435 0.768845i \(-0.720832\pi\)
−0.639435 + 0.768845i \(0.720832\pi\)
\(38\) 0 0
\(39\) −79917.0 −0.215731
\(40\) 0 0
\(41\) 181640. 0.411594 0.205797 0.978595i \(-0.434021\pi\)
0.205797 + 0.978595i \(0.434021\pi\)
\(42\) 0 0
\(43\) 555492. 1.06546 0.532731 0.846285i \(-0.321166\pi\)
0.532731 + 0.846285i \(0.321166\pi\)
\(44\) 0 0
\(45\) 162627. 0.266041
\(46\) 0 0
\(47\) 669993. 0.941299 0.470649 0.882320i \(-0.344020\pi\)
0.470649 + 0.882320i \(0.344020\pi\)
\(48\) 0 0
\(49\) 1.50646e6 1.82924
\(50\) 0 0
\(51\) −543856. −0.574101
\(52\) 0 0
\(53\) 772991. 0.713196 0.356598 0.934258i \(-0.383937\pi\)
0.356598 + 0.934258i \(0.383937\pi\)
\(54\) 0 0
\(55\) −357870. −0.290039
\(56\) 0 0
\(57\) −209132. −0.149575
\(58\) 0 0
\(59\) 1.30252e6 0.825659 0.412830 0.910808i \(-0.364541\pi\)
0.412830 + 0.910808i \(0.364541\pi\)
\(60\) 0 0
\(61\) 1.68711e6 0.951677 0.475838 0.879533i \(-0.342145\pi\)
0.475838 + 0.879533i \(0.342145\pi\)
\(62\) 0 0
\(63\) −1.11277e6 −0.560678
\(64\) 0 0
\(65\) −660299. −0.298225
\(66\) 0 0
\(67\) 1.08822e6 0.442032 0.221016 0.975270i \(-0.429063\pi\)
0.221016 + 0.975270i \(0.429063\pi\)
\(68\) 0 0
\(69\) 1.92992e6 0.707240
\(70\) 0 0
\(71\) −859754. −0.285082 −0.142541 0.989789i \(-0.545527\pi\)
−0.142541 + 0.989789i \(0.545527\pi\)
\(72\) 0 0
\(73\) 2.61965e6 0.788158 0.394079 0.919077i \(-0.371064\pi\)
0.394079 + 0.919077i \(0.371064\pi\)
\(74\) 0 0
\(75\) −765699. −0.209577
\(76\) 0 0
\(77\) 2.44871e6 0.611253
\(78\) 0 0
\(79\) 6.33710e6 1.44609 0.723046 0.690800i \(-0.242741\pi\)
0.723046 + 0.690800i \(0.242741\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −8.43435e6 −1.61912 −0.809559 0.587038i \(-0.800294\pi\)
−0.809559 + 0.587038i \(0.800294\pi\)
\(84\) 0 0
\(85\) −4.49351e6 −0.793632
\(86\) 0 0
\(87\) −679899. −0.110695
\(88\) 0 0
\(89\) −1.18695e7 −1.78471 −0.892354 0.451337i \(-0.850947\pi\)
−0.892354 + 0.451337i \(0.850947\pi\)
\(90\) 0 0
\(91\) 4.51807e6 0.628504
\(92\) 0 0
\(93\) 7.30435e6 0.941654
\(94\) 0 0
\(95\) −1.72791e6 −0.206771
\(96\) 0 0
\(97\) 4.05179e6 0.450761 0.225380 0.974271i \(-0.427638\pi\)
0.225380 + 0.974271i \(0.427638\pi\)
\(98\) 0 0
\(99\) −1.16947e6 −0.121134
\(100\) 0 0
\(101\) 4.68684e6 0.452643 0.226321 0.974053i \(-0.427330\pi\)
0.226321 + 0.974053i \(0.427330\pi\)
\(102\) 0 0
\(103\) 1.57360e7 1.41894 0.709468 0.704737i \(-0.248935\pi\)
0.709468 + 0.704737i \(0.248935\pi\)
\(104\) 0 0
\(105\) −9.19405e6 −0.775076
\(106\) 0 0
\(107\) −1.17323e7 −0.925849 −0.462924 0.886398i \(-0.653200\pi\)
−0.462924 + 0.886398i \(0.653200\pi\)
\(108\) 0 0
\(109\) −1.64178e6 −0.121429 −0.0607144 0.998155i \(-0.519338\pi\)
−0.0607144 + 0.998155i \(0.519338\pi\)
\(110\) 0 0
\(111\) −1.06389e7 −0.738356
\(112\) 0 0
\(113\) −1.65935e7 −1.08184 −0.540921 0.841073i \(-0.681924\pi\)
−0.540921 + 0.841073i \(0.681924\pi\)
\(114\) 0 0
\(115\) 1.59456e7 0.977682
\(116\) 0 0
\(117\) −2.15776e6 −0.124552
\(118\) 0 0
\(119\) 3.07467e7 1.67257
\(120\) 0 0
\(121\) −1.69137e7 −0.867940
\(122\) 0 0
\(123\) 4.90429e6 0.237634
\(124\) 0 0
\(125\) −2.37548e7 −1.08784
\(126\) 0 0
\(127\) −2.38888e7 −1.03486 −0.517431 0.855725i \(-0.673111\pi\)
−0.517431 + 0.855725i \(0.673111\pi\)
\(128\) 0 0
\(129\) 1.49983e7 0.615145
\(130\) 0 0
\(131\) 3.03484e7 1.17947 0.589735 0.807597i \(-0.299232\pi\)
0.589735 + 0.807597i \(0.299232\pi\)
\(132\) 0 0
\(133\) 1.18232e7 0.435767
\(134\) 0 0
\(135\) 4.39093e6 0.153599
\(136\) 0 0
\(137\) 3.84774e7 1.27845 0.639225 0.769020i \(-0.279255\pi\)
0.639225 + 0.769020i \(0.279255\pi\)
\(138\) 0 0
\(139\) 5.72666e7 1.80863 0.904315 0.426867i \(-0.140383\pi\)
0.904315 + 0.426867i \(0.140383\pi\)
\(140\) 0 0
\(141\) 1.80898e7 0.543459
\(142\) 0 0
\(143\) 4.74827e6 0.135787
\(144\) 0 0
\(145\) −5.61754e6 −0.153024
\(146\) 0 0
\(147\) 4.06743e7 1.05611
\(148\) 0 0
\(149\) 5.12193e7 1.26847 0.634237 0.773139i \(-0.281314\pi\)
0.634237 + 0.773139i \(0.281314\pi\)
