Properties

Label 384.9.b.b.319.6
Level $384$
Weight $9$
Character 384.319
Analytic conductor $156.433$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,9,Mod(319,384)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4826x^{6} + 8748877x^{4} + 7060845096x^{2} + 2140819627716 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.6
Root \(-1.73205 + 35.9608i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.9.b.b.319.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+46.7654 q^{3} -197.353i q^{5} -3794.06i q^{7} +2187.00 q^{9} +O(q^{10})\) \(q+46.7654 q^{3} -197.353i q^{5} -3794.06i q^{7} +2187.00 q^{9} -16480.4 q^{11} -11110.6i q^{13} -9229.26i q^{15} +66738.9 q^{17} -20481.8 q^{19} -177431. i q^{21} -44505.2i q^{23} +351677. q^{25} +102276. q^{27} -725037. i q^{29} -44205.0i q^{31} -770713. q^{33} -748768. q^{35} -2.85851e6i q^{37} -519592. i q^{39} -10702.9 q^{41} +374259. q^{43} -431610. i q^{45} -2.11469e6i q^{47} -8.63010e6 q^{49} +3.12107e6 q^{51} -5.99442e6i q^{53} +3.25245e6i q^{55} -957837. q^{57} -1.11289e7 q^{59} +2.30232e7i q^{61} -8.29761e6i q^{63} -2.19271e6 q^{65} +1.77776e7 q^{67} -2.08130e6i q^{69} -2.44380e7i q^{71} +3.70522e6 q^{73} +1.64463e7 q^{75} +6.25277e7i q^{77} +5.98403e7i q^{79} +4.78297e6 q^{81} -5.83719e7 q^{83} -1.31711e7i q^{85} -3.39066e7i q^{87} -8.20248e7 q^{89} -4.21544e7 q^{91} -2.06726e6i q^{93} +4.04213e6i q^{95} -5.81980e7 q^{97} -3.60427e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 17496 q^{9} + 276848 q^{17} - 2841976 q^{25} + 775008 q^{33} + 5826832 q^{41} - 16599928 q^{49} - 42366240 q^{57} - 132963072 q^{65} - 57759760 q^{73} + 38263752 q^{81} - 258778768 q^{89} - 771489424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 46.7654 0.577350
\(4\) 0 0
\(5\) − 197.353i − 0.315764i −0.987458 0.157882i \(-0.949533\pi\)
0.987458 0.157882i \(-0.0504666\pi\)
\(6\) 0 0
\(7\) − 3794.06i − 1.58020i −0.612978 0.790100i \(-0.710028\pi\)
0.612978 0.790100i \(-0.289972\pi\)
\(8\) 0 0
\(9\) 2187.00 0.333333
\(10\) 0 0
\(11\) −16480.4 −1.12563 −0.562817 0.826581i \(-0.690282\pi\)
−0.562817 + 0.826581i \(0.690282\pi\)
\(12\) 0 0
\(13\) − 11110.6i − 0.389014i −0.980901 0.194507i \(-0.937689\pi\)
0.980901 0.194507i \(-0.0623106\pi\)
\(14\) 0 0
\(15\) − 9229.26i − 0.182306i
\(16\) 0 0
\(17\) 66738.9 0.799067 0.399534 0.916719i \(-0.369172\pi\)
0.399534 + 0.916719i \(0.369172\pi\)
\(18\) 0 0
\(19\) −20481.8 −0.157164 −0.0785820 0.996908i \(-0.525039\pi\)
−0.0785820 + 0.996908i \(0.525039\pi\)
\(20\) 0 0
\(21\) − 177431.i − 0.912329i
\(22\) 0 0
\(23\) − 44505.2i − 0.159037i −0.996833 0.0795187i \(-0.974662\pi\)
0.996833 0.0795187i \(-0.0253383\pi\)
\(24\) 0 0
\(25\) 351677. 0.900293
\(26\) 0 0
\(27\) 102276. 0.192450
\(28\) 0 0
\(29\) − 725037.i − 1.02510i −0.858656 0.512552i \(-0.828700\pi\)
0.858656 0.512552i \(-0.171300\pi\)
\(30\) 0 0
\(31\) − 44205.0i − 0.0478657i −0.999714 0.0239329i \(-0.992381\pi\)
0.999714 0.0239329i \(-0.00761880\pi\)
\(32\) 0 0
\(33\) −770713. −0.649885
\(34\) 0 0
\(35\) −748768. −0.498971
\(36\) 0 0
\(37\) − 2.85851e6i − 1.52522i −0.646857 0.762611i \(-0.723917\pi\)
0.646857 0.762611i \(-0.276083\pi\)
\(38\) 0 0
\(39\) − 519592.i − 0.224597i
\(40\) 0 0
\(41\) −10702.9 −0.00378761 −0.00189381 0.999998i \(-0.500603\pi\)
−0.00189381 + 0.999998i \(0.500603\pi\)
\(42\) 0 0
\(43\) 374259. 0.109471 0.0547354 0.998501i \(-0.482568\pi\)
0.0547354 + 0.998501i \(0.482568\pi\)
\(44\) 0 0
\(45\) − 431610.i − 0.105255i
\(46\) 0 0
\(47\) − 2.11469e6i − 0.433365i −0.976242 0.216683i \(-0.930476\pi\)
0.976242 0.216683i \(-0.0695237\pi\)
\(48\) 0 0
\(49\) −8.63010e6 −1.49703
\(50\) 0 0
\(51\) 3.12107e6 0.461342
\(52\) 0 0
\(53\) − 5.99442e6i − 0.759703i −0.925047 0.379852i \(-0.875975\pi\)
0.925047 0.379852i \(-0.124025\pi\)
\(54\) 0 0
\(55\) 3.25245e6i 0.355435i
\(56\) 0 0
\(57\) −957837. −0.0907387
\(58\) 0 0
\(59\) −1.11289e7 −0.918424 −0.459212 0.888327i \(-0.651868\pi\)
−0.459212 + 0.888327i \(0.651868\pi\)
\(60\) 0 0
\(61\) 2.30232e7i 1.66282i 0.555657 + 0.831411i \(0.312467\pi\)
−0.555657 + 0.831411i \(0.687533\pi\)
\(62\) 0 0
\(63\) − 8.29761e6i − 0.526734i
\(64\) 0 0
\(65\) −2.19271e6 −0.122837
\(66\) 0 0
\(67\) 1.77776e7 0.882212 0.441106 0.897455i \(-0.354586\pi\)
0.441106 + 0.897455i \(0.354586\pi\)
\(68\) 0 0
\(69\) − 2.08130e6i − 0.0918203i
\(70\) 0 0
\(71\) − 2.44380e7i − 0.961682i −0.876808 0.480841i \(-0.840331\pi\)
0.876808 0.480841i \(-0.159669\pi\)
\(72\) 0 0
