Properties

Label 24.28.a.d.1.1
Level $24$
Weight $28$
Character 24.1
Self dual yes
Analytic conductor $110.845$
Analytic rank $0$
Dimension $4$
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] = [24,28,Mod(1,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 24.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: \(110.845337961\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{8}\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-263517.\) of defining polynomial
Character \(\chi\) \(=\) 24.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.59432e6 q^{3} -2.94911e9 q^{5} -2.52860e11 q^{7} +2.54187e12 q^{9} +O(q^{10})\) \(q+1.59432e6 q^{3} -2.94911e9 q^{5} -2.52860e11 q^{7} +2.54187e12 q^{9} -1.51354e14 q^{11} +9.28080e13 q^{13} -4.70183e15 q^{15} +1.39004e16 q^{17} -1.70678e17 q^{19} -4.03140e17 q^{21} -4.09502e18 q^{23} +1.24666e18 q^{25} +4.05256e18 q^{27} +4.32334e19 q^{29} -1.99593e20 q^{31} -2.41307e20 q^{33} +7.45710e20 q^{35} +1.37767e21 q^{37} +1.47966e20 q^{39} +6.82134e21 q^{41} +9.39297e21 q^{43} -7.49624e21 q^{45} -2.13749e21 q^{47} -1.77440e21 q^{49} +2.21618e22 q^{51} -2.93406e23 q^{53} +4.46358e23 q^{55} -2.72116e23 q^{57} +8.14136e23 q^{59} +5.59522e23 q^{61} -6.42735e23 q^{63} -2.73701e23 q^{65} -6.51018e24 q^{67} -6.52878e24 q^{69} +3.69127e24 q^{71} +1.27012e24 q^{73} +1.98757e24 q^{75} +3.82712e25 q^{77} -1.00060e25 q^{79} +6.46108e24 q^{81} +9.33551e25 q^{83} -4.09938e25 q^{85} +6.89281e25 q^{87} +3.02863e25 q^{89} -2.34674e25 q^{91} -3.18215e26 q^{93} +5.03347e26 q^{95} -6.11382e26 q^{97} -3.84721e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6377292 q^{3} + 4949050424 q^{5} - 88249731552 q^{7} + 10167463313316 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6377292 q^{3} + 4949050424 q^{5} - 88249731552 q^{7} + 10167463313316 q^{9} - 111189633716848 q^{11} - 7339657642664 q^{13} + 78\!\cdots\!52 q^{15}+ \cdots - 28\!\cdots\!92 q^{99}+O(q^{100}) \) 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
\(3\) 1.59432e6 0.577350
\(4\) 0 0
\(5\) −2.94911e9 −1.08043 −0.540214 0.841528i \(-0.681657\pi\)
−0.540214 + 0.841528i \(0.681657\pi\)
\(6\) 0 0
\(7\) −2.52860e11 −0.986406 −0.493203 0.869914i \(-0.664174\pi\)
−0.493203 + 0.869914i \(0.664174\pi\)
\(8\) 0 0
\(9\) 2.54187e12 0.333333
\(10\) 0 0
\(11\) −1.51354e14 −1.32188 −0.660940 0.750439i \(-0.729842\pi\)
−0.660940 + 0.750439i \(0.729842\pi\)
\(12\) 0 0
\(13\) 9.28080e13 0.0849865 0.0424933 0.999097i \(-0.486470\pi\)
0.0424933 + 0.999097i \(0.486470\pi\)
\(14\) 0 0
\(15\) −4.70183e15 −0.623785
\(16\) 0 0
\(17\) 1.39004e16 0.340383 0.170191 0.985411i \(-0.445561\pi\)
0.170191 + 0.985411i \(0.445561\pi\)
\(18\) 0 0
\(19\) −1.70678e17 −0.931115 −0.465558 0.885018i \(-0.654146\pi\)
−0.465558 + 0.885018i \(0.654146\pi\)
\(20\) 0 0
\(21\) −4.03140e17 −0.569502
\(22\) 0 0
\(23\) −4.09502e18 −1.69407 −0.847033 0.531541i \(-0.821613\pi\)
−0.847033 + 0.531541i \(0.821613\pi\)
\(24\) 0 0
\(25\) 1.24666e18 0.167324
\(26\) 0 0
\(27\) 4.05256e18 0.192450
\(28\) 0 0
\(29\) 4.32334e19 0.782433 0.391216 0.920299i \(-0.372054\pi\)
0.391216 + 0.920299i \(0.372054\pi\)
\(30\) 0 0
\(31\) −1.99593e20 −1.46812 −0.734060 0.679085i \(-0.762377\pi\)
−0.734060 + 0.679085i \(0.762377\pi\)
\(32\) 0 0
\(33\) −2.41307e20 −0.763187
\(34\) 0 0
\(35\) 7.45710e20 1.06574
\(36\) 0 0
\(37\) 1.37767e21 0.929867 0.464933 0.885346i \(-0.346078\pi\)
0.464933 + 0.885346i \(0.346078\pi\)
\(38\) 0 0
\(39\) 1.47966e20 0.0490670
\(40\) 0 0
\(41\) 6.82134e21 1.15156 0.575781 0.817604i \(-0.304698\pi\)
0.575781 + 0.817604i \(0.304698\pi\)
\(42\) 0 0
\(43\) 9.39297e21 0.833641 0.416820 0.908989i \(-0.363144\pi\)
0.416820 + 0.908989i \(0.363144\pi\)
\(44\) 0 0
\(45\) −7.49624e21 −0.360142
\(46\) 0 0
\(47\) −2.13749e21 −0.0570929 −0.0285464 0.999592i \(-0.509088\pi\)