\(150\) 0 0
\(151\) 2.07825e6 0.0491223 0.0245612 0.999698i \(-0.492181\pi\)
0.0245612 + 0.999698i \(0.492181\pi\)
\(152\) 0 0
\(153\) −1.46841e7 −0.331457
\(154\) 0 0
\(155\) 6.03508e7 1.30173
\(156\) 0 0
\(157\) 7.49433e7 1.54555 0.772777 0.634678i \(-0.218867\pi\)
0.772777 + 0.634678i \(0.218867\pi\)
\(158\) 0 0
\(159\) 2.08707e7 0.411764
\(160\) 0 0
\(161\) −1.09107e8 −2.06045
\(162\) 0 0
\(163\) −2.32916e7 −0.421252 −0.210626 0.977567i \(-0.567550\pi\)
−0.210626 + 0.977567i \(0.567550\pi\)
\(164\) 0 0
\(165\) −9.66250e6 −0.167454
\(166\) 0 0
\(167\) 4.24114e7 0.704653 0.352326 0.935877i \(-0.385391\pi\)
0.352326 + 0.935877i \(0.385391\pi\)
\(168\) 0 0
\(169\) −5.39876e7 −0.860380
\(170\) 0 0
\(171\) −5.64657e6 −0.0863571
\(172\) 0 0
\(173\) −7.97832e7 −1.17152 −0.585760 0.810484i \(-0.699204\pi\)
−0.585760 + 0.810484i \(0.699204\pi\)
\(174\) 0 0
\(175\) 4.32885e7 0.610575
\(176\) 0 0
\(177\) 3.51679e7 0.476694
\(178\) 0 0
\(179\) −2.19077e6 −0.0285503 −0.0142751 0.999898i \(-0.504544\pi\)
−0.0142751 + 0.999898i \(0.504544\pi\)
\(180\) 0 0
\(181\) 5.36384e7 0.672358 0.336179 0.941798i \(-0.390865\pi\)
0.336179 + 0.941798i \(0.390865\pi\)
\(182\) 0 0
\(183\) 4.55520e7 0.549451
\(184\) 0 0
\(185\) −8.79018e7 −1.02070
\(186\) 0 0
\(187\) 3.23132e7 0.361356
\(188\) 0 0
\(189\) −3.00448e7 −0.323708
\(190\) 0 0
\(191\) 1.09646e8 1.13861 0.569305 0.822127i \(-0.307213\pi\)
0.569305 + 0.822127i \(0.307213\pi\)
\(192\) 0 0
\(193\) −1.15412e8 −1.15558 −0.577788 0.816187i \(-0.696084\pi\)
−0.577788 + 0.816187i \(0.696084\pi\)
\(194\) 0 0
\(195\) −1.78281e7 −0.172180
\(196\) 0 0
\(197\) 1.78654e8 1.66487 0.832435 0.554122i \(-0.186946\pi\)
0.832435 + 0.554122i \(0.186946\pi\)
\(198\) 0 0
\(199\) −1.06694e8 −0.959745 −0.479873 0.877338i \(-0.659317\pi\)
−0.479873 + 0.877338i \(0.659317\pi\)
\(200\) 0 0
\(201\) 2.93819e7 0.255208
\(202\) 0 0
\(203\) 3.84378e7 0.322495
\(204\) 0 0
\(205\) 4.05208e7 0.328503
\(206\) 0 0
\(207\) 5.21078e7 0.408325
\(208\) 0 0
\(209\) 1.24256e7 0.0941467
\(210\) 0 0
\(211\) −1.79426e8 −1.31491 −0.657456 0.753493i \(-0.728368\pi\)
−0.657456 + 0.753493i \(0.728368\pi\)
\(212\) 0 0
\(213\) −2.32133e7 −0.164592
\(214\) 0 0
\(215\) 1.23920e8 0.850371
\(216\) 0 0
\(217\) −4.12948e8 −2.74338
\(218\) 0 0
\(219\) 7.07305e7 0.455043
\(220\) 0 0
\(221\) 5.96205e7 0.371554
\(222\) 0 0
\(223\) 2.37823e8 1.43611 0.718054 0.695988i \(-0.245033\pi\)
0.718054 + 0.695988i \(0.245033\pi\)
\(224\) 0 0
\(225\) −2.06739e7 −0.120999
\(226\) 0 0
\(227\) 2.34015e8 1.32786 0.663932 0.747793i \(-0.268886\pi\)
0.663932 + 0.747793i \(0.268886\pi\)
\(228\) 0 0
\(229\) 2.47626e8 1.36261 0.681306 0.731998i \(-0.261412\pi\)
0.681306 + 0.731998i \(0.261412\pi\)
\(230\) 0 0
\(231\) 6.61153e7 0.352907
\(232\) 0 0
\(233\) 2.29473e8 1.18846 0.594231 0.804295i \(-0.297457\pi\)
0.594231 + 0.804295i \(0.297457\pi\)
\(234\) 0 0
\(235\) 1.49464e8 0.751273
\(236\) 0 0
\(237\) 1.71102e8 0.834902
\(238\) 0 0
\(239\) −2.21111e7 −0.104766 −0.0523828 0.998627i \(-0.516682\pi\)
−0.0523828 + 0.998627i \(0.516682\pi\)
\(240\) 0 0
\(241\) 2.31724e8 1.06638 0.533189 0.845996i \(-0.320994\pi\)
0.533189 + 0.845996i \(0.320994\pi\)
\(242\) 0 0
\(243\) 1.43489e7 0.0641500
\(244\) 0 0
\(245\) 3.36064e8 1.45996
\(246\) 0 0
\(247\) 2.29262e7 0.0968039
\(248\) 0 0
\(249\) −2.27728e8 −0.934798
\(250\) 0 0
\(251\) 6.22516e6 0.0248481 0.0124240 0.999923i \(-0.496045\pi\)
0.0124240 + 0.999923i \(0.496045\pi\)
\(252\) 0 0
\(253\) −1.14666e8 −0.445157
\(254\) 0 0
\(255\) −1.21325e8 −0.458204
\(256\) 0 0
\(257\) 2.02916e8 0.745675 0.372837 0.927897i \(-0.378385\pi\)
0.372837 + 0.927897i \(0.378385\pi\)
\(258\) 0 0
\(259\) 6.01465e8 2.15110
\(260\) 0 0
\(261\) −1.83573e7 −0.0639097
\(262\) 0 0
\(263\) 8.91567e7 0.302210 0.151105 0.988518i \(-0.451717\pi\)
0.151105 + 0.988518i \(0.451717\pi\)
\(264\) 0 0
\(265\) 1.72441e8 0.569219
\(266\) 0 0
\(267\) −3.20476e8 −1.03040