\(73\) 3.70522e6 0.130474 0.0652368 0.997870i \(-0.479220\pi\)
0.0652368 + 0.997870i \(0.479220\pi\)
\(74\) 0 0
\(75\) 1.64463e7 0.519784
\(76\) 0 0
\(77\) 6.25277e7i 1.77873i
\(78\) 0 0
\(79\) 5.98403e7i 1.53633i 0.640251 + 0.768166i \(0.278831\pi\)
−0.640251 + 0.768166i \(0.721169\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) −5.83719e7 −1.22996 −0.614981 0.788542i \(-0.710836\pi\)
−0.614981 + 0.788542i \(0.710836\pi\)
\(84\) 0 0
\(85\) − 1.31711e7i − 0.252317i
\(86\) 0 0
\(87\) − 3.39066e7i − 0.591844i
\(88\) 0 0
\(89\) −8.20248e7 −1.30733 −0.653665 0.756784i \(-0.726769\pi\)
−0.653665 + 0.756784i \(0.726769\pi\)
\(90\) 0 0
\(91\) −4.21544e7 −0.614719
\(92\) 0 0
\(93\) − 2.06726e6i − 0.0276353i
\(94\) 0 0
\(95\) 4.04213e6i 0.0496267i
\(96\) 0 0
\(97\) −5.81980e7 −0.657387 −0.328694 0.944437i \(-0.606608\pi\)
−0.328694 + 0.944437i \(0.606608\pi\)
\(98\) 0 0
\(99\) −3.60427e7 −0.375211
\(100\) 0 0
\(101\) − 7.64434e7i − 0.734606i −0.930101 0.367303i \(-0.880281\pi\)
0.930101 0.367303i \(-0.119719\pi\)
\(102\) 0 0
\(103\) 1.91556e8i 1.70195i 0.525207 + 0.850974i \(0.323988\pi\)
−0.525207 + 0.850974i \(0.676012\pi\)
\(104\) 0 0
\(105\) −3.50164e7 −0.288081
\(106\) 0 0
\(107\) 5.44812e7 0.415635 0.207817 0.978168i \(-0.433364\pi\)
0.207817 + 0.978168i \(0.433364\pi\)
\(108\) 0 0
\(109\) − 1.72940e8i − 1.22515i −0.790413 0.612574i \(-0.790134\pi\)
0.790413 0.612574i \(-0.209866\pi\)
\(110\) 0 0
\(111\) − 1.33679e8i − 0.880588i
\(112\) 0 0
\(113\) −1.07735e8 −0.660761 −0.330380 0.943848i \(-0.607177\pi\)
−0.330380 + 0.943848i \(0.607177\pi\)
\(114\) 0 0
\(115\) −8.78321e6 −0.0502183
\(116\) 0 0
\(117\) − 2.42989e7i − 0.129671i
\(118\) 0 0
\(119\) − 2.53212e8i − 1.26269i
\(120\) 0 0
\(121\) 5.72450e7 0.267052
\(122\) 0 0
\(123\) −500524. −0.00218678
\(124\) 0 0
\(125\) − 1.46495e8i − 0.600044i
\(126\) 0 0
\(127\) 2.05573e8i 0.790225i 0.918633 + 0.395113i \(0.129294\pi\)
−0.918633 + 0.395113i \(0.870706\pi\)
\(128\) 0 0
\(129\) 1.75024e7 0.0632030
\(130\) 0 0
\(131\) −3.21055e7 −0.109017 −0.0545084 0.998513i \(-0.517359\pi\)
−0.0545084 + 0.998513i \(0.517359\pi\)
\(132\) 0 0
\(133\) 7.77091e7i 0.248351i
\(134\) 0 0
\(135\) − 2.01844e7i − 0.0607688i
\(136\) 0 0
\(137\) 6.85467e8 1.94583 0.972913 0.231171i \(-0.0742558\pi\)
0.972913 + 0.231171i \(0.0742558\pi\)
\(138\) 0 0
\(139\) −5.81121e8 −1.55671 −0.778354 0.627826i \(-0.783945\pi\)
−0.778354 + 0.627826i \(0.783945\pi\)
\(140\) 0 0
\(141\) − 9.88940e7i − 0.250204i
\(142\) 0 0
\(143\) 1.83108e8i 0.437887i
\(144\) 0 0
\(145\) −1.43088e8 −0.323691
\(146\) 0 0
\(147\) −4.03590e8 −0.864313
\(148\) 0 0
\(149\) 8.86338e8i 1.79827i 0.437673 + 0.899134i \(0.355803\pi\)
−0.437673 + 0.899134i \(0.644197\pi\)
\(150\) 0 0
\(151\) 2.60061e8i 0.500227i 0.968217 + 0.250114i \(0.0804680\pi\)
−0.968217 + 0.250114i \(0.919532\pi\)
\(152\) 0 0
\(153\) 1.45958e8 0.266356
\(154\) 0 0
\(155\) −8.72397e6 −0.0151143
\(156\) 0 0
\(157\) − 8.08233e7i − 0.133026i −0.997786 0.0665132i \(-0.978813\pi\)
0.997786 0.0665132i \(-0.0211875\pi\)
\(158\) 0 0
\(159\) − 2.80331e8i − 0.438615i
\(160\) 0 0
\(161\) −1.68855e8 −0.251311
\(162\) 0 0
\(163\) −4.11086e8 −0.582348 −0.291174 0.956670i \(-0.594046\pi\)
−0.291174 + 0.956670i \(0.594046\pi\)
\(164\) 0 0
\(165\) 1.52102e8i 0.205210i
\(166\) 0 0
\(167\) − 7.36188e8i − 0.946504i −0.880927 0.473252i \(-0.843080\pi\)
0.880927 0.473252i \(-0.156920\pi\)
\(168\) 0 0
\(169\) 6.92285e8 0.848668
\(170\) 0 0
\(171\) −4.47936e7 −0.0523880
\(172\) 0 0
\(173\) 6.33738e8i 0.707498i 0.935340 + 0.353749i \(0.115093\pi\)
−0.935340 + 0.353749i \(0.884907\pi\)
\(174\) 0 0
\(175\) − 1.33428e9i − 1.42264i
\(176\) 0 0
\(177\) −5.20446e8 −0.530252
\(178\) 0 0
\(179\) −1.50585e9 −1.46679 −0.733396 0.679802i \(-0.762066\pi\)
−0.733396 + 0.679802i \(0.762066\pi\)
\(180\) 0 0
\(181\) 8.25523e8i 0.769157i 0.923092 + 0.384579i \(0.125653\pi\)
−0.923092 + 0.384579i \(0.874347\pi\)
\(182\) 0 0
\(183\) 1.07669e9i 0.960031i
\(184\) 0 0
\(185\) −5.64135e8 −0.481611
\(186\) 0 0
\(187\) −1.09988e9 −0.899458
\(188\) 0 0
\(189\) − 3.88041e8i − 0.304110i
\(190\) 0 0
\(191\) 1.94421e9i 1.46086i 0.682986 + 0.730431i \(0.260681\pi\)
−0.682986 + 0.730431i \(0.739319\pi\)
\(192\) 0 0
\(193\) 9.05725e8 0.652781 0.326390 0.945235i \(-0.394168\pi\)
0.326390 + 0.945235i \(0.394168\pi\)
\(194\) 0 0
\(195\) −1.02543e8 −0.0709197
\(196\) 0 0
\(197\) − 2.17129e9i − 1.44163i −0.693130 0.720813i \(-0.743769\pi\)