−0.0285464 + 0.999592i \(0.509088\pi\)
\(48\) 0 0
\(49\) −1.77440e21 −0.0270025
\(50\) 0 0
\(51\) 2.21618e22 0.196520
\(52\) 0 0
\(53\) −2.93406e23 −1.54791 −0.773953 0.633244i \(-0.781723\pi\)
−0.773953 + 0.633244i \(0.781723\pi\)
\(54\) 0 0
\(55\) 4.46358e23 1.42819
\(56\) 0 0
\(57\) −2.72116e23 −0.537580
\(58\) 0 0
\(59\) 8.14136e23 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(60\) 0 0
\(61\) 5.59522e23 0.442450 0.221225 0.975223i \(-0.428994\pi\)
0.221225 + 0.975223i \(0.428994\pi\)
\(62\) 0 0
\(63\) −6.42735e23 −0.328802
\(64\) 0 0
\(65\) −2.73701e23 −0.0918218
\(66\) 0 0
\(67\) −6.51018e24 −1.45072 −0.725359 0.688371i \(-0.758326\pi\)
−0.725359 + 0.688371i \(0.758326\pi\)
\(68\) 0 0
\(69\) −6.52878e24 −0.978069
\(70\) 0 0
\(71\) 3.69127e24 0.376000 0.188000 0.982169i \(-0.439799\pi\)
0.188000 + 0.982169i \(0.439799\pi\)
\(72\) 0 0
\(73\) 1.27012e24 0.0889176 0.0444588 0.999011i \(-0.485844\pi\)
0.0444588 + 0.999011i \(0.485844\pi\)
\(74\) 0 0
\(75\) 1.98757e24 0.0966043
\(76\) 0 0
\(77\) 3.82712e25 1.30391
\(78\) 0 0
\(79\) −1.00060e25 −0.241154 −0.120577 0.992704i \(-0.538474\pi\)
−0.120577 + 0.992704i \(0.538474\pi\)
\(80\) 0 0
\(81\) 6.46108e24 0.111111
\(82\) 0 0
\(83\) 9.33551e25 1.15500 0.577502 0.816389i \(-0.304027\pi\)
0.577502 + 0.816389i \(0.304027\pi\)
\(84\) 0 0
\(85\) −4.09938e25 −0.367759
\(86\) 0 0
\(87\) 6.89281e25 0.451738
\(88\) 0 0
\(89\) 3.02863e25 0.146043 0.0730215 0.997330i \(-0.476736\pi\)
0.0730215 + 0.997330i \(0.476736\pi\)
\(90\) 0 0
\(91\) −2.34674e25 −0.0838312
\(92\) 0 0
\(93\) −3.18215e26 −0.847619
\(94\) 0 0
\(95\) 5.03347e26 1.00600
\(96\) 0 0
\(97\) −6.11382e26 −0.922347 −0.461174 0.887310i \(-0.652571\pi\)
−0.461174 + 0.887310i \(0.652571\pi\)
\(98\) 0 0
\(99\) −3.84721e26 −0.440626
\(100\) 0 0
\(101\) 3.53903e26 0.309418 0.154709 0.987960i \(-0.450556\pi\)
0.154709 + 0.987960i \(0.450556\pi\)
\(102\) 0 0
\(103\) 2.86663e27 1.92339 0.961697 0.274114i \(-0.0883845\pi\)
0.961697 + 0.274114i \(0.0883845\pi\)
\(104\) 0 0
\(105\) 1.18890e27 0.615306
\(106\) 0 0
\(107\) −3.77279e27 −1.51350 −0.756748 0.653706i \(-0.773213\pi\)
−0.756748 + 0.653706i \(0.773213\pi\)
\(108\) 0 0
\(109\) 1.19634e27 0.373764 0.186882 0.982382i \(-0.440162\pi\)
0.186882 + 0.982382i \(0.440162\pi\)
\(110\) 0 0
\(111\) 2.19644e27 0.536859
\(112\) 0 0
\(113\) 6.03773e27 1.15962 0.579809 0.814753i \(-0.303127\pi\)
0.579809 + 0.814753i \(0.303127\pi\)
\(114\) 0 0
\(115\) 1.20767e28 1.83031
\(116\) 0 0
\(117\) 2.35905e26 0.0283288
\(118\) 0 0
\(119\) −3.51485e27 −0.335756
\(120\) 0 0
\(121\) 9.79795e27 0.747365
\(122\) 0 0
\(123\) 1.08754e28 0.664855
\(124\) 0 0
\(125\) 1.82960e28 0.899647
\(126\) 0 0
\(127\) −2.19122e28 −0.869631 −0.434815 0.900520i \(-0.643186\pi\)
−0.434815 + 0.900520i \(0.643186\pi\)
\(128\) 0 0
\(129\) 1.49754e28 0.481303
\(130\) 0 0
\(131\) 4.13755e28 1.08039 0.540196 0.841539i \(-0.318350\pi\)
0.540196 + 0.841539i \(0.318350\pi\)
\(132\) 0 0
\(133\) 4.31575e28 0.918458
\(134\) 0 0
\(135\) −1.19514e28 −0.207928
\(136\) 0 0
\(137\) 1.31599e29 1.87726 0.938632 0.344921i \(-0.112094\pi\)
0.938632 + 0.344921i \(0.112094\pi\)
\(138\) 0 0
\(139\) 1.31016e29 1.53682 0.768411 0.639956i \(-0.221047\pi\)
0.768411 + 0.639956i \(0.221047\pi\)
\(140\) 0 0
\(141\) −3.40784e27 −0.0329626
\(142\) 0 0
\(143\) −1.40468e28 −0.112342
\(144\) 0 0
\(145\) −1.27500e29 −0.845362
\(146\) 0 0
\(147\) −2.82896e27 −0.0155899
\(148\) 0 0
\(149\) −9.91784e28 −0.455410 −0.227705 0.973730i \(-0.573122\pi\)
−0.227705 + 0.973730i \(0.573122\pi\)
\(150\) 0 0
\(151\) 3.24364e29 1.24407 0.622035 0.782990i \(-0.286306\pi\)
0.622035 + 0.782990i \(0.286306\pi\)
\(152\) 0 0
\(153\) 3.53330e28 0.113461
\(154\) 0 0
\(155\) 5.88620e29 1.58620
\(156\) 0 0
\(157\) 3.35983e29 0.761503 0.380752 0.924677i \(-0.375665\pi\)