\(268\) 0 0
\(269\) 3.29913e8 1.03339 0.516697 0.856168i \(-0.327161\pi\)
0.516697 + 0.856168i \(0.327161\pi\)
\(270\) 0 0
\(271\) 2.51563e8 0.767810 0.383905 0.923372i \(-0.374579\pi\)
0.383905 + 0.923372i \(0.374579\pi\)
\(272\) 0 0
\(273\) 1.21988e8 0.362867
\(274\) 0 0
\(275\) 4.54940e7 0.131914
\(276\) 0 0
\(277\) −3.19740e8 −0.903894 −0.451947 0.892045i \(-0.649270\pi\)
−0.451947 + 0.892045i \(0.649270\pi\)
\(278\) 0 0
\(279\) 1.97218e8 0.543664
\(280\) 0 0
\(281\) −5.79730e8 −1.55867 −0.779334 0.626609i \(-0.784442\pi\)
−0.779334 + 0.626609i \(0.784442\pi\)
\(282\) 0 0
\(283\) 1.52928e8 0.401082 0.200541 0.979685i \(-0.435730\pi\)
0.200541 + 0.979685i \(0.435730\pi\)
\(284\) 0 0
\(285\) −4.66537e7 −0.119379
\(286\) 0 0
\(287\) −2.77262e8 −0.692315
\(288\) 0 0
\(289\) −4.60590e6 −0.0112246
\(290\) 0 0
\(291\) 1.09398e8 0.260247
\(292\) 0 0
\(293\) 6.33108e7 0.147042 0.0735210 0.997294i \(-0.476576\pi\)
0.0735210 + 0.997294i \(0.476576\pi\)
\(294\) 0 0
\(295\) 2.90568e8 0.658978
\(296\) 0 0
\(297\) −3.15756e7 −0.0699365
\(298\) 0 0
\(299\) −2.11568e8 −0.457721
\(300\) 0 0
\(301\) −8.47921e8 −1.79214
\(302\) 0 0
\(303\) 1.26545e8 0.261333
\(304\) 0 0
\(305\) 3.76365e8 0.759556
\(306\) 0 0
\(307\) 1.53352e8 0.302485 0.151243 0.988497i \(-0.451673\pi\)
0.151243 + 0.988497i \(0.451673\pi\)
\(308\) 0 0
\(309\) 4.24871e8 0.819224
\(310\) 0 0
\(311\) −2.53440e8 −0.477764 −0.238882 0.971049i \(-0.576781\pi\)
−0.238882 + 0.971049i \(0.576781\pi\)
\(312\) 0 0
\(313\) −4.34566e8 −0.801033 −0.400517 0.916290i \(-0.631169\pi\)
−0.400517 + 0.916290i \(0.631169\pi\)
\(314\) 0 0
\(315\) −2.48239e8 −0.447491
\(316\) 0 0
\(317\) −5.39819e8 −0.951789 −0.475895 0.879502i \(-0.657876\pi\)
−0.475895 + 0.879502i \(0.657876\pi\)
\(318\) 0 0
\(319\) 4.03963e7 0.0696745
\(320\) 0 0
\(321\) −3.16772e8 −0.534539
\(322\) 0 0
\(323\) 1.56019e8 0.257613
\(324\) 0 0
\(325\) 8.39401e7 0.135637
\(326\) 0 0
\(327\) −4.43280e7 −0.0701069
\(328\) 0 0
\(329\) −1.02270e9 −1.58330
\(330\) 0 0
\(331\) −5.78667e7 −0.0877062 −0.0438531 0.999038i \(-0.513963\pi\)
−0.0438531 + 0.999038i \(0.513963\pi\)
\(332\) 0 0
\(333\) −2.87250e8 −0.426290
\(334\) 0 0
\(335\) 2.42762e8 0.352797
\(336\) 0 0
\(337\) 1.27253e8 0.181118 0.0905592 0.995891i \(-0.471135\pi\)
0.0905592 + 0.995891i \(0.471135\pi\)
\(338\) 0 0
\(339\) −4.48025e8 −0.624602
\(340\) 0 0
\(341\) −4.33989e8 −0.592704
\(342\) 0 0
\(343\) −1.04242e9 −1.39481
\(344\) 0 0
\(345\) 4.30531e8 0.564465
\(346\) 0 0
\(347\) −8.70640e8 −1.11863 −0.559314 0.828956i \(-0.688935\pi\)
−0.559314 + 0.828956i \(0.688935\pi\)
\(348\) 0 0
\(349\) −6.54583e8 −0.824282 −0.412141 0.911120i \(-0.635219\pi\)
−0.412141 + 0.911120i \(0.635219\pi\)
\(350\) 0 0
\(351\) −5.82595e7 −0.0719104
\(352\) 0 0
\(353\) 1.18126e9 1.42933 0.714667 0.699464i \(-0.246578\pi\)
0.714667 + 0.699464i \(0.246578\pi\)
\(354\) 0 0
\(355\) −1.91796e8 −0.227531
\(356\) 0 0
\(357\) 8.30160e8 0.965657
\(358\) 0 0
\(359\) −2.73290e8 −0.311740 −0.155870 0.987778i \(-0.549818\pi\)
−0.155870 + 0.987778i \(0.549818\pi\)
\(360\) 0 0
\(361\) −8.33877e8 −0.932882
\(362\) 0 0
\(363\) −4.56670e8 −0.501105
\(364\) 0 0
\(365\) 5.84398e8 0.629048
\(366\) 0 0
\(367\) −1.33000e9 −1.40449 −0.702246 0.711935i \(-0.747819\pi\)
−0.702246 + 0.711935i \(0.747819\pi\)
\(368\) 0 0
\(369\) 1.32416e8 0.137198
\(370\) 0 0
\(371\) −1.17992e9 −1.19962
\(372\) 0 0
\(373\) 4.52311e8 0.451290 0.225645 0.974210i \(-0.427551\pi\)
0.225645 + 0.974210i \(0.427551\pi\)
\(374\) 0 0
\(375\) −6.41379e8 −0.628066
\(376\) 0 0
\(377\) 7.45343e7 0.0716409
\(378\) 0 0
\(379\) 1.64402e9 1.55121 0.775603 0.631221i \(-0.217446\pi\)
0.775603 + 0.631221i \(0.217446\pi\)
\(380\) 0 0
\(381\) −6.44999e8 −0.597477
\(382\) 0 0
\(383\) 6.98969e7 0.0635714 0.0317857 0.999495i \(-0.489881\pi\)
0.0317857 + 0.999495i \(0.489881\pi\)
\(384\) 0 0
\(385\) 5.46265e8 0.487855
\(386\) 0 0
\(387\) 4.04953e8 0.355154