0.693130 0.720813i \(-0.256231\pi\)
\(198\) 0 0
\(199\) − 5.33204e8i − 0.340002i −0.985444 0.170001i \(-0.945623\pi\)
0.985444 0.170001i \(-0.0543771\pi\)
\(200\) 0 0
\(201\) 8.31374e8 0.509345
\(202\) 0 0
\(203\) −2.75083e9 −1.61987
\(204\) 0 0
\(205\) 2.11224e6i 0.00119599i
\(206\) 0 0
\(207\) − 9.73328e7i − 0.0530125i
\(208\) 0 0
\(209\) 3.37548e8 0.176909
\(210\) 0 0
\(211\) −1.80735e9 −0.911828 −0.455914 0.890024i \(-0.650688\pi\)
−0.455914 + 0.890024i \(0.650688\pi\)
\(212\) 0 0
\(213\) − 1.14285e9i − 0.555227i
\(214\) 0 0
\(215\) − 7.38610e7i − 0.0345670i
\(216\) 0 0
\(217\) −1.67717e8 −0.0756375
\(218\) 0 0
\(219\) 1.73276e8 0.0753289
\(220\) 0 0
\(221\) − 7.41510e8i − 0.310848i
\(222\) 0 0
\(223\) − 3.30606e9i − 1.33688i −0.743767 0.668439i \(-0.766963\pi\)
0.743767 0.668439i \(-0.233037\pi\)
\(224\) 0 0
\(225\) 7.69118e8 0.300098
\(226\) 0 0
\(227\) −4.39061e9 −1.65357 −0.826783 0.562521i \(-0.809831\pi\)
−0.826783 + 0.562521i \(0.809831\pi\)
\(228\) 0 0
\(229\) − 4.51374e9i − 1.64133i −0.571413 0.820663i \(-0.693605\pi\)
0.571413 0.820663i \(-0.306395\pi\)
\(230\) 0 0
\(231\) 2.92413e9i 1.02695i
\(232\) 0 0
\(233\) 1.03865e9 0.352408 0.176204 0.984354i \(-0.443618\pi\)
0.176204 + 0.984354i \(0.443618\pi\)
\(234\) 0 0
\(235\) −4.17338e8 −0.136841
\(236\) 0 0
\(237\) 2.79845e9i 0.887002i
\(238\) 0 0
\(239\) 4.05969e9i 1.24423i 0.782925 + 0.622116i \(0.213727\pi\)
−0.782925 + 0.622116i \(0.786273\pi\)
\(240\) 0 0
\(241\) −1.83737e9 −0.544663 −0.272331 0.962204i \(-0.587795\pi\)
−0.272331 + 0.962204i \(0.587795\pi\)
\(242\) 0 0
\(243\) 2.23677e8 0.0641500
\(244\) 0 0
\(245\) 1.70317e9i 0.472710i
\(246\) 0 0
\(247\) 2.27565e8i 0.0611389i
\(248\) 0 0
\(249\) −2.72979e9 −0.710119
\(250\) 0 0
\(251\) −2.99843e9 −0.755437 −0.377719 0.925920i \(-0.623291\pi\)
−0.377719 + 0.925920i \(0.623291\pi\)
\(252\) 0 0
\(253\) 7.33464e8i 0.179018i
\(254\) 0 0
\(255\) − 6.15951e8i − 0.145675i
\(256\) 0 0
\(257\) −4.06684e9 −0.932233 −0.466117 0.884723i \(-0.654347\pi\)
−0.466117 + 0.884723i \(0.654347\pi\)
\(258\) 0 0
\(259\) −1.08454e10 −2.41016
\(260\) 0 0
\(261\) − 1.58566e9i − 0.341701i
\(262\) 0 0
\(263\) − 3.67554e9i − 0.768243i −0.923283 0.384122i \(-0.874504\pi\)
0.923283 0.384122i \(-0.125496\pi\)
\(264\) 0 0
\(265\) −1.18301e9 −0.239887
\(266\) 0 0
\(267\) −3.83592e9 −0.754787
\(268\) 0 0
\(269\) 4.57397e9i 0.873542i 0.899573 + 0.436771i \(0.143878\pi\)
−0.899573 + 0.436771i \(0.856122\pi\)
\(270\) 0 0
\(271\) − 1.80061e9i − 0.333843i −0.985970 0.166922i \(-0.946617\pi\)
0.985970 0.166922i \(-0.0533827\pi\)
\(272\) 0 0
\(273\) −1.97136e9 −0.354908
\(274\) 0 0
\(275\) −5.79578e9 −1.01340
\(276\) 0 0
\(277\) 1.08581e10i 1.84431i 0.386823 + 0.922154i \(0.373572\pi\)
−0.386823 + 0.922154i \(0.626428\pi\)
\(278\) 0 0
\(279\) − 9.66764e7i − 0.0159552i
\(280\) 0 0
\(281\) −2.87260e9 −0.460734 −0.230367 0.973104i \(-0.573993\pi\)
−0.230367 + 0.973104i \(0.573993\pi\)
\(282\) 0 0
\(283\) −5.08258e8 −0.0792389 −0.0396195 0.999215i \(-0.512615\pi\)
−0.0396195 + 0.999215i \(0.512615\pi\)
\(284\) 0 0
\(285\) 1.89032e8i 0.0286520i
\(286\) 0 0
\(287\) 4.06074e7i 0.00598519i
\(288\) 0 0
\(289\) −2.52168e9 −0.361491
\(290\) 0 0
\(291\) −2.72165e9 −0.379543
\(292\) 0 0
\(293\) − 2.13811e9i − 0.290108i −0.989424 0.145054i \(-0.953664\pi\)
0.989424 0.145054i \(-0.0463356\pi\)
\(294\) 0 0
\(295\) 2.19631e9i 0.290005i
\(296\) 0 0
\(297\) −1.68555e9 −0.216628
\(298\) 0 0
\(299\) −4.94480e8 −0.0618677
\(300\) 0 0
\(301\) − 1.41996e9i − 0.172986i
\(302\) 0 0
\(303\) − 3.57490e9i − 0.424125i
\(304\) 0 0
\(305\) 4.54368e9 0.525060
\(306\) 0 0
\(307\) −1.39905e10 −1.57499 −0.787497 0.616318i \(-0.788623\pi\)
−0.787497 + 0.616318i \(0.788623\pi\)
\(308\) 0 0
\(309\) 8.95818e9i 0.982621i
\(310\) 0 0
\(311\) − 5.84680e9i − 0.624995i −0.949919 0.312497i \(-0.898834\pi\)
0.949919 0.312497i \(-0.101166\pi\)
\(312\) 0 0
\(313\) 8.27511e9 0.862177 0.431088 0.902310i \(-0.358130\pi\)
0.431088 + 0.902310i \(0.358130\pi\)
\(314\) 0 0
\(315\) −1.63756e9 −0.166324
\(316\) 0 0
\(317\) − 1.37723e10i − 1.36386i −0.731419 0.681928i \(-0.761142\pi\)
0.731419 0.681928i \(-0.238858\pi\)
\(318\) 0 0
\(319\) 1.19489e10i 1.15389i
\(320\) 0 0
\(321\) 2.54784e9 0.239967
\(322\) 0 0
\(323\) −1.36693e9 −0.125585
\(324\) 0 0
\(325\) − 3.90735e9i − 0.350226i
\(326\) 0 0
\(327\) − 8.08759e9i − 0.707340i
\(328\) 0 0
\(329\) −8.02325e9 −0.684804
\(330\) 0 0
\(331\) 2.73215e9 0.227611 0.113806 0.993503i \(-0.463696\pi\)