0.380752 + 0.924677i \(0.375665\pi\)
\(158\) 0 0
\(159\) −4.67783e29 −0.893683
\(160\) 0 0
\(161\) 1.03546e30 1.67104
\(162\) 0 0
\(163\) 2.71722e29 0.371187 0.185594 0.982627i \(-0.440579\pi\)
0.185594 + 0.982627i \(0.440579\pi\)
\(164\) 0 0
\(165\) 7.11640e29 0.824569
\(166\) 0 0
\(167\) −8.17343e29 −0.804882 −0.402441 0.915446i \(-0.631838\pi\)
−0.402441 + 0.915446i \(0.631838\pi\)
\(168\) 0 0
\(169\) −1.18392e30 −0.992777
\(170\) 0 0
\(171\) −4.33840e29 −0.310372
\(172\) 0 0
\(173\) −2.06734e30 −1.26413 −0.632063 0.774917i \(-0.717791\pi\)
−0.632063 + 0.774917i \(0.717791\pi\)
\(174\) 0 0
\(175\) −3.15229e29 −0.165049
\(176\) 0 0
\(177\) 1.29800e30 0.582954
\(178\) 0 0
\(179\) −2.84130e30 −1.09648 −0.548241 0.836320i \(-0.684703\pi\)
−0.548241 + 0.836320i \(0.684703\pi\)
\(180\) 0 0
\(181\) −4.08892e30 −1.35815 −0.679074 0.734069i \(-0.737619\pi\)
−0.679074 + 0.734069i \(0.737619\pi\)
\(182\) 0 0
\(183\) 8.92058e29 0.255449
\(184\) 0 0
\(185\) −4.06289e30 −1.00465
\(186\) 0 0
\(187\) −2.10388e30 −0.449945
\(188\) 0 0
\(189\) −1.02473e30 −0.189834
\(190\) 0 0
\(191\) −2.18709e30 −0.351493 −0.175746 0.984435i \(-0.556234\pi\)
−0.175746 + 0.984435i \(0.556234\pi\)
\(192\) 0 0
\(193\) 9.97718e30 1.39311 0.696553 0.717505i \(-0.254716\pi\)
0.696553 + 0.717505i \(0.254716\pi\)
\(194\) 0 0
\(195\) −4.36367e29 −0.0530133
\(196\) 0 0
\(197\) 2.89658e30 0.306614 0.153307 0.988179i \(-0.451008\pi\)
0.153307 + 0.988179i \(0.451008\pi\)
\(198\) 0 0
\(199\) −6.81123e30 −0.629085 −0.314542 0.949243i \(-0.601851\pi\)
−0.314542 + 0.949243i \(0.601851\pi\)
\(200\) 0 0
\(201\) −1.03793e31 −0.837572
\(202\) 0 0
\(203\) −1.09320e31 −0.771796
\(204\) 0 0
\(205\) −2.01169e31 −1.24418
\(206\) 0 0
\(207\) −1.04090e31 −0.564688
\(208\) 0 0
\(209\) 2.58327e31 1.23082
\(210\) 0 0
\(211\) −3.13701e31 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(212\) 0 0
\(213\) 5.88508e30 0.217084
\(214\) 0 0
\(215\) −2.77009e31 −0.900688
\(216\) 0 0
\(217\) 5.04689e31 1.44816
\(218\) 0 0
\(219\) 2.02499e30 0.0513366
\(220\) 0 0
\(221\) 1.29007e30 0.0289279
\(222\) 0 0
\(223\) −1.13345e31 −0.225054 −0.112527 0.993649i \(-0.535894\pi\)
−0.112527 + 0.993649i \(0.535894\pi\)
\(224\) 0 0
\(225\) 3.16884e30 0.0557745
\(226\) 0 0
\(227\) 1.38884e31 0.216922 0.108461 0.994101i \(-0.465408\pi\)
0.108461 + 0.994101i \(0.465408\pi\)
\(228\) 0 0
\(229\) −4.10593e31 −0.569682 −0.284841 0.958575i \(-0.591941\pi\)
−0.284841 + 0.958575i \(0.591941\pi\)
\(230\) 0 0
\(231\) 6.10167e31 0.752813
\(232\) 0 0
\(233\) 8.01474e31 0.880207 0.440103 0.897947i \(-0.354942\pi\)
0.440103 + 0.897947i \(0.354942\pi\)
\(234\) 0 0
\(235\) 6.30367e30 0.0616847
\(236\) 0 0
\(237\) −1.59528e31 −0.139230
\(238\) 0 0
\(239\) 1.22012e32 0.950670 0.475335 0.879805i \(-0.342327\pi\)
0.475335 + 0.879805i \(0.342327\pi\)
\(240\) 0 0
\(241\) 1.79059e32 1.24671 0.623357 0.781937i \(-0.285768\pi\)
0.623357 + 0.781937i \(0.285768\pi\)
\(242\) 0 0
\(243\) 1.03011e31 0.0641500
\(244\) 0 0
\(245\) 5.23289e30 0.0291742
\(246\) 0 0
\(247\) −1.58403e31 −0.0791322
\(248\) 0 0
\(249\) 1.48838e32 0.666842
\(250\) 0 0
\(251\) −2.98031e31 −0.119858 −0.0599290 0.998203i \(-0.519087\pi\)
−0.0599290 + 0.998203i \(0.519087\pi\)
\(252\) 0 0
\(253\) 6.19796e32 2.23935
\(254\) 0 0
\(255\) −6.53574e31 −0.212326
\(256\) 0 0
\(257\) −4.62356e31 −0.135170 −0.0675849 0.997714i \(-0.521529\pi\)
−0.0675849 + 0.997714i \(0.521529\pi\)
\(258\) 0 0
\(259\) −3.48356e32 −0.917226
\(260\) 0 0
\(261\) 1.09894e32 0.260811
\(262\) 0 0
\(263\) −3.44300e32 −0.737113 −0.368556 0.929605i \(-0.620148\pi\)
−0.368556 + 0.929605i \(0.620148\pi\)
\(264\) 0 0
\(265\) 8.65285e32 1.67240
\(266\) 0 0
\(267\) 4.82861e31 0.0843180
\(268\) 0 0
\(269\) 1.03375e33 1.63214 0.816072 0.577950i \(-0.196147\pi\)
0.816072 + 0.577950i \(0.196147\pi\)