\(388\) 0 0
\(389\) 8.76666e8 0.755110 0.377555 0.925987i \(-0.376765\pi\)
0.377555 + 0.925987i \(0.376765\pi\)
\(390\) 0 0
\(391\) −1.43978e9 −1.21808
\(392\) 0 0
\(393\) 8.19408e8 0.680967
\(394\) 0 0
\(395\) 1.41370e9 1.15416
\(396\) 0 0
\(397\) −1.95321e9 −1.56669 −0.783344 0.621588i \(-0.786488\pi\)
−0.783344 + 0.621588i \(0.786488\pi\)
\(398\) 0 0
\(399\) 3.19226e8 0.251590
\(400\) 0 0
\(401\) 1.56622e9 1.21296 0.606481 0.795098i \(-0.292580\pi\)
0.606481 + 0.795098i \(0.292580\pi\)
\(402\) 0 0
\(403\) −8.00743e8 −0.609432
\(404\) 0 0
\(405\) 1.18555e8 0.0886805
\(406\) 0 0
\(407\) 6.32110e8 0.464742
\(408\) 0 0
\(409\) 1.15083e9 0.831724 0.415862 0.909428i \(-0.363480\pi\)
0.415862 + 0.909428i \(0.363480\pi\)
\(410\) 0 0
\(411\) 1.03889e9 0.738113
\(412\) 0 0
\(413\) −1.98820e9 −1.38879
\(414\) 0 0
\(415\) −1.88156e9 −1.29226
\(416\) 0 0
\(417\) 1.54620e9 1.04421
\(418\) 0 0
\(419\) −1.67886e9 −1.11498 −0.557488 0.830185i \(-0.688235\pi\)
−0.557488 + 0.830185i \(0.688235\pi\)
\(420\) 0 0
\(421\) 1.42077e9 0.927976 0.463988 0.885842i \(-0.346418\pi\)
0.463988 + 0.885842i \(0.346418\pi\)
\(422\) 0 0
\(423\) 4.88425e8 0.313766
\(424\) 0 0
\(425\) 5.71234e8 0.360955
\(426\) 0 0
\(427\) −2.57526e9 −1.60075
\(428\) 0 0
\(429\) 1.28203e8 0.0783969
\(430\) 0 0
\(431\) 5.76731e8 0.346978 0.173489 0.984836i \(-0.444496\pi\)
0.173489 + 0.984836i \(0.444496\pi\)
\(432\) 0 0
\(433\) −1.29171e9 −0.764642 −0.382321 0.924030i \(-0.624875\pi\)
−0.382321 + 0.924030i \(0.624875\pi\)
\(434\) 0 0
\(435\) −1.51674e8 −0.0883482
\(436\) 0 0
\(437\) −5.53645e8 −0.317356
\(438\) 0 0
\(439\) −1.60873e9 −0.907523 −0.453761 0.891123i \(-0.649918\pi\)
−0.453761 + 0.891123i \(0.649918\pi\)
\(440\) 0 0
\(441\) 1.09821e9 0.609746
\(442\) 0 0
\(443\) −2.37709e9 −1.29907 −0.649534 0.760333i \(-0.725036\pi\)
−0.649534 + 0.760333i \(0.725036\pi\)
\(444\) 0 0
\(445\) −2.64787e9 −1.42442
\(446\) 0 0
\(447\) 1.38292e9 0.732353
\(448\) 0 0
\(449\) 3.25417e9 1.69659 0.848297 0.529520i \(-0.177628\pi\)
0.848297 + 0.529520i \(0.177628\pi\)
\(450\) 0 0
\(451\) −2.91389e8 −0.149574
\(452\) 0 0
\(453\) 5.61128e7 0.0283608
\(454\) 0 0
\(455\) 1.00790e9 0.501624
\(456\) 0 0
\(457\) −3.30664e9 −1.62062 −0.810309 0.586003i \(-0.800701\pi\)
−0.810309 + 0.586003i \(0.800701\pi\)
\(458\) 0 0
\(459\) −3.96471e8 −0.191367
\(460\) 0 0
\(461\) −4.00472e9 −1.90379 −0.951894 0.306428i \(-0.900866\pi\)
−0.951894 + 0.306428i \(0.900866\pi\)
\(462\) 0 0
\(463\) −1.35702e9 −0.635407 −0.317703 0.948190i \(-0.602912\pi\)
−0.317703 + 0.948190i \(0.602912\pi\)
\(464\) 0 0
\(465\) 1.62947e9 0.751557
\(466\) 0 0
\(467\) −3.42473e9 −1.55603 −0.778014 0.628247i \(-0.783773\pi\)
−0.778014 + 0.628247i \(0.783773\pi\)
\(468\) 0 0
\(469\) −1.66109e9 −0.743513
\(470\) 0 0
\(471\) 2.02347e9 0.892326
\(472\) 0 0
\(473\) −8.91124e8 −0.387190
\(474\) 0 0
\(475\) 2.19660e8 0.0940423
\(476\) 0 0
\(477\) 5.63510e8 0.237732
\(478\) 0 0
\(479\) 1.35267e8 0.0562366 0.0281183 0.999605i \(-0.491048\pi\)
0.0281183 + 0.999605i \(0.491048\pi\)
\(480\) 0 0
\(481\) 1.16629e9 0.477859
\(482\) 0 0
\(483\) −2.94589e9 −1.18960
\(484\) 0 0
\(485\) 9.03884e8 0.359763
\(486\) 0 0
\(487\) 1.21620e9 0.477148 0.238574 0.971124i \(-0.423320\pi\)
0.238574 + 0.971124i \(0.423320\pi\)
\(488\) 0 0
\(489\) −6.28873e8 −0.243210
\(490\) 0 0
\(491\) 1.04610e8 0.0398831 0.0199416 0.999801i \(-0.493652\pi\)
0.0199416 + 0.999801i \(0.493652\pi\)
\(492\) 0 0
\(493\) 5.07225e8 0.190650
\(494\) 0 0
\(495\) −2.60888e8 −0.0966797
\(496\) 0 0
\(497\) 1.31236e9 0.479517
\(498\) 0 0
\(499\) 4.71953e9 1.70038 0.850192 0.526473i \(-0.176486\pi\)
0.850192 + 0.526473i \(0.176486\pi\)
\(500\) 0 0
\(501\) 1.14511e9 0.406831
\(502\) 0 0
\(503\) 2.98374e9 1.04538 0.522689 0.852524i \(-0.324929\pi\)
0.522689 + 0.852524i \(0.324929\pi\)
\(504\) 0 0
\(505\) 1.04555e9 0.361265
\(506\) 0 0
\(507\) −1.45766e9 −0.496741
\(508\) 0 0