0.113806 + 0.993503i \(0.463696\pi\)
\(332\) 0 0
\(333\) − 6.25157e9i − 0.508408i
\(334\) 0 0
\(335\) − 3.50845e9i − 0.278571i
\(336\) 0 0
\(337\) 2.19153e10 1.69913 0.849567 0.527481i \(-0.176863\pi\)
0.849567 + 0.527481i \(0.176863\pi\)
\(338\) 0 0
\(339\) −5.03828e9 −0.381490
\(340\) 0 0
\(341\) 7.28517e8i 0.0538793i
\(342\) 0 0
\(343\) 1.08711e10i 0.785414i
\(344\) 0 0
\(345\) −4.10750e8 −0.0289935
\(346\) 0 0
\(347\) 9.17879e9 0.633093 0.316547 0.948577i \(-0.397477\pi\)
0.316547 + 0.948577i \(0.397477\pi\)
\(348\) 0 0
\(349\) − 1.80632e10i − 1.21757i −0.793335 0.608785i \(-0.791657\pi\)
0.793335 0.608785i \(-0.208343\pi\)
\(350\) 0 0
\(351\) − 1.13635e9i − 0.0748657i
\(352\) 0 0
\(353\) 4.40105e9 0.283438 0.141719 0.989907i \(-0.454737\pi\)
0.141719 + 0.989907i \(0.454737\pi\)
\(354\) 0 0
\(355\) −4.82289e9 −0.303665
\(356\) 0 0
\(357\) − 1.18415e10i − 0.729013i
\(358\) 0 0
\(359\) − 6.13107e8i − 0.0369113i −0.999830 0.0184556i \(-0.994125\pi\)
0.999830 0.0184556i \(-0.00587494\pi\)
\(360\) 0 0
\(361\) −1.65641e10 −0.975299
\(362\) 0 0
\(363\) 2.67709e9 0.154183
\(364\) 0 0
\(365\) − 7.31234e8i − 0.0411989i
\(366\) 0 0
\(367\) 1.46445e10i 0.807253i 0.914924 + 0.403626i \(0.132250\pi\)
−0.914924 + 0.403626i \(0.867750\pi\)
\(368\) 0 0
\(369\) −2.34072e7 −0.00126254
\(370\) 0 0
\(371\) −2.27432e10 −1.20048
\(372\) 0 0
\(373\) 1.37672e10i 0.711230i 0.934633 + 0.355615i \(0.115728\pi\)
−0.934633 + 0.355615i \(0.884272\pi\)
\(374\) 0 0
\(375\) − 6.85090e9i − 0.346436i
\(376\) 0 0
\(377\) −8.05561e9 −0.398779
\(378\) 0 0
\(379\) 4.02771e8 0.0195210 0.00976048 0.999952i \(-0.496893\pi\)
0.00976048 + 0.999952i \(0.496893\pi\)
\(380\) 0 0
\(381\) 9.61369e9i 0.456237i
\(382\) 0 0
\(383\) − 2.88378e10i − 1.34019i −0.742274 0.670097i \(-0.766253\pi\)
0.742274 0.670097i \(-0.233747\pi\)
\(384\) 0 0
\(385\) 1.23400e10 0.561658
\(386\) 0 0
\(387\) 8.18505e8 0.0364903
\(388\) 0 0
\(389\) − 1.15555e10i − 0.504649i −0.967643 0.252325i \(-0.918805\pi\)
0.967643 0.252325i \(-0.0811951\pi\)
\(390\) 0 0
\(391\) − 2.97023e9i − 0.127082i
\(392\) 0 0
\(393\) −1.50142e9 −0.0629409
\(394\) 0 0
\(395\) 1.18096e10 0.485118
\(396\) 0 0
\(397\) 2.13939e10i 0.861247i 0.902532 + 0.430623i \(0.141706\pi\)
−0.902532 + 0.430623i \(0.858294\pi\)
\(398\) 0 0
\(399\) 3.63409e9i 0.143385i
\(400\) 0 0
\(401\) 2.83492e9 0.109639 0.0548194 0.998496i \(-0.482542\pi\)
0.0548194 + 0.998496i \(0.482542\pi\)
\(402\) 0 0
\(403\) −4.91145e8 −0.0186204
\(404\) 0 0
\(405\) − 9.43931e8i − 0.0350849i
\(406\) 0 0
\(407\) 4.71095e10i 1.71684i
\(408\) 0 0
\(409\) 3.93438e10 1.40599 0.702997 0.711193i \(-0.251845\pi\)
0.702997 + 0.711193i \(0.251845\pi\)
\(410\) 0 0
\(411\) 3.20561e10 1.12342
\(412\) 0 0
\(413\) 4.22236e10i 1.45129i
\(414\) 0 0
\(415\) 1.15199e10i 0.388378i
\(416\) 0 0
\(417\) −2.71763e10 −0.898766
\(418\) 0 0
\(419\) −3.17809e10 −1.03112 −0.515561 0.856853i \(-0.672417\pi\)
−0.515561 + 0.856853i \(0.672417\pi\)
\(420\) 0 0
\(421\) − 2.53716e9i − 0.0807643i −0.999184 0.0403821i \(-0.987142\pi\)
0.999184 0.0403821i \(-0.0128575\pi\)
\(422\) 0 0
\(423\) − 4.62482e9i − 0.144455i
\(424\) 0 0
\(425\) 2.34705e10 0.719395
\(426\) 0 0
\(427\) 8.73514e10 2.62759
\(428\) 0 0
\(429\) 8.56309e9i 0.252814i
\(430\) 0 0
\(431\) − 5.95521e9i − 0.172579i −0.996270 0.0862894i \(-0.972499\pi\)
0.996270 0.0862894i \(-0.0275010\pi\)
\(432\) 0 0
\(433\) 3.61421e10 1.02816 0.514081 0.857741i \(-0.328133\pi\)
0.514081 + 0.857741i \(0.328133\pi\)
\(434\) 0 0
\(435\) −6.69156e9 −0.186883
\(436\) 0 0
\(437\) 9.11545e8i 0.0249949i
\(438\) 0 0
\(439\) − 3.40899e10i − 0.917840i −0.888478 0.458920i \(-0.848236\pi\)
0.888478 0.458920i \(-0.151764\pi\)
\(440\) 0 0
\(441\) −1.88740e10 −0.499011
\(442\) 0 0
\(443\) 7.28042e10 1.89035 0.945174 0.326567i \(-0.105892\pi\)
0.945174 + 0.326567i \(0.105892\pi\)
\(444\) 0 0
\(445\) 1.61878e10i 0.412808i
\(446\) 0 0
\(447\) 4.14499e10i 1.03823i
\(448\) 0 0
\(449\) −3.63688e10 −0.894836 −0.447418 0.894325i \(-0.647656\pi\)
−0.447418 + 0.894325i \(0.647656\pi\)
\(450\) 0 0
\(451\) 1.76388e8 0.00426346
\(452\) 0 0
\(453\) 1.21618e10i 0.288806i
\(454\) 0 0
\(455\) 8.31927e9i 0.194106i
\(456\) 0 0
\(457\) 2.39306e10 0.548643 0.274321 0.961638i \(-0.411547\pi\)
0.274321 + 0.961638i \(0.411547\pi\)
\(458\) 0 0
\(459\) 6.82578e9 0.153781
\(460\) 0 0
\(461\) 1.37304e10i 0.304004i 0.988380 + 0.152002i \(0.0485720\pi\)
−0.988380 + 0.152002i \(0.951428\pi\)
\(462\) 0 0