\(270\) 0 0
\(271\) −9.90254e32 −1.41469 −0.707345 0.706869i \(-0.750107\pi\)
−0.707345 + 0.706869i \(0.750107\pi\)
\(272\) 0 0
\(273\) −3.74146e31 −0.0484000
\(274\) 0 0
\(275\) −1.88686e32 −0.221182
\(276\) 0 0
\(277\) −1.71524e33 −1.82326 −0.911631 0.411010i \(-0.865176\pi\)
−0.911631 + 0.411010i \(0.865176\pi\)
\(278\) 0 0
\(279\) −5.07337e32 −0.489373
\(280\) 0 0
\(281\) 3.27334e32 0.286719 0.143359 0.989671i \(-0.454210\pi\)
0.143359 + 0.989671i \(0.454210\pi\)
\(282\) 0 0
\(283\) −2.25086e32 −0.179156 −0.0895782 0.995980i \(-0.528552\pi\)
−0.0895782 + 0.995980i \(0.528552\pi\)
\(284\) 0 0
\(285\) 8.02498e32 0.580816
\(286\) 0 0
\(287\) −1.72484e33 −1.13591
\(288\) 0 0
\(289\) −1.47449e33 −0.884140
\(290\) 0 0
\(291\) −9.74741e32 −0.532517
\(292\) 0 0
\(293\) 3.63890e33 1.81241 0.906207 0.422833i \(-0.138964\pi\)
0.906207 + 0.422833i \(0.138964\pi\)
\(294\) 0 0
\(295\) −2.40097e33 −1.09091
\(296\) 0 0
\(297\) −6.13369e32 −0.254396
\(298\) 0 0
\(299\) −3.80050e32 −0.143973
\(300\) 0 0
\(301\) −2.37510e33 −0.822308
\(302\) 0 0
\(303\) 5.64236e32 0.178643
\(304\) 0 0
\(305\) −1.65009e33 −0.478035
\(306\) 0 0
\(307\) 3.46198e33 0.918241 0.459120 0.888374i \(-0.348165\pi\)
0.459120 + 0.888374i \(0.348165\pi\)
\(308\) 0 0
\(309\) 4.57033e33 1.11047
\(310\) 0 0
\(311\) −7.31854e33 −1.62989 −0.814947 0.579536i \(-0.803234\pi\)
−0.814947 + 0.579536i \(0.803234\pi\)
\(312\) 0 0
\(313\) 6.01872e32 0.122929 0.0614646 0.998109i \(-0.480423\pi\)
0.0614646 + 0.998109i \(0.480423\pi\)
\(314\) 0 0
\(315\) 1.89550e33 0.355247
\(316\) 0 0
\(317\) −5.16567e33 −0.888847 −0.444423 0.895817i \(-0.646591\pi\)
−0.444423 + 0.895817i \(0.646591\pi\)
\(318\) 0 0
\(319\) −6.54354e33 −1.03428
\(320\) 0 0
\(321\) −6.01504e33 −0.873818
\(322\) 0 0
\(323\) −2.37249e33 −0.316936
\(324\) 0 0
\(325\) 1.15700e32 0.0142202
\(326\) 0 0
\(327\) 1.90736e33 0.215793
\(328\) 0 0
\(329\) 5.40484e32 0.0563168
\(330\) 0 0
\(331\) −1.73903e34 −1.66967 −0.834834 0.550502i \(-0.814436\pi\)
−0.834834 + 0.550502i \(0.814436\pi\)
\(332\) 0 0
\(333\) 3.50184e33 0.309956
\(334\) 0 0
\(335\) 1.91992e34 1.56740
\(336\) 0 0
\(337\) 2.88416e33 0.217278 0.108639 0.994081i \(-0.465351\pi\)
0.108639 + 0.994081i \(0.465351\pi\)
\(338\) 0 0
\(339\) 9.62609e33 0.669505
\(340\) 0 0
\(341\) 3.02091e34 1.94068
\(342\) 0 0
\(343\) 1.70647e34 1.01304
\(344\) 0 0
\(345\) 1.92541e34 1.05673
\(346\) 0 0
\(347\) 2.65372e34 1.34712 0.673562 0.739131i \(-0.264763\pi\)
0.673562 + 0.739131i \(0.264763\pi\)
\(348\) 0 0
\(349\) −1.77192e34 −0.832341 −0.416170 0.909287i \(-0.636628\pi\)
−0.416170 + 0.909287i \(0.636628\pi\)
\(350\) 0 0
\(351\) 3.76109e32 0.0163557
\(352\) 0 0
\(353\) −1.17206e34 −0.472053 −0.236026 0.971747i \(-0.575845\pi\)
−0.236026 + 0.971747i \(0.575845\pi\)
\(354\) 0 0
\(355\) −1.08860e34 −0.406241
\(356\) 0 0
\(357\) −5.60381e33 −0.193849
\(358\) 0 0
\(359\) 3.36159e34 1.07838 0.539188 0.842186i \(-0.318731\pi\)
0.539188 + 0.842186i \(0.318731\pi\)
\(360\) 0 0
\(361\) −4.46969e33 −0.133024
\(362\) 0 0
\(363\) 1.56211e34 0.431491
\(364\) 0 0
\(365\) −3.74573e33 −0.0960690
\(366\) 0 0
\(367\) −6.24921e34 −1.48879 −0.744395 0.667740i \(-0.767262\pi\)
−0.744395 + 0.667740i \(0.767262\pi\)
\(368\) 0 0
\(369\) 1.73389e34 0.383854
\(370\) 0 0
\(371\) 7.41904e34 1.52686
\(372\) 0 0
\(373\) −1.81566e34 −0.347509 −0.173754 0.984789i \(-0.555590\pi\)
−0.173754 + 0.984789i \(0.555590\pi\)
\(374\) 0 0
\(375\) 2.91698e34 0.519411
\(376\) 0 0
\(377\) 4.01241e33 0.0664962
\(378\) 0 0
\(379\) 6.01249e34 0.927738 0.463869 0.885904i \(-0.346461\pi\)
0.463869 + 0.885904i \(0.346461\pi\)
\(380\) 0 0
\(381\) −3.49350e34 −0.502081
\(382\) 0 0
\(383\) −1.27205e34 −0.170342 −0.0851709 0.996366i \(-0.527144\pi\)
−0.0851709 + 0.996366i \(0.527144\pi\)
\(384\) 0 0