\(509\) 3.51373e9 1.18102 0.590509 0.807031i \(-0.298927\pi\)
0.590509 + 0.807031i \(0.298927\pi\)
\(510\) 0 0
\(511\) −3.99872e9 −1.32571
\(512\) 0 0
\(513\) −1.52457e8 −0.0498583
\(514\) 0 0
\(515\) 3.51042e9 1.13249
\(516\) 0 0
\(517\) −1.07481e9 −0.342069
\(518\) 0 0
\(519\) −2.15415e9 −0.676378
\(520\) 0 0
\(521\) −1.74420e7 −0.00540335 −0.00270168 0.999996i \(-0.500860\pi\)
−0.00270168 + 0.999996i \(0.500860\pi\)
\(522\) 0 0
\(523\) −2.15669e9 −0.659222 −0.329611 0.944117i \(-0.606918\pi\)
−0.329611 + 0.944117i \(0.606918\pi\)
\(524\) 0 0
\(525\) 1.16879e9 0.352516
\(526\) 0 0
\(527\) −5.44927e9 −1.62181
\(528\) 0 0
\(529\) 1.70434e9 0.500565
\(530\) 0 0
\(531\) 9.49534e8 0.275220
\(532\) 0 0
\(533\) −5.37635e8 −0.153795
\(534\) 0 0
\(535\) −2.61727e9 −0.738942
\(536\) 0 0
\(537\) −5.91507e7 −0.0164835
\(538\) 0 0
\(539\) −2.41667e9 −0.664746
\(540\) 0 0
\(541\) −4.97673e9 −1.35131 −0.675653 0.737220i \(-0.736138\pi\)
−0.675653 + 0.737220i \(0.736138\pi\)
\(542\) 0 0
\(543\) 1.44824e9 0.388186
\(544\) 0 0
\(545\) −3.66252e8 −0.0969152
\(546\) 0 0
\(547\) 3.97379e9 1.03812 0.519062 0.854737i \(-0.326281\pi\)
0.519062 + 0.854737i \(0.326281\pi\)
\(548\) 0 0
\(549\) 1.22990e9 0.317226
\(550\) 0 0
\(551\) 1.95046e8 0.0496715
\(552\) 0 0
\(553\) −9.67316e9 −2.43238
\(554\) 0 0
\(555\) −2.37335e9 −0.589299
\(556\) 0 0
\(557\) 3.21200e9 0.787558 0.393779 0.919205i \(-0.371167\pi\)
0.393779 + 0.919205i \(0.371167\pi\)
\(558\) 0 0
\(559\) −1.64419e9 −0.398118
\(560\) 0 0
\(561\) 8.72457e8 0.208629
\(562\) 0 0
\(563\) −7.69409e9 −1.81710 −0.908548 0.417781i \(-0.862808\pi\)
−0.908548 + 0.417781i \(0.862808\pi\)
\(564\) 0 0
\(565\) −3.70172e9 −0.863444
\(566\) 0 0
\(567\) −8.11209e8 −0.186893
\(568\) 0 0
\(569\) 7.21967e9 1.64295 0.821475 0.570244i \(-0.193152\pi\)
0.821475 + 0.570244i \(0.193152\pi\)
\(570\) 0 0
\(571\) 6.66999e9 1.49934 0.749668 0.661814i \(-0.230213\pi\)
0.749668 + 0.661814i \(0.230213\pi\)
\(572\) 0 0
\(573\) 2.96043e9 0.657376
\(574\) 0 0
\(575\) −2.02707e9 −0.444664
\(576\) 0 0
\(577\) 1.52193e9 0.329822 0.164911 0.986308i \(-0.447266\pi\)
0.164911 + 0.986308i \(0.447266\pi\)
\(578\) 0 0
\(579\) −3.11611e9 −0.667172
\(580\) 0 0
\(581\) 1.28745e10 2.72341
\(582\) 0 0
\(583\) −1.24004e9 −0.259176
\(584\) 0 0
\(585\) −4.81358e8 −0.0994083
\(586\) 0 0
\(587\) 9.26742e9 1.89115 0.945574 0.325408i \(-0.105502\pi\)
0.945574 + 0.325408i \(0.105502\pi\)
\(588\) 0 0
\(589\) −2.09544e9 −0.422543
\(590\) 0 0
\(591\) 4.82365e9 0.961214
\(592\) 0 0
\(593\) −3.62133e9 −0.713142 −0.356571 0.934268i \(-0.616054\pi\)
−0.356571 + 0.934268i \(0.616054\pi\)
\(594\) 0 0
\(595\) 6.85904e9 1.33492
\(596\) 0 0
\(597\) −2.88075e9 −0.554109
\(598\) 0 0
\(599\) −6.87148e9 −1.30634 −0.653170 0.757211i \(-0.726561\pi\)
−0.653170 + 0.757211i \(0.726561\pi\)
\(600\) 0 0
\(601\) −4.37454e9 −0.822001 −0.411000 0.911635i \(-0.634820\pi\)
−0.411000 + 0.911635i \(0.634820\pi\)
\(602\) 0 0
\(603\) 7.93311e8 0.147344
\(604\) 0 0
\(605\) −3.77315e9 −0.692724
\(606\) 0 0
\(607\) 1.79016e9 0.324887 0.162443 0.986718i \(-0.448063\pi\)
0.162443 + 0.986718i \(0.448063\pi\)
\(608\) 0 0
\(609\) 1.03782e9 0.186192
\(610\) 0 0
\(611\) −1.98310e9 −0.351723
\(612\) 0 0
\(613\) −7.18426e9 −1.25971 −0.629855 0.776713i \(-0.716886\pi\)
−0.629855 + 0.776713i \(0.716886\pi\)
\(614\) 0 0
\(615\) 1.09406e9 0.189661
\(616\) 0 0
\(617\) −4.00365e9 −0.686212 −0.343106 0.939297i \(-0.611479\pi\)
−0.343106 + 0.939297i \(0.611479\pi\)
\(618\) 0 0
\(619\) 8.14329e9 1.38001 0.690006 0.723804i \(-0.257608\pi\)
0.690006 + 0.723804i \(0.257608\pi\)
\(620\) 0 0
\(621\) 1.40691e9 0.235747
\(622\) 0 0
\(623\) 1.81180e10 3.00194
\(624\) 0 0
\(625\) −3.08371e9 −0.505235
\(626\) 0 0
\(627\) 3.35491e8 0.0543556
\(628\) 0 0
\(629\) 7.93693e9 1.27167
\(630\) 0 0
\(631\) 4.92407e9 0.780229 0.390114 0.920766i \(-0.372435\pi\)
0.390114 + 0.920766i \(0.372435\pi\)
\(632\) 0 0