\(463\) − 5.34930e10i − 1.16405i −0.813170 0.582026i \(-0.802260\pi\)
0.813170 0.582026i \(-0.197740\pi\)
\(464\) 0 0
\(465\) −4.07980e8 −0.00872623
\(466\) 0 0
\(467\) 2.41485e8 0.00507718 0.00253859 0.999997i \(-0.499192\pi\)
0.00253859 + 0.999997i \(0.499192\pi\)
\(468\) 0 0
\(469\) − 6.74492e10i − 1.39407i
\(470\) 0 0
\(471\) − 3.77973e9i − 0.0768029i
\(472\) 0 0
\(473\) −6.16794e9 −0.123224
\(474\) 0 0
\(475\) −7.20297e9 −0.141494
\(476\) 0 0
\(477\) − 1.31098e10i − 0.253234i
\(478\) 0 0
\(479\) − 1.58734e10i − 0.301528i −0.988570 0.150764i \(-0.951827\pi\)
0.988570 0.150764i \(-0.0481734\pi\)
\(480\) 0 0
\(481\) −3.17598e10 −0.593332
\(482\) 0 0
\(483\) −7.89659e9 −0.145094
\(484\) 0 0
\(485\) 1.14855e10i 0.207579i
\(486\) 0 0
\(487\) − 6.42808e10i − 1.14279i −0.820676 0.571393i \(-0.806403\pi\)
0.820676 0.571393i \(-0.193597\pi\)
\(488\) 0 0
\(489\) −1.92246e10 −0.336219
\(490\) 0 0
\(491\) 1.06597e11 1.83408 0.917040 0.398795i \(-0.130571\pi\)
0.917040 + 0.398795i \(0.130571\pi\)
\(492\) 0 0
\(493\) − 4.83882e10i − 0.819127i
\(494\) 0 0
\(495\) 7.11311e9i 0.118478i
\(496\) 0 0
\(497\) −9.27191e10 −1.51965
\(498\) 0 0
\(499\) 7.59553e10 1.22506 0.612528 0.790449i \(-0.290153\pi\)
0.612528 + 0.790449i \(0.290153\pi\)
\(500\) 0 0
\(501\) − 3.44281e10i − 0.546465i
\(502\) 0 0
\(503\) 6.63220e10i 1.03606i 0.855362 + 0.518031i \(0.173335\pi\)
−0.855362 + 0.518031i \(0.826665\pi\)
\(504\) 0 0
\(505\) −1.50863e10 −0.231962
\(506\) 0 0
\(507\) 3.23750e10 0.489979
\(508\) 0 0
\(509\) − 6.87347e10i − 1.02401i −0.858982 0.512006i \(-0.828903\pi\)
0.858982 0.512006i \(-0.171097\pi\)
\(510\) 0 0
\(511\) − 1.40578e10i − 0.206174i
\(512\) 0 0
\(513\) −2.09479e9 −0.0302462
\(514\) 0 0
\(515\) 3.78040e10 0.537414
\(516\) 0 0
\(517\) 3.48509e10i 0.487811i
\(518\) 0 0
\(519\) 2.96370e10i 0.408474i
\(520\) 0 0
\(521\) −7.24827e10 −0.983747 −0.491874 0.870667i \(-0.663688\pi\)
−0.491874 + 0.870667i \(0.663688\pi\)
\(522\) 0 0
\(523\) 1.21199e11 1.61991 0.809957 0.586489i \(-0.199490\pi\)
0.809957 + 0.586489i \(0.199490\pi\)
\(524\) 0 0
\(525\) − 6.23983e10i − 0.821364i
\(526\) 0 0
\(527\) − 2.95019e9i − 0.0382479i
\(528\) 0 0
\(529\) 7.63303e10 0.974707
\(530\) 0 0
\(531\) −2.43388e10 −0.306141
\(532\) 0 0
\(533\) 1.18916e8i 0.00147343i
\(534\) 0 0
\(535\) − 1.07520e10i − 0.131243i
\(536\) 0 0
\(537\) −7.04215e10 −0.846853
\(538\) 0 0
\(539\) 1.42228e11 1.68511
\(540\) 0 0
\(541\) 3.08763e10i 0.360442i 0.983626 + 0.180221i \(0.0576813\pi\)
−0.983626 + 0.180221i \(0.942319\pi\)
\(542\) 0 0
\(543\) 3.86059e10i 0.444073i
\(544\) 0 0
\(545\) −3.41301e10 −0.386858
\(546\) 0 0
\(547\) −1.59776e11 −1.78468 −0.892341 0.451361i \(-0.850939\pi\)
−0.892341 + 0.451361i \(0.850939\pi\)
\(548\) 0 0
\(549\) 5.03517e10i 0.554274i
\(550\) 0 0
\(551\) 1.48500e10i 0.161109i
\(552\) 0 0
\(553\) 2.27038e11 2.42771
\(554\) 0 0
\(555\) −2.63820e10 −0.278058
\(556\) 0 0
\(557\) − 4.27428e10i − 0.444061i −0.975040 0.222030i \(-0.928732\pi\)
0.975040 0.222030i \(-0.0712684\pi\)
\(558\) 0 0
\(559\) − 4.15825e9i − 0.0425856i
\(560\) 0 0
\(561\) −5.14365e10 −0.519302
\(562\) 0 0
\(563\) 2.01627e10 0.200685 0.100343 0.994953i \(-0.468006\pi\)
0.100343 + 0.994953i \(0.468006\pi\)
\(564\) 0 0
\(565\) 2.12618e10i 0.208644i
\(566\) 0 0
\(567\) − 1.81469e10i − 0.175578i
\(568\) 0 0
\(569\) −6.07200e10 −0.579272 −0.289636 0.957137i \(-0.593534\pi\)
−0.289636 + 0.957137i \(0.593534\pi\)
\(570\) 0 0
\(571\) 1.73024e11 1.62765 0.813825 0.581110i \(-0.197381\pi\)
0.813825 + 0.581110i \(0.197381\pi\)
\(572\) 0 0
\(573\) 9.09216e10i 0.843429i
\(574\) 0 0
\(575\) − 1.56514e10i − 0.143180i
\(576\) 0 0
\(577\) 8.41388e10 0.759089 0.379545 0.925173i \(-0.376081\pi\)
0.379545 + 0.925173i \(0.376081\pi\)
\(578\) 0 0
\(579\) 4.23566e10 0.376883
\(580\) 0 0
\(581\) 2.21467e11i 1.94359i
\(582\) 0 0
\(583\) 9.87905e10i 0.855148i
\(584\) 0 0
\(585\) −4.79545e9 −0.0409455
\(586\) 0 0
\(587\) 2.93365e10 0.247091 0.123545 0.992339i \(-0.460574\pi\)
0.123545 + 0.992339i \(0.460574\pi\)
\(588\) 0 0
\(589\) 9.05397e8i 0.00752277i
\(590\) 0 0
\(591\) − 1.01541e11i − 0.832323i
\(592\) 0 0
\(593\) 6.15955e10 0.498116 0.249058 0.968489i \(-0.419879\pi\)
0.249058 + 0.968489i \(0.419879\pi\)
\(594\) 0 0
\(595\) −4.99719e10 −0.398711
\(596\) 0 0
\(597\) − 2.49355e10i − 0.196300i
\(598\) 0 0
\(599\) − 1.86673e11i − 1.45002i −0.688740 0.725008i \(-0.741836\pi\)
0.688740 0.725008i \(-0.258164\pi\)
\(600\) 0 0