\(385\) −1.12866e35 −1.40878
\(386\) 0 0
\(387\) 2.38757e34 0.277880
\(388\) 0 0
\(389\) −1.29479e34 −0.140566 −0.0702832 0.997527i \(-0.522390\pi\)
−0.0702832 + 0.997527i \(0.522390\pi\)
\(390\) 0 0
\(391\) −5.69225e34 −0.576631
\(392\) 0 0
\(393\) 6.59660e34 0.623764
\(394\) 0 0
\(395\) 2.95087e34 0.260549
\(396\) 0 0
\(397\) −2.21092e35 −1.82348 −0.911741 0.410765i \(-0.865262\pi\)
−0.911741 + 0.410765i \(0.865262\pi\)
\(398\) 0 0
\(399\) 6.88070e34 0.530272
\(400\) 0 0
\(401\) −1.64003e34 −0.118142 −0.0590708 0.998254i \(-0.518814\pi\)
−0.0590708 + 0.998254i \(0.518814\pi\)
\(402\) 0 0
\(403\) −1.85238e34 −0.124770
\(404\) 0 0
\(405\) −1.90544e34 −0.120047
\(406\) 0 0
\(407\) −2.08515e35 −1.22917
\(408\) 0 0
\(409\) −1.30759e35 −0.721451 −0.360726 0.932672i \(-0.617471\pi\)
−0.360726 + 0.932672i \(0.617471\pi\)
\(410\) 0 0
\(411\) 2.09811e35 1.08384
\(412\) 0 0
\(413\) −2.05862e35 −0.995980
\(414\) 0 0
\(415\) −2.75314e35 −1.24790
\(416\) 0 0
\(417\) 2.08881e35 0.887285
\(418\) 0 0
\(419\) 3.17393e35 1.26389 0.631943 0.775015i \(-0.282258\pi\)
0.631943 + 0.775015i \(0.282258\pi\)
\(420\) 0 0
\(421\) −4.30484e35 −1.60749 −0.803746 0.594973i \(-0.797163\pi\)
−0.803746 + 0.594973i \(0.797163\pi\)
\(422\) 0 0
\(423\) −5.43320e33 −0.0190310
\(424\) 0 0
\(425\) 1.73291e34 0.0569540
\(426\) 0 0
\(427\) −1.41480e35 −0.436435
\(428\) 0 0
\(429\) −2.23952e34 −0.0648606
\(430\) 0 0
\(431\) 5.03774e35 1.37023 0.685114 0.728436i \(-0.259752\pi\)
0.685114 + 0.728436i \(0.259752\pi\)
\(432\) 0 0
\(433\) −7.57241e35 −1.93485 −0.967426 0.253154i \(-0.918532\pi\)
−0.967426 + 0.253154i \(0.918532\pi\)
\(434\) 0 0
\(435\) −2.03276e35 −0.488070
\(436\) 0 0
\(437\) 6.98929e35 1.57737
\(438\) 0 0
\(439\) 4.23974e35 0.899638 0.449819 0.893120i \(-0.351489\pi\)
0.449819 + 0.893120i \(0.351489\pi\)
\(440\) 0 0
\(441\) −4.51028e33 −0.00900083
\(442\) 0 0
\(443\) −8.43326e35 −1.58323 −0.791617 0.611017i \(-0.790761\pi\)
−0.791617 + 0.611017i \(0.790761\pi\)
\(444\) 0 0
\(445\) −8.93175e34 −0.157789
\(446\) 0 0
\(447\) −1.58122e35 −0.262931
\(448\) 0 0
\(449\) 5.18891e35 0.812364 0.406182 0.913792i \(-0.366860\pi\)
0.406182 + 0.913792i \(0.366860\pi\)
\(450\) 0 0
\(451\) −1.03243e36 −1.52223
\(452\) 0 0
\(453\) 5.17142e35 0.718264
\(454\) 0 0
\(455\) 6.92078e34 0.0905736
\(456\) 0 0
\(457\) 9.52583e35 1.17499 0.587495 0.809228i \(-0.300114\pi\)
0.587495 + 0.809228i \(0.300114\pi\)
\(458\) 0 0
\(459\) 5.63322e34 0.0655067
\(460\) 0 0
\(461\) 1.04963e36 1.15099 0.575497 0.817804i \(-0.304809\pi\)
0.575497 + 0.817804i \(0.304809\pi\)
\(462\) 0 0
\(463\) −5.25044e35 −0.543066 −0.271533 0.962429i \(-0.587531\pi\)
−0.271533 + 0.962429i \(0.587531\pi\)
\(464\) 0 0
\(465\) 9.38450e35 0.915791
\(466\) 0 0
\(467\) 1.53832e36 1.41667 0.708336 0.705876i \(-0.249446\pi\)
0.708336 + 0.705876i \(0.249446\pi\)
\(468\) 0 0
\(469\) 1.64616e36 1.43100
\(470\) 0 0
\(471\) 5.35665e35 0.439654
\(472\) 0 0
\(473\) −1.42166e36 −1.10197
\(474\) 0 0
\(475\) −2.12777e35 −0.155797
\(476\) 0 0
\(477\) −7.45798e35 −0.515968
\(478\) 0 0
\(479\) 1.71162e35 0.111912 0.0559561 0.998433i \(-0.482179\pi\)
0.0559561 + 0.998433i \(0.482179\pi\)
\(480\) 0 0
\(481\) 1.27858e35 0.0790261
\(482\) 0 0
\(483\) 1.65087e36 0.964774
\(484\) 0 0
\(485\) 1.80303e36 0.996529
\(486\) 0 0
\(487\) 1.41363e35 0.0739084 0.0369542 0.999317i \(-0.488234\pi\)
0.0369542 + 0.999317i \(0.488234\pi\)
\(488\) 0 0
\(489\) 4.33213e35 0.214305
\(490\) 0 0
\(491\) 1.38190e35 0.0646959 0.0323480 0.999477i \(-0.489702\pi\)
0.0323480 + 0.999477i \(0.489702\pi\)
\(492\) 0 0
\(493\) 6.00963e35 0.266327
\(494\) 0 0
\(495\) 1.13458e36 0.476065
\(496\) 0 0
\(497\) −9.33374e35 −0.370889
\(498\) 0 0
\(499\) −2.66489e36 −1.00305 −0.501524 0.865144i \(-0.667227\pi\)
−0.501524 + 0.865144i \(0.667227\pi\)
\(500\) 0 0