\(633\) −4.84450e9 −0.759165
\(634\) 0 0
\(635\) −5.32918e9 −0.825948
\(636\) 0 0
\(637\) −4.45894e9 −0.683508
\(638\) 0 0
\(639\) −6.26760e8 −0.0950273
\(640\) 0 0
\(641\) −4.25248e9 −0.637734 −0.318867 0.947800i \(-0.603302\pi\)
−0.318867 + 0.947800i \(0.603302\pi\)
\(642\) 0 0
\(643\) −1.99844e9 −0.296452 −0.148226 0.988954i \(-0.547356\pi\)
−0.148226 + 0.988954i \(0.547356\pi\)
\(644\) 0 0
\(645\) 3.34585e9 0.490962
\(646\) 0 0
\(647\) 4.98078e9 0.722990 0.361495 0.932374i \(-0.382267\pi\)
0.361495 + 0.932374i \(0.382267\pi\)
\(648\) 0 0
\(649\) −2.08950e9 −0.300045
\(650\) 0 0
\(651\) −1.11496e10 −1.58389
\(652\) 0 0
\(653\) 9.24503e8 0.129931 0.0649654 0.997888i \(-0.479306\pi\)
0.0649654 + 0.997888i \(0.479306\pi\)
\(654\) 0 0
\(655\) 6.77020e9 0.941363
\(656\) 0 0
\(657\) 1.90972e9 0.262719
\(658\) 0 0
\(659\) 9.40009e9 1.27948 0.639740 0.768592i \(-0.279042\pi\)
0.639740 + 0.768592i \(0.279042\pi\)
\(660\) 0 0
\(661\) −1.20065e10 −1.61700 −0.808501 0.588495i \(-0.799721\pi\)
−0.808501 + 0.588495i \(0.799721\pi\)
\(662\) 0 0
\(663\) 1.60975e9 0.214517
\(664\) 0 0
\(665\) 2.63755e9 0.347796
\(666\) 0 0
\(667\) −1.79993e9 −0.234863
\(668\) 0 0
\(669\) 6.42122e9 0.829137
\(670\) 0 0
\(671\) −2.70648e9 −0.345840
\(672\) 0 0
\(673\) −1.23701e10 −1.56431 −0.782154 0.623086i \(-0.785879\pi\)
−0.782154 + 0.623086i \(0.785879\pi\)
\(674\) 0 0
\(675\) −5.58194e8 −0.0698590
\(676\) 0 0
\(677\) −4.27900e9 −0.530007 −0.265004 0.964247i \(-0.585373\pi\)
−0.265004 + 0.964247i \(0.585373\pi\)
\(678\) 0 0
\(679\) −6.18479e9 −0.758195
\(680\) 0 0
\(681\) 6.31841e9 0.766643
\(682\) 0 0
\(683\) 1.62834e10 1.95557 0.977786 0.209607i \(-0.0672185\pi\)
0.977786 + 0.209607i \(0.0672185\pi\)
\(684\) 0 0
\(685\) 8.58363e9 1.02036
\(686\) 0 0
\(687\) 6.68591e9 0.786705
\(688\) 0 0
\(689\) −2.28797e9 −0.266491
\(690\) 0 0
\(691\) 1.32503e10 1.52775 0.763875 0.645364i \(-0.223294\pi\)
0.763875 + 0.645364i \(0.223294\pi\)
\(692\) 0 0
\(693\) 1.78511e9 0.203751
\(694\) 0 0
\(695\) 1.27752e10 1.44351
\(696\) 0 0
\(697\) −3.65875e9 −0.409277
\(698\) 0 0
\(699\) 6.19576e9 0.686159
\(700\) 0 0
\(701\) 4.06887e9 0.446130 0.223065 0.974804i \(-0.428394\pi\)
0.223065 + 0.974804i \(0.428394\pi\)
\(702\) 0 0
\(703\) 3.05203e9 0.331318
\(704\) 0 0
\(705\) 4.03552e9 0.433748
\(706\) 0 0
\(707\) −7.15415e9 −0.761360
\(708\) 0 0
\(709\) 1.59855e10 1.68447 0.842236 0.539109i \(-0.181239\pi\)
0.842236 + 0.539109i \(0.181239\pi\)
\(710\) 0 0
\(711\) 4.61975e9 0.482031
\(712\) 0 0
\(713\) 1.93372e10 1.99793
\(714\) 0 0
\(715\) 1.05926e9 0.108375
\(716\) 0 0
\(717\) −5.97001e8 −0.0604864
\(718\) 0 0
\(719\) −4.52111e9 −0.453622 −0.226811 0.973939i \(-0.572830\pi\)
−0.226811 + 0.973939i \(0.572830\pi\)
\(720\) 0 0
\(721\) −2.40199e10 −2.38670
\(722\) 0 0
\(723\) 6.25654e9 0.615673
\(724\) 0 0
\(725\) 7.14126e8 0.0695972
\(726\) 0 0
\(727\) −1.67194e10 −1.61380 −0.806899 0.590689i \(-0.798856\pi\)
−0.806899 + 0.590689i \(0.798856\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) −1.11892e10 −1.05947
\(732\) 0 0
\(733\) 9.84678e9 0.923486 0.461743 0.887014i \(-0.347224\pi\)
0.461743 + 0.887014i \(0.347224\pi\)
\(734\) 0 0
\(735\) 9.07372e9 0.842907
\(736\) 0 0
\(737\) −1.74573e9 −0.160635
\(738\) 0 0
\(739\) 4.02442e9 0.366816 0.183408 0.983037i \(-0.441287\pi\)
0.183408 + 0.983037i \(0.441287\pi\)
\(740\) 0 0
\(741\) 6.19007e8 0.0558897
\(742\) 0 0
\(743\) 1.10258e10 0.986161 0.493080 0.869984i \(-0.335871\pi\)
0.493080 + 0.869984i \(0.335871\pi\)
\(744\) 0 0
\(745\) 1.14261e10 1.01240
\(746\) 0 0
\(747\) −6.14864e9 −0.539706
\(748\) 0 0
\(749\) 1.79086e10 1.55731
\(750\) 0 0
\(751\) −6.87188e9 −0.592019 −0.296010 0.955185i \(-0.595656\pi\)
−0.296010 + 0.955185i \(0.595656\pi\)
\(752\) 0 0
\(753\) 1.68079e8 0.0143460
\(754\) 0 0
\(755\) 4.63622e8 0.0392057
\(756\) 0 0
\(757\) 2.04576e10 1.71404 0.857018 0.515287i \(-0.172315\pi\)
0.857018 + 0.515287i \(0.172315\pi\)