\(601\) 7.79316e10 0.597332 0.298666 0.954358i \(-0.403458\pi\)
0.298666 + 0.954358i \(0.403458\pi\)
\(602\) 0 0
\(603\) 3.88795e10 0.294071
\(604\) 0 0
\(605\) − 1.12975e10i − 0.0843255i
\(606\) 0 0
\(607\) 1.32801e11i 0.978242i 0.872216 + 0.489121i \(0.162682\pi\)
−0.872216 + 0.489121i \(0.837318\pi\)
\(608\) 0 0
\(609\) −1.28644e11 −0.935233
\(610\) 0 0
\(611\) −2.34955e10 −0.168585
\(612\) 0 0
\(613\) 1.41461e11i 1.00183i 0.865496 + 0.500916i \(0.167003\pi\)
−0.865496 + 0.500916i \(0.832997\pi\)
\(614\) 0 0
\(615\) 9.87797e7i 0 0.000690506i
\(616\) 0 0
\(617\) 1.08559e11 0.749076 0.374538 0.927212i \(-0.377801\pi\)
0.374538 + 0.927212i \(0.377801\pi\)
\(618\) 0 0
\(619\) 1.65925e11 1.13019 0.565093 0.825027i \(-0.308840\pi\)
0.565093 + 0.825027i \(0.308840\pi\)
\(620\) 0 0
\(621\) − 4.55181e9i − 0.0306068i
\(622\) 0 0
\(623\) 3.11207e11i 2.06584i
\(624\) 0 0
\(625\) 1.08463e11 0.710821
\(626\) 0 0
\(627\) 1.57856e10 0.102139
\(628\) 0 0
\(629\) − 1.90774e11i − 1.21876i
\(630\) 0 0
\(631\) 8.74514e10i 0.551632i 0.961210 + 0.275816i \(0.0889480\pi\)
−0.961210 + 0.275816i \(0.911052\pi\)
\(632\) 0 0
\(633\) −8.45215e10 −0.526444
\(634\) 0 0
\(635\) 4.05703e10 0.249525
\(636\) 0 0
\(637\) 9.58858e10i 0.582367i
\(638\) 0 0
\(639\) − 5.34458e10i − 0.320561i
\(640\) 0 0
\(641\) −2.77089e11 −1.64129 −0.820647 0.571435i \(-0.806387\pi\)
−0.820647 + 0.571435i \(0.806387\pi\)
\(642\) 0 0
\(643\) 1.47094e10 0.0860498 0.0430249 0.999074i \(-0.486301\pi\)
0.0430249 + 0.999074i \(0.486301\pi\)
\(644\) 0 0
\(645\) − 3.45414e9i − 0.0199572i
\(646\) 0 0
\(647\) − 2.14667e11i − 1.22503i −0.790458 0.612517i \(-0.790157\pi\)
0.790458 0.612517i \(-0.209843\pi\)
\(648\) 0 0
\(649\) 1.83408e11 1.03381
\(650\) 0 0
\(651\) −7.84333e9 −0.0436693
\(652\) 0 0
\(653\) 8.93518e10i 0.491418i 0.969344 + 0.245709i \(0.0790207\pi\)
−0.969344 + 0.245709i \(0.920979\pi\)
\(654\) 0 0
\(655\) 6.33609e9i 0.0344236i
\(656\) 0 0
\(657\) 8.10331e9 0.0434912
\(658\) 0 0
\(659\) −1.96460e11 −1.04167 −0.520837 0.853656i \(-0.674380\pi\)
−0.520837 + 0.853656i \(0.674380\pi\)
\(660\) 0 0
\(661\) − 2.55340e11i − 1.33756i −0.743460 0.668780i \(-0.766817\pi\)
0.743460 0.668780i \(-0.233183\pi\)
\(662\) 0 0
\(663\) − 3.46770e10i − 0.179468i
\(664\) 0 0
\(665\) 1.53361e10 0.0784202
\(666\) 0 0
\(667\) −3.22679e10 −0.163030
\(668\) 0 0
\(669\) − 1.54609e11i − 0.771847i
\(670\) 0 0
\(671\) − 3.79431e11i − 1.87173i
\(672\) 0 0
\(673\) 3.16723e11 1.54390 0.771950 0.635683i \(-0.219282\pi\)
0.771950 + 0.635683i \(0.219282\pi\)
\(674\) 0 0
\(675\) 3.59681e10 0.173261
\(676\) 0 0
\(677\) − 2.24064e11i − 1.06664i −0.845914 0.533319i \(-0.820945\pi\)
0.845914 0.533319i \(-0.179055\pi\)
\(678\) 0 0
\(679\) 2.20807e11i 1.03880i
\(680\) 0 0
\(681\) −2.05328e11 −0.954686
\(682\) 0 0
\(683\) −2.78839e11 −1.28136 −0.640679 0.767809i \(-0.721347\pi\)
−0.640679 + 0.767809i \(0.721347\pi\)
\(684\) 0 0
\(685\) − 1.35279e11i − 0.614422i
\(686\) 0 0
\(687\) − 2.11087e11i − 0.947620i
\(688\) 0 0
\(689\) −6.66017e10 −0.295535
\(690\) 0 0
\(691\) −2.74979e11 −1.20611 −0.603056 0.797699i \(-0.706051\pi\)
−0.603056 + 0.797699i \(0.706051\pi\)
\(692\) 0 0
\(693\) 1.36748e11i 0.592909i
\(694\) 0 0
\(695\) 1.14686e11i 0.491552i
\(696\) 0 0
\(697\) −7.14299e8 −0.00302656
\(698\) 0 0
\(699\) 4.85728e10 0.203463
\(700\) 0 0
\(701\) 1.21009e11i 0.501123i 0.968101 + 0.250562i \(0.0806152\pi\)
−0.968101 + 0.250562i \(0.919385\pi\)
\(702\) 0 0
\(703\) 5.85474e10i 0.239710i
\(704\) 0 0
\(705\) −1.95170e10 −0.0790053
\(706\) 0 0
\(707\) −2.90031e11 −1.16082
\(708\) 0 0
\(709\) − 3.24092e11i − 1.28258i −0.767301 0.641288i \(-0.778401\pi\)
0.767301 0.641288i \(-0.221599\pi\)
\(710\) 0 0
\(711\) 1.30871e11i 0.512111i
\(712\) 0 0
\(713\) −1.96735e9 −0.00761244
\(714\) 0 0
\(715\) 3.61367e10 0.138269
\(716\) 0 0
\(717\) 1.89853e11i 0.718358i
\(718\) 0 0
\(719\) 4.10633e11i 1.53652i 0.640138 + 0.768260i \(0.278877\pi\)
−0.640138 + 0.768260i \(0.721123\pi\)
\(720\) 0 0
\(721\) 7.26775e11 2.68942
\(722\) 0 0
\(723\) −8.59252e10 −0.314461
\(724\) 0 0
\(725\) − 2.54979e11i − 0.922894i
\(726\) 0 0
\(727\) − 1.06772e10i − 0.0382225i −0.999817 0.0191112i \(-0.993916\pi\)
0.999817 0.0191112i \(-0.00608367\pi\)
\(728\) 0 0
\(729\) 1.04604e10 0.0370370
\(730\) 0 0
\(731\) 2.49776e10 0.0874746
\(732\) 0 0
\(733\) 2.08286e11i 0.721515i 0.932660 + 0.360757i \(0.117482\pi\)
−0.932660 + 0.360757i \(0.882518\pi\)
\(734\) 0 0
\(735\) 7.96495e10i 0.272919i