\(501\) −1.30311e36 −0.464699
\(502\) 0 0
\(503\) −1.79792e36 −0.607578 −0.303789 0.952739i \(-0.598252\pi\)
−0.303789 + 0.952739i \(0.598252\pi\)
\(504\) 0 0
\(505\) −1.04370e36 −0.334304
\(506\) 0 0
\(507\) −1.88755e36 −0.573180
\(508\) 0 0
\(509\) 2.19200e36 0.631176 0.315588 0.948896i \(-0.397798\pi\)
0.315588 + 0.948896i \(0.397798\pi\)
\(510\) 0 0
\(511\) −3.21163e35 −0.0877089
\(512\) 0 0
\(513\) −6.91681e35 −0.179193
\(514\) 0 0
\(515\) −8.45399e36 −2.07809
\(516\) 0 0
\(517\) 3.23516e35 0.0754699
\(518\) 0 0
\(519\) −3.29601e36 −0.729843
\(520\) 0 0
\(521\) 1.51669e36 0.318852 0.159426 0.987210i \(-0.449036\pi\)
0.159426 + 0.987210i \(0.449036\pi\)
\(522\) 0 0
\(523\) −3.10531e36 −0.619915 −0.309958 0.950750i \(-0.600315\pi\)
−0.309958 + 0.950750i \(0.600315\pi\)
\(524\) 0 0
\(525\) −5.02577e35 −0.0952911
\(526\) 0 0
\(527\) −2.77442e36 −0.499723
\(528\) 0 0
\(529\) 1.09260e37 1.86986
\(530\) 0 0
\(531\) 2.06942e36 0.336569
\(532\) 0 0
\(533\) 6.33074e35 0.0978673
\(534\) 0 0
\(535\) 1.11264e37 1.63522
\(536\) 0 0
\(537\) −4.52996e36 −0.633055
\(538\) 0 0
\(539\) 2.68562e35 0.0356940
\(540\) 0 0
\(541\) −1.30208e37 −1.64617 −0.823087 0.567916i \(-0.807750\pi\)
−0.823087 + 0.567916i \(0.807750\pi\)
\(542\) 0 0
\(543\) −6.51905e36 −0.784128
\(544\) 0 0
\(545\) −3.52814e36 −0.403825
\(546\) 0 0
\(547\) −4.39612e36 −0.478895 −0.239448 0.970909i \(-0.576966\pi\)
−0.239448 + 0.970909i \(0.576966\pi\)
\(548\) 0 0
\(549\) 1.42223e36 0.147483
\(550\) 0 0
\(551\) −7.37899e36 −0.728535
\(552\) 0 0
\(553\) 2.53011e36 0.237875
\(554\) 0 0
\(555\) −6.47755e36 −0.580037
\(556\) 0 0
\(557\) 3.10948e36 0.265242 0.132621 0.991167i \(-0.457661\pi\)
0.132621 + 0.991167i \(0.457661\pi\)
\(558\) 0 0
\(559\) 8.71743e35 0.0708482
\(560\) 0 0
\(561\) −3.35426e36 −0.259776
\(562\) 0 0
\(563\) 9.78955e36 0.722603 0.361301 0.932449i \(-0.382333\pi\)
0.361301 + 0.932449i \(0.382333\pi\)
\(564\) 0 0
\(565\) −1.78059e37 −1.25288
\(566\) 0 0
\(567\) −1.63375e36 −0.109601
\(568\) 0 0
\(569\) −2.38541e37 −1.52597 −0.762986 0.646415i \(-0.776267\pi\)
−0.762986 + 0.646415i \(0.776267\pi\)
\(570\) 0 0
\(571\) 2.74250e37 1.67324 0.836622 0.547781i \(-0.184527\pi\)
0.836622 + 0.547781i \(0.184527\pi\)
\(572\) 0 0
\(573\) −3.48693e36 −0.202934
\(574\) 0 0
\(575\) −5.10509e36 −0.283457
\(576\) 0 0
\(577\) 3.11547e37 1.65063 0.825316 0.564672i \(-0.190997\pi\)
0.825316 + 0.564672i \(0.190997\pi\)
\(578\) 0 0
\(579\) 1.59069e37 0.804310
\(580\) 0 0
\(581\) −2.36057e37 −1.13930
\(582\) 0 0
\(583\) 4.44080e37 2.04614
\(584\) 0 0
\(585\) −6.95710e35 −0.0306073
\(586\) 0 0
\(587\) −4.69405e36 −0.197212 −0.0986059 0.995127i \(-0.531438\pi\)
−0.0986059 + 0.995127i \(0.531438\pi\)
\(588\) 0 0
\(589\) 3.40660e37 1.36699
\(590\) 0 0
\(591\) 4.61809e36 0.177024
\(592\) 0 0
\(593\) 3.58866e37 1.31430 0.657149 0.753761i \(-0.271762\pi\)
0.657149 + 0.753761i \(0.271762\pi\)
\(594\) 0 0
\(595\) 1.03657e37 0.362760
\(596\) 0 0
\(597\) −1.08593e37 −0.363202
\(598\) 0 0
\(599\) 1.35541e37 0.433321 0.216660 0.976247i \(-0.430484\pi\)
0.216660 + 0.976247i \(0.430484\pi\)
\(600\) 0 0
\(601\) −2.72933e37 −0.834163 −0.417081 0.908869i \(-0.636947\pi\)
−0.417081 + 0.908869i \(0.636947\pi\)
\(602\) 0 0
\(603\) −1.65480e37 −0.483573
\(604\) 0 0
\(605\) −2.88952e37 −0.807474
\(606\) 0 0
\(607\) 6.35953e37 1.69972 0.849861 0.527007i \(-0.176686\pi\)
0.849861 + 0.527007i \(0.176686\pi\)
\(608\) 0 0
\(609\) −1.74291e37 −0.445597
\(610\) 0 0
\(611\) −1.98376e35 −0.00485212
\(612\) 0 0
\(613\) 4.82418e37 1.12903 0.564517 0.825421i \(-0.309062\pi\)
0.564517 + 0.825421i \(0.309062\pi\)
\(614\) 0 0
\(615\) −3.20728e37 −0.718327
\(616\) 0 0
\(617\) 2.26067e37 0.484604 0.242302 0.970201i \(-0.422097\pi\)
0.242302 + 0.970201i \(0.422097\pi\)
\(618\) 0 0
\(619\) 2.23444e37 0.458506 0.229253 0.973367i \(-0.426372\pi\)