\(758\) 0 0
\(759\) −3.09599e9 −0.257012
\(760\) 0 0
\(761\) −2.23380e10 −1.83737 −0.918686 0.394989i \(-0.870748\pi\)
−0.918686 + 0.394989i \(0.870748\pi\)
\(762\) 0 0
\(763\) 2.50606e9 0.204247
\(764\) 0 0
\(765\) −3.27577e9 −0.264544
\(766\) 0 0
\(767\) −3.85530e9 −0.308514
\(768\) 0 0
\(769\) −4.68174e9 −0.371249 −0.185624 0.982621i \(-0.559431\pi\)
−0.185624 + 0.982621i \(0.559431\pi\)
\(770\) 0 0
\(771\) 5.47872e9 0.430515
\(772\) 0 0
\(773\) −5.45602e9 −0.424862 −0.212431 0.977176i \(-0.568138\pi\)
−0.212431 + 0.977176i \(0.568138\pi\)
\(774\) 0 0
\(775\) −7.67206e9 −0.592047
\(776\) 0 0
\(777\) 1.62395e10 1.24194
\(778\) 0 0
\(779\) −1.40692e9 −0.106632
\(780\) 0 0
\(781\) 1.37922e9 0.103599
\(782\) 0 0
\(783\) −4.95646e8 −0.0368983
\(784\) 0 0
\(785\) 1.67185e10 1.23354
\(786\) 0 0
\(787\) 4.83754e9 0.353763 0.176882 0.984232i \(-0.443399\pi\)
0.176882 + 0.984232i \(0.443399\pi\)
\(788\) 0 0
\(789\) 2.40723e9 0.174481
\(790\) 0 0
\(791\) 2.53289e10 1.81969
\(792\) 0 0
\(793\) −4.99366e9 −0.355601
\(794\) 0 0
\(795\) 4.65590e9 0.328639
\(796\) 0 0
\(797\) 2.10631e10 1.47373 0.736865 0.676040i \(-0.236305\pi\)
0.736865 + 0.676040i \(0.236305\pi\)
\(798\) 0 0
\(799\) −1.34955e10 −0.936001
\(800\) 0 0
\(801\) −8.65286e9 −0.594902
\(802\) 0 0
\(803\) −4.20246e9 −0.286417
\(804\) 0 0
\(805\) −2.43399e10 −1.64449
\(806\) 0 0
\(807\) 8.90764e9 0.596630
\(808\) 0 0
\(809\) 1.32648e10 0.880810 0.440405 0.897799i \(-0.354835\pi\)
0.440405 + 0.897799i \(0.354835\pi\)
\(810\) 0 0
\(811\) −1.24665e10 −0.820676 −0.410338 0.911933i \(-0.634589\pi\)
−0.410338 + 0.911933i \(0.634589\pi\)
\(812\) 0 0
\(813\) 6.79220e9 0.443296
\(814\) 0 0
\(815\) −5.19594e9 −0.336212
\(816\) 0 0
\(817\) −4.30263e9 −0.276031
\(818\) 0 0
\(819\) 3.29367e9 0.209501
\(820\) 0 0
\(821\) −1.83660e10 −1.15828 −0.579141 0.815228i \(-0.696612\pi\)
−0.579141 + 0.815228i \(0.696612\pi\)
\(822\) 0 0
\(823\) −4.65247e9 −0.290927 −0.145463 0.989364i \(-0.546467\pi\)
−0.145463 + 0.989364i \(0.546467\pi\)
\(824\) 0 0
\(825\) 1.22834e9 0.0761605
\(826\) 0 0
\(827\) −6.62501e9 −0.407302 −0.203651 0.979044i \(-0.565281\pi\)
−0.203651 + 0.979044i \(0.565281\pi\)
\(828\) 0 0
\(829\) −6.24995e9 −0.381009 −0.190505 0.981686i \(-0.561012\pi\)
−0.190505 + 0.981686i \(0.561012\pi\)
\(830\) 0 0
\(831\) −8.63297e9 −0.521863
\(832\) 0 0
\(833\) −3.03442e10 −1.81894
\(834\) 0 0
\(835\) 9.46124e9 0.562400
\(836\) 0 0
\(837\) 5.32487e9 0.313885
\(838\) 0 0
\(839\) 2.30459e10 1.34719 0.673593 0.739103i \(-0.264750\pi\)
0.673593 + 0.739103i \(0.264750\pi\)
\(840\) 0 0
\(841\) −1.66158e10 −0.963240
\(842\) 0 0
\(843\) −1.56527e10 −0.899897
\(844\) 0 0
\(845\) −1.20437e10 −0.686690
\(846\) 0 0
\(847\) 2.58176e10 1.45990
\(848\) 0 0
\(849\) 4.12905e9 0.231565
\(850\) 0 0
\(851\) −2.81648e10 −1.56658
\(852\) 0 0
\(853\) −3.01575e10 −1.66369 −0.831847 0.555006i \(-0.812716\pi\)
−0.831847 + 0.555006i \(0.812716\pi\)
\(854\) 0 0
\(855\) −1.25965e9 −0.0689237
\(856\) 0 0
\(857\) −1.33465e10 −0.724329 −0.362165 0.932114i \(-0.617962\pi\)
−0.362165 + 0.932114i \(0.617962\pi\)
\(858\) 0 0
\(859\) 1.93560e10 1.04193 0.520965 0.853578i \(-0.325572\pi\)
0.520965 + 0.853578i \(0.325572\pi\)
\(860\) 0 0
\(861\) −7.48607e9 −0.399708
\(862\) 0 0
\(863\) −3.05481e9 −0.161788 −0.0808940 0.996723i \(-0.525778\pi\)
−0.0808940 + 0.996723i \(0.525778\pi\)
\(864\) 0 0
\(865\) −1.77982e10 −0.935019
\(866\) 0 0
\(867\) −1.24359e8 −0.00648055
\(868\) 0 0
\(869\) −1.01660e10 −0.525511
\(870\) 0 0
\(871\) −3.22100e9 −0.165169
\(872\) 0 0
\(873\) 2.95376e9 0.150254
\(874\) 0 0
\(875\) 3.62601e10 1.82979
\(876\) 0 0
\(877\) 1.65091e10 0.826465 0.413233 0.910626i \(-0.364400\pi\)
0.413233 + 0.910626i \(0.364400\pi\)
\(878\) 0 0
\(879\) 1.70939e9 0.0848947
\(880\) 0 0
\(881\) 5.79401e9 0.285472 0.142736 0.989761i \(-0.454410\pi\)
0.142736 + 0.989761i \(0.454410\pi\)
\(882\) 0 0