\(736\) 0 0
\(737\) −2.92981e11 −0.993048
\(738\) 0 0
\(739\) −1.82789e11 −0.612875 −0.306437 0.951891i \(-0.599137\pi\)
−0.306437 + 0.951891i \(0.599137\pi\)
\(740\) 0 0
\(741\) 1.06422e10i 0.0352986i
\(742\) 0 0
\(743\) − 5.51013e11i − 1.80803i −0.427498 0.904016i \(-0.640605\pi\)
0.427498 0.904016i \(-0.359395\pi\)
\(744\) 0 0
\(745\) 1.74921e11 0.567829
\(746\) 0 0
\(747\) −1.27659e11 −0.409987
\(748\) 0 0
\(749\) − 2.06705e11i − 0.656786i
\(750\) 0 0
\(751\) − 4.25385e11i − 1.33728i −0.743586 0.668640i \(-0.766877\pi\)
0.743586 0.668640i \(-0.233123\pi\)
\(752\) 0 0
\(753\) −1.40223e11 −0.436152
\(754\) 0 0
\(755\) 5.13237e10 0.157954
\(756\) 0 0
\(757\) 4.55433e11i 1.38689i 0.720512 + 0.693443i \(0.243907\pi\)
−0.720512 + 0.693443i \(0.756093\pi\)
\(758\) 0 0
\(759\) 3.43007e10i 0.103356i
\(760\) 0 0
\(761\) −5.60243e11 −1.67047 −0.835233 0.549897i \(-0.814667\pi\)
−0.835233 + 0.549897i \(0.814667\pi\)
\(762\) 0 0
\(763\) −6.56144e11 −1.93598
\(764\) 0 0
\(765\) − 2.88052e10i − 0.0841056i
\(766\) 0 0
\(767\) 1.23649e11i 0.357279i
\(768\) 0 0
\(769\) 2.49195e11 0.712581 0.356290 0.934375i \(-0.384041\pi\)
0.356290 + 0.934375i \(0.384041\pi\)
\(770\) 0 0
\(771\) −1.90187e11 −0.538225
\(772\) 0 0
\(773\) 5.13512e11i 1.43824i 0.694884 + 0.719121i \(0.255456\pi\)
−0.694884 + 0.719121i \(0.744544\pi\)
\(774\) 0 0
\(775\) − 1.55459e10i − 0.0430932i
\(776\) 0 0
\(777\) −5.07188e11 −1.39151
\(778\) 0 0
\(779\) 2.19214e8 0.000595276 0
\(780\) 0 0
\(781\) 4.02747e11i 1.08250i
\(782\) 0 0
\(783\) − 7.41538e10i − 0.197281i
\(784\) 0 0
\(785\) −1.59507e10 −0.0420050
\(786\) 0 0
\(787\) 1.88847e11 0.492279 0.246139 0.969234i \(-0.420838\pi\)
0.246139 + 0.969234i \(0.420838\pi\)
\(788\) 0 0
\(789\) − 1.71888e11i − 0.443545i
\(790\) 0 0
\(791\) 4.08754e11i 1.04413i
\(792\) 0 0
\(793\) 2.55802e11 0.646861
\(794\) 0 0
\(795\) −5.53241e10 −0.138499
\(796\) 0 0
\(797\) − 4.98231e8i − 0.00123480i −1.00000 0.000617401i \(-0.999803\pi\)
1.00000 0.000617401i \(-0.000196525\pi\)
\(798\) 0 0
\(799\) − 1.41132e11i − 0.346288i
\(800\) 0 0
\(801\) −1.79388e11 −0.435777
\(802\) 0 0
\(803\) −6.10635e10 −0.146865
\(804\) 0 0
\(805\) 3.33240e10i 0.0793550i
\(806\) 0 0
\(807\) 2.13903e11i 0.504340i
\(808\) 0 0
\(809\) −4.76114e11 −1.11152 −0.555760 0.831343i \(-0.687573\pi\)
−0.555760 + 0.831343i \(0.687573\pi\)
\(810\) 0 0
\(811\) −7.86403e11 −1.81787 −0.908933 0.416942i \(-0.863102\pi\)
−0.908933 + 0.416942i \(0.863102\pi\)
\(812\) 0 0
\(813\) − 8.42063e10i − 0.192745i
\(814\) 0 0
\(815\) 8.11289e10i 0.183885i
\(816\) 0 0
\(817\) −7.66549e9 −0.0172049
\(818\) 0 0
\(819\) −9.21916e10 −0.204906
\(820\) 0 0
\(821\) 7.14924e11i 1.57357i 0.617224 + 0.786787i \(0.288257\pi\)
−0.617224 + 0.786787i \(0.711743\pi\)
\(822\) 0 0
\(823\) − 3.90745e11i − 0.851715i −0.904790 0.425858i \(-0.859972\pi\)
0.904790 0.425858i \(-0.140028\pi\)
\(824\) 0 0
\(825\) −2.71042e11 −0.585087
\(826\) 0 0
\(827\) −4.83361e11 −1.03336 −0.516678 0.856180i \(-0.672831\pi\)
−0.516678 + 0.856180i \(0.672831\pi\)
\(828\) 0 0
\(829\) − 6.59993e11i − 1.39740i −0.715414 0.698701i \(-0.753762\pi\)
0.715414 0.698701i \(-0.246238\pi\)
\(830\) 0 0
\(831\) 5.07782e11i 1.06481i
\(832\) 0 0
\(833\) −5.75964e11 −1.19623
\(834\) 0 0
\(835\) −1.45288e11 −0.298872
\(836\) 0 0
\(837\) − 4.52111e9i − 0.00921176i
\(838\) 0 0
\(839\) 8.61080e11i 1.73778i 0.495003 + 0.868891i \(0.335167\pi\)
−0.495003 + 0.868891i \(0.664833\pi\)
\(840\) 0 0
\(841\) −2.54319e10 −0.0508387
\(842\) 0 0
\(843\) −1.34338e11 −0.266005
\(844\) 0 0
\(845\) − 1.36624e11i − 0.267979i
\(846\) 0 0
\(847\) − 2.17191e11i − 0.421996i
\(848\) 0 0
\(849\) −2.37689e10 −0.0457486
\(850\) 0 0
\(851\) −1.27219e11 −0.242567
\(852\) 0 0
\(853\) 5.20403e11i 0.982977i 0.870884 + 0.491489i \(0.163547\pi\)
−0.870884 + 0.491489i \(0.836453\pi\)
\(854\) 0 0
\(855\) 8.84014e9i 0.0165422i
\(856\) 0 0
\(857\) 8.53509e11 1.58229 0.791143 0.611632i \(-0.209486\pi\)
0.791143 + 0.611632i \(0.209486\pi\)
\(858\) 0 0
\(859\) −2.12057e11 −0.389476 −0.194738 0.980855i \(-0.562386\pi\)
−0.194738 + 0.980855i \(0.562386\pi\)
\(860\) 0 0
\(861\) 1.89902e9i 0.00345555i
\(862\) 0 0
\(863\) − 4.67543e11i − 0.842905i −0.906851 0.421452i \(-0.861520\pi\)
0.906851 0.421452i \(-0.138480\pi\)
\(864\) 0 0
\(865\) 1.25070e11 0.223403
\(866\) 0 0
\(867\) −1.17927e11 −0.208707
\(868\) 0 0
\(869\) − 9.86192e11i − 1.72935i
\(870\) 0 0
\(871\) − 1.97520e11i − 0.343192i
\(872\) 0 0
\(873\) −1.27279e11 −0.219129