0.229253 + 0.973367i \(0.426372\pi\)
\(620\) 0 0
\(621\) −1.65953e37 −0.326023
\(622\) 0 0
\(623\) −7.65817e36 −0.144058
\(624\) 0 0
\(625\) −6.32453e37 −1.13933
\(626\) 0 0
\(627\) 4.11857e37 0.710616
\(628\) 0 0
\(629\) 1.91501e37 0.316511
\(630\) 0 0
\(631\) 5.33027e37 0.844021 0.422010 0.906591i \(-0.361325\pi\)
0.422010 + 0.906591i \(0.361325\pi\)
\(632\) 0 0
\(633\) −5.00140e37 −0.758825
\(634\) 0 0
\(635\) 6.46213e37 0.939573
\(636\) 0 0
\(637\) −1.64678e35 −0.00229485
\(638\) 0 0
\(639\) 9.38272e36 0.125333
\(640\) 0 0
\(641\) 1.19868e38 1.53504 0.767522 0.641023i \(-0.221490\pi\)
0.767522 + 0.641023i \(0.221490\pi\)
\(642\) 0 0
\(643\) −8.41805e37 −1.03363 −0.516813 0.856098i \(-0.672882\pi\)
−0.516813 + 0.856098i \(0.672882\pi\)
\(644\) 0 0
\(645\) −4.41642e37 −0.520013
\(646\) 0 0
\(647\) −1.54778e38 −1.74784 −0.873919 0.486072i \(-0.838429\pi\)
−0.873919 + 0.486072i \(0.838429\pi\)
\(648\) 0 0
\(649\) −1.23222e38 −1.33471
\(650\) 0 0
\(651\) 8.04637e37 0.836097
\(652\) 0 0
\(653\) −1.36035e38 −1.35620 −0.678099 0.734970i \(-0.737196\pi\)
−0.678099 + 0.734970i \(0.737196\pi\)
\(654\) 0 0
\(655\) −1.22021e38 −1.16728
\(656\) 0 0
\(657\) 3.22849e36 0.0296392
\(658\) 0 0
\(659\) −1.52293e38 −1.34192 −0.670959 0.741494i \(-0.734117\pi\)
−0.670959 + 0.741494i \(0.734117\pi\)
\(660\) 0 0
\(661\) 1.83977e38 1.55612 0.778062 0.628188i \(-0.216203\pi\)
0.778062 + 0.628188i \(0.216203\pi\)
\(662\) 0 0
\(663\) 2.05679e36 0.0167016
\(664\) 0 0
\(665\) −1.27276e38 −0.992327
\(666\) 0 0
\(667\) −1.77042e38 −1.32549
\(668\) 0 0
\(669\) −1.80708e37 −0.129935
\(670\) 0 0
\(671\) −8.46857e37 −0.584866
\(672\) 0 0
\(673\) 9.36686e37 0.621428 0.310714 0.950503i \(-0.399432\pi\)
0.310714 + 0.950503i \(0.399432\pi\)
\(674\) 0 0
\(675\) 5.05215e36 0.0322014
\(676\) 0 0
\(677\) 2.69907e38 1.65298 0.826488 0.562954i \(-0.190335\pi\)
0.826488 + 0.562954i \(0.190335\pi\)
\(678\) 0 0
\(679\) 1.54594e38 0.909809
\(680\) 0 0
\(681\) 2.21427e37 0.125240
\(682\) 0 0
\(683\) 1.91566e38 1.04145 0.520726 0.853724i \(-0.325661\pi\)
0.520726 + 0.853724i \(0.325661\pi\)
\(684\) 0 0
\(685\) −3.88100e38 −2.02825
\(686\) 0 0
\(687\) −6.54618e37 −0.328906
\(688\) 0 0
\(689\) −2.72304e37 −0.131551
\(690\) 0 0
\(691\) 6.26889e37 0.291230 0.145615 0.989341i \(-0.453484\pi\)
0.145615 + 0.989341i \(0.453484\pi\)
\(692\) 0 0
\(693\) 9.72803e37 0.434637
\(694\) 0 0
\(695\) −3.86380e38 −1.66043
\(696\) 0 0
\(697\) 9.48195e37 0.391972
\(698\) 0 0
\(699\) 1.27781e38 0.508188
\(700\) 0 0
\(701\) 1.06445e38 0.407317 0.203659 0.979042i \(-0.434717\pi\)
0.203659 + 0.979042i \(0.434717\pi\)
\(702\) 0 0
\(703\) −2.35137e38 −0.865813
\(704\) 0 0
\(705\) 1.00501e37 0.0356137
\(706\) 0 0
\(707\) −8.94878e37 −0.305212
\(708\) 0 0
\(709\) 7.15469e37 0.234891 0.117446 0.993079i \(-0.462529\pi\)
0.117446 + 0.993079i \(0.462529\pi\)
\(710\) 0 0
\(711\) −2.54338e37 −0.0803845
\(712\) 0 0
\(713\) 8.17335e38 2.48709
\(714\) 0 0
\(715\) 4.14256e37 0.121377
\(716\) 0 0
\(717\) 1.94526e38 0.548870
\(718\) 0 0
\(719\) −2.71010e38 −0.736453 −0.368227 0.929736i \(-0.620035\pi\)
−0.368227 + 0.929736i \(0.620035\pi\)
\(720\) 0 0
\(721\) −7.24854e38 −1.89725
\(722\) 0 0
\(723\) 2.85478e38 0.719791
\(724\) 0 0
\(725\) 5.38973e37 0.130919
\(726\) 0 0
\(727\) 2.02421e38 0.473741 0.236870 0.971541i \(-0.423878\pi\)
0.236870 + 0.971541i \(0.423878\pi\)
\(728\) 0 0
\(729\) 1.64232e37 0.0370370
\(730\) 0 0
\(731\) 1.30566e38 0.283757
\(732\) 0 0
\(733\) −3.86048e38 −0.808607 −0.404304 0.914625i \(-0.632486\pi\)
−0.404304 + 0.914625i \(0.632486\pi\)
\(734\) 0 0
\(735\) 8.34292e36 0.0168438
\(736\) 0 0
\(737\) 9.85340e38 1.91767
\(738\) 0 0
\(739\) −6.27102e38 −1.17663 −0.588313 0.808633i \(-0.700208\pi\)
−0.588313 + 0.808633i \(0.700208\pi\)
\(740\) 0 0