\(883\) −2.02986e10 −0.992209 −0.496104 0.868263i \(-0.665237\pi\)
−0.496104 + 0.868263i \(0.665237\pi\)
\(884\) 0 0
\(885\) 7.84534e9 0.380461
\(886\) 0 0
\(887\) −1.71123e10 −0.823334 −0.411667 0.911334i \(-0.635053\pi\)
−0.411667 + 0.911334i \(0.635053\pi\)
\(888\) 0 0
\(889\) 3.64647e10 1.74067
\(890\) 0 0
\(891\) −8.52541e8 −0.0403779
\(892\) 0 0
\(893\) −5.18952e9 −0.243863
\(894\) 0 0
\(895\) −4.88722e8 −0.0227867
\(896\) 0 0
\(897\) −5.71234e9 −0.264265
\(898\) 0 0
\(899\) −6.81238e9 −0.312709
\(900\) 0 0
\(901\) −1.55702e10 −0.709182
\(902\) 0 0
\(903\) −2.28939e10 −1.03469
\(904\) 0 0
\(905\) 1.19658e10 0.536625
\(906\) 0 0
\(907\) 6.34285e9 0.282266 0.141133 0.989991i \(-0.454925\pi\)
0.141133 + 0.989991i \(0.454925\pi\)
\(908\) 0 0
\(909\) 3.41671e9 0.150881
\(910\) 0 0
\(911\) −1.93974e10 −0.850018 −0.425009 0.905189i \(-0.639729\pi\)
−0.425009 + 0.905189i \(0.639729\pi\)
\(912\) 0 0
\(913\) 1.35305e10 0.588389
\(914\) 0 0
\(915\) 1.01619e10 0.438530
\(916\) 0 0
\(917\) −4.63249e10 −1.98391
\(918\) 0 0
\(919\) −2.42437e10 −1.03038 −0.515188 0.857077i \(-0.672278\pi\)
−0.515188 + 0.857077i \(0.672278\pi\)
\(920\) 0 0
\(921\) 4.14050e9 0.174640
\(922\) 0 0
\(923\) 2.54477e9 0.106523
\(924\) 0 0
\(925\) 1.11745e10 0.464227
\(926\) 0 0
\(927\) 1.14715e10 0.472979
\(928\) 0 0
\(929\) 2.27725e10 0.931872 0.465936 0.884818i \(-0.345718\pi\)
0.465936 + 0.884818i \(0.345718\pi\)
\(930\) 0 0
\(931\) −1.16684e10 −0.473903
\(932\) 0 0
\(933\) −6.84288e9 −0.275837
\(934\) 0 0
\(935\) 7.20852e9 0.288407
\(936\) 0 0
\(937\) 2.68538e10 1.06639 0.533196 0.845992i \(-0.320991\pi\)
0.533196 + 0.845992i \(0.320991\pi\)
\(938\) 0 0
\(939\) −1.17333e10 −0.462477
\(940\) 0 0
\(941\) 3.92886e10 1.53710 0.768552 0.639788i \(-0.220978\pi\)
0.768552 + 0.639788i \(0.220978\pi\)
\(942\) 0 0
\(943\) 1.29834e10 0.504193
\(944\) 0 0
\(945\) −6.70246e9 −0.258359
\(946\) 0 0
\(947\) −9.61412e9 −0.367862 −0.183931 0.982939i \(-0.558882\pi\)
−0.183931 + 0.982939i \(0.558882\pi\)
\(948\) 0 0
\(949\) −7.75387e9 −0.294501
\(950\) 0 0
\(951\) −1.45751e10 −0.549516
\(952\) 0 0
\(953\) 6.03874e9 0.226007 0.113003 0.993595i \(-0.463953\pi\)
0.113003 + 0.993595i \(0.463953\pi\)
\(954\) 0 0
\(955\) 2.44600e10 0.908751
\(956\) 0 0
\(957\) 1.09070e9 0.0402266
\(958\) 0 0
\(959\) −5.87332e10 −2.15040
\(960\) 0 0
\(961\) 4.56747e10 1.66014
\(962\) 0 0
\(963\) −8.55285e9 −0.308616
\(964\) 0 0
\(965\) −2.57463e10 −0.922293
\(966\) 0 0
\(967\) 2.51151e9 0.0893187 0.0446593 0.999002i \(-0.485780\pi\)
0.0446593 + 0.999002i \(0.485780\pi\)
\(968\) 0 0
\(969\) 4.21251e9 0.148733
\(970\) 0 0
\(971\) 8.52174e9 0.298718 0.149359 0.988783i \(-0.452279\pi\)
0.149359 + 0.988783i \(0.452279\pi\)
\(972\) 0 0
\(973\) −8.74136e10 −3.04217
\(974\) 0 0
\(975\) 2.26638e9 0.0783100
\(976\) 0 0
\(977\) −1.00947e10 −0.346307 −0.173153 0.984895i \(-0.555396\pi\)
−0.173153 + 0.984895i \(0.555396\pi\)
\(978\) 0 0
\(979\) 1.90411e10 0.648564
\(980\) 0 0
\(981\) −1.19686e9 −0.0404762
\(982\) 0 0
\(983\) −2.91948e10 −0.980322 −0.490161 0.871632i \(-0.663062\pi\)
−0.490161 + 0.871632i \(0.663062\pi\)
\(984\) 0 0
\(985\) 3.98545e10 1.32877
\(986\) 0 0
\(987\) −2.76129e10 −0.914117
\(988\) 0 0
\(989\) 3.97057e10 1.30516
\(990\) 0 0
\(991\) −1.49553e10 −0.488132 −0.244066 0.969759i \(-0.578481\pi\)
−0.244066 + 0.969759i \(0.578481\pi\)
\(992\) 0 0
\(993\) −1.56240e9 −0.0506372
\(994\) 0 0
\(995\) −2.38017e10 −0.765996
\(996\) 0 0
\(997\) 5.03923e9 0.161039 0.0805196 0.996753i \(-0.474342\pi\)
0.0805196 + 0.996753i \(0.474342\pi\)
\(998\) 0 0
\(999\) −7.75575e9 −0.246119
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 384.8.a.l.1.2 yes 3
4.3 odd 2 384.8.a.j.1.2 yes 3
8.3 odd 2 384.8.a.k.1.2 yes 3
8.5 even 2 384.8.a.i.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.a.i.1.2 3 8.5 even 2
384.8.a.j.1.2 yes 3 4.3 odd 2
384.8.a.k.1.2 yes 3 8.3 odd 2
384.8.a.l.1.2 yes 3 1.1 even 1 trivial