\(874\) 0 0
\(875\) −5.55812e11 −0.948190
\(876\) 0 0
\(877\) − 5.09851e11i − 0.861876i −0.902382 0.430938i \(-0.858183\pi\)
0.902382 0.430938i \(-0.141817\pi\)
\(878\) 0 0
\(879\) − 9.99895e10i − 0.167494i
\(880\) 0 0
\(881\) −7.67923e10 −0.127472 −0.0637359 0.997967i \(-0.520302\pi\)
−0.0637359 + 0.997967i \(0.520302\pi\)
\(882\) 0 0
\(883\) 2.22619e11 0.366201 0.183101 0.983094i \(-0.441387\pi\)
0.183101 + 0.983094i \(0.441387\pi\)
\(884\) 0 0
\(885\) 1.02711e11i 0.167435i
\(886\) 0 0
\(887\) − 5.12545e11i − 0.828014i −0.910274 0.414007i \(-0.864129\pi\)
0.910274 0.414007i \(-0.135871\pi\)
\(888\) 0 0
\(889\) 7.79956e11 1.24871
\(890\) 0 0
\(891\) −7.88253e10 −0.125070
\(892\) 0 0
\(893\) 4.33125e10i 0.0681094i
\(894\) 0 0
\(895\) 2.97183e11i 0.463160i
\(896\) 0 0
\(897\) −2.31245e10 −0.0357193
\(898\) 0 0
\(899\) −3.20503e10 −0.0490674
\(900\) 0 0
\(901\) − 4.00061e11i − 0.607054i
\(902\) 0 0
\(903\) − 6.64050e10i − 0.0998735i
\(904\) 0 0
\(905\) 1.62919e11 0.242872
\(906\) 0 0
\(907\) −6.10957e11 −0.902779 −0.451390 0.892327i \(-0.649072\pi\)
−0.451390 + 0.892327i \(0.649072\pi\)
\(908\) 0 0
\(909\) − 1.67182e11i − 0.244869i
\(910\) 0 0
\(911\) 9.42463e11i 1.36833i 0.729328 + 0.684165i \(0.239833\pi\)
−0.729328 + 0.684165i \(0.760167\pi\)
\(912\) 0 0
\(913\) 9.61994e11 1.38449
\(914\) 0 0
\(915\) 2.12487e11 0.303143
\(916\) 0 0
\(917\) 1.21810e11i 0.172269i
\(918\) 0 0
\(919\) − 8.39974e11i − 1.17762i −0.808273 0.588808i \(-0.799597\pi\)
0.808273 0.588808i \(-0.200403\pi\)
\(920\) 0 0
\(921\) −6.54270e11 −0.909323
\(922\) 0 0
\(923\) −2.71521e11 −0.374107
\(924\) 0 0
\(925\) − 1.00527e12i − 1.37315i
\(926\) 0 0
\(927\) 4.18933e11i 0.567316i
\(928\) 0 0
\(929\) −1.01174e12 −1.35834 −0.679169 0.733982i \(-0.737660\pi\)
−0.679169 + 0.733982i \(0.737660\pi\)
\(930\) 0 0
\(931\) 1.76760e11 0.235280
\(932\) 0 0
\(933\) − 2.73428e11i − 0.360841i
\(934\) 0 0
\(935\) 2.17065e11i 0.284016i
\(936\) 0 0
\(937\) 1.06278e12 1.37875 0.689374 0.724406i \(-0.257886\pi\)
0.689374 + 0.724406i \(0.257886\pi\)
\(938\) 0 0
\(939\) 3.86989e11 0.497778
\(940\) 0 0
\(941\) 2.62998e11i 0.335424i 0.985836 + 0.167712i \(0.0536379\pi\)
−0.985836 + 0.167712i \(0.946362\pi\)
\(942\) 0 0
\(943\) 4.76334e8i 0 0.000602372i
\(944\) 0 0
\(945\) −7.65809e10 −0.0960269
\(946\) 0 0
\(947\) 4.60464e11 0.572526 0.286263 0.958151i \(-0.407587\pi\)
0.286263 + 0.958151i \(0.407587\pi\)
\(948\) 0 0
\(949\) − 4.11673e10i − 0.0507560i
\(950\) 0 0
\(951\) − 6.44066e11i − 0.787423i
\(952\) 0 0
\(953\) −1.73103e11 −0.209862 −0.104931 0.994480i \(-0.533462\pi\)
−0.104931 + 0.994480i \(0.533462\pi\)
\(954\) 0 0
\(955\) 3.83694e11 0.461288
\(956\) 0 0
\(957\) 5.58795e11i 0.666200i
\(958\) 0 0
\(959\) − 2.60070e12i − 3.07480i
\(960\) 0 0
\(961\) 8.50937e11 0.997709
\(962\) 0 0
\(963\) 1.19150e11 0.138545
\(964\) 0 0
\(965\) − 1.78747e11i − 0.206125i
\(966\) 0 0
\(967\) 6.52116e11i 0.745794i 0.927873 + 0.372897i \(0.121635\pi\)
−0.927873 + 0.372897i \(0.878365\pi\)
\(968\) 0 0
\(969\) −6.39250e10 −0.0725063
\(970\) 0 0
\(971\) −7.61429e11 −0.856549 −0.428275 0.903649i \(-0.640878\pi\)
−0.428275 + 0.903649i \(0.640878\pi\)
\(972\) 0 0
\(973\) 2.20481e12i 2.45991i
\(974\) 0 0
\(975\) − 1.82729e11i − 0.202203i
\(976\) 0 0
\(977\) −1.01203e12 −1.11075 −0.555374 0.831600i \(-0.687425\pi\)
−0.555374 + 0.831600i \(0.687425\pi\)
\(978\) 0 0
\(979\) 1.35180e12 1.47158
\(980\) 0 0
\(981\) − 3.78219e11i − 0.408383i
\(982\) 0 0
\(983\) 1.43816e12i 1.54026i 0.637886 + 0.770131i \(0.279809\pi\)
−0.637886 + 0.770131i \(0.720191\pi\)
\(984\) 0 0
\(985\) −4.28509e11 −0.455214
\(986\) 0 0
\(987\) −3.75210e11 −0.395372
\(988\) 0 0
\(989\) − 1.66565e10i − 0.0174100i
\(990\) 0 0
\(991\) 1.76628e12i 1.83132i 0.401953 + 0.915660i \(0.368331\pi\)
−0.401953 + 0.915660i \(0.631669\pi\)
\(992\) 0 0
\(993\) 1.27770e11 0.131411
\(994\) 0 0
\(995\) −1.05229e11 −0.107360
\(996\) 0 0
\(997\) − 9.33822e11i − 0.945112i −0.881301 0.472556i \(-0.843331\pi\)
0.881301 0.472556i \(-0.156669\pi\)
\(998\) 0 0
\(999\) − 2.92357e11i − 0.293529i
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 384.9.b.b.319.6 yes 8
4.3 odd 2 inner 384.9.b.b.319.2 8
8.3 odd 2 inner 384.9.b.b.319.7 yes 8
8.5 even 2 inner 384.9.b.b.319.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.9.b.b.319.2 8 4.3 odd 2 inner
384.9.b.b.319.3 yes 8 8.5 even 2 inner
384.9.b.b.319.6 yes 8 1.1 even 1 trivial
384.9.b.b.319.7 yes 8 8.3 odd 2 inner