\(741\) −2.52545e37 −0.0456870
\(742\) 0 0
\(743\) 7.62105e38 1.32943 0.664716 0.747096i \(-0.268553\pi\)
0.664716 + 0.747096i \(0.268553\pi\)
\(744\) 0 0
\(745\) 2.92488e38 0.492037
\(746\) 0 0
\(747\) 2.37296e38 0.385002
\(748\) 0 0
\(749\) 9.53986e38 1.49292
\(750\) 0 0
\(751\) 6.28777e38 0.949200 0.474600 0.880202i \(-0.342593\pi\)
0.474600 + 0.880202i \(0.342593\pi\)
\(752\) 0 0
\(753\) −4.75158e37 −0.0692000
\(754\) 0 0
\(755\) −9.56586e38 −1.34413
\(756\) 0 0
\(757\) −8.24507e38 −1.11789 −0.558946 0.829204i \(-0.688794\pi\)
−0.558946 + 0.829204i \(0.688794\pi\)
\(758\) 0 0
\(759\) 9.88156e38 1.29289
\(760\) 0 0
\(761\) 1.41435e39 1.78593 0.892963 0.450130i \(-0.148622\pi\)
0.892963 + 0.450130i \(0.148622\pi\)
\(762\) 0 0
\(763\) −3.02507e38 −0.368684
\(764\) 0 0
\(765\) −1.04201e38 −0.122586
\(766\) 0 0
\(767\) 7.55583e37 0.0858113
\(768\) 0 0
\(769\) 7.77007e38 0.851960 0.425980 0.904732i \(-0.359929\pi\)
0.425980 + 0.904732i \(0.359929\pi\)
\(770\) 0 0
\(771\) −7.37145e37 −0.0780403
\(772\) 0 0
\(773\) −4.61034e38 −0.471314 −0.235657 0.971836i \(-0.575724\pi\)
−0.235657 + 0.971836i \(0.575724\pi\)
\(774\) 0 0
\(775\) −2.48823e38 −0.245651
\(776\) 0 0
\(777\) −5.55392e38 −0.529561
\(778\) 0 0
\(779\) −1.16425e39 −1.07224
\(780\) 0 0
\(781\) −5.58688e38 −0.497027
\(782\) 0 0
\(783\) 1.75206e38 0.150579
\(784\) 0 0
\(785\) −9.90850e38 −0.822749
\(786\) 0 0
\(787\) −1.74475e38 −0.139983 −0.0699914 0.997548i \(-0.522297\pi\)
−0.0699914 + 0.997548i \(0.522297\pi\)
\(788\) 0 0
\(789\) −5.48925e38 −0.425572
\(790\) 0 0
\(791\) −1.52670e39 −1.14385
\(792\) 0 0
\(793\) 5.19281e37 0.0376023
\(794\) 0 0
\(795\) 1.37954e39 0.965560
\(796\) 0 0
\(797\) −6.47929e38 −0.438369 −0.219185 0.975683i \(-0.570340\pi\)
−0.219185 + 0.975683i \(0.570340\pi\)
\(798\) 0 0
\(799\) −2.97119e37 −0.0194334
\(800\) 0 0
\(801\) 7.69836e37 0.0486810
\(802\) 0 0
\(803\) −1.92238e38 −0.117538
\(804\) 0 0
\(805\) −3.05370e39 −1.80543
\(806\) 0 0
\(807\) 1.64813e39 0.942319
\(808\) 0 0
\(809\) −2.06930e39 −1.14424 −0.572122 0.820168i \(-0.693880\pi\)
−0.572122 + 0.820168i \(0.693880\pi\)
\(810\) 0 0
\(811\) −4.70862e38 −0.251833 −0.125916 0.992041i \(-0.540187\pi\)
−0.125916 + 0.992041i \(0.540187\pi\)
\(812\) 0 0
\(813\) −1.57878e39 −0.816771
\(814\) 0 0
\(815\) −8.01339e38 −0.401041
\(816\) 0 0
\(817\) −1.60317e39 −0.776216
\(818\) 0 0
\(819\) −5.96509e37 −0.0279437
\(820\) 0 0
\(821\) 1.51803e39 0.688093 0.344046 0.938953i \(-0.388202\pi\)
0.344046 + 0.938953i \(0.388202\pi\)
\(822\) 0 0
\(823\) 3.42558e39 1.50257 0.751287 0.659975i \(-0.229433\pi\)
0.751287 + 0.659975i \(0.229433\pi\)
\(824\) 0 0
\(825\) −3.00827e38 −0.127699
\(826\) 0 0
\(827\) 3.46077e39 1.42183 0.710916 0.703277i \(-0.248281\pi\)
0.710916 + 0.703277i \(0.248281\pi\)
\(828\) 0 0
\(829\) −2.65696e39 −1.05657 −0.528286 0.849067i \(-0.677165\pi\)
−0.528286 + 0.849067i \(0.677165\pi\)
\(830\) 0 0
\(831\) −2.73465e39 −1.05266
\(832\) 0 0
\(833\) −2.46649e37 −0.00919118
\(834\) 0 0
\(835\) 2.41043e39 0.869616
\(836\) 0 0
\(837\) −8.08860e38 −0.282540
\(838\) 0 0
\(839\) 2.26620e39 0.766499 0.383249 0.923645i \(-0.374805\pi\)
0.383249 + 0.923645i \(0.374805\pi\)
\(840\) 0 0
\(841\) −1.18400e39 −0.387799
\(842\) 0 0
\(843\) 5.21877e38 0.165537
\(844\) 0 0
\(845\) 3.49151e39 1.07262
\(846\) 0 0
\(847\) −2.47751e39 −0.737206
\(848\) 0 0
\(849\) −3.58860e38 −0.103436
\(850\) 0 0
\(851\) −5.64157e39 −1.57526
\(852\) 0 0
\(853\) −7.54148e38 −0.204007 −0.102003 0.994784i \(-0.532525\pi\)
−0.102003 + 0.994784i \(0.532525\pi\)
\(854\) 0 0
\(855\) 1.27944e39 0.335334
\(856\) 0 0
\(857\) −1.01461e39 −0.257666 −0.128833 0.991666i \(-0.541123\pi\)
−0.128833 + 0.991666i \(0.541123\pi\)
\(858\) 0 0
\(859\) 7.17513e39 1.76572 0.882861 0.469635i \(-0.155614\pi\)
0.882861 + 0.469635i \(0.155614\pi\)
\(860\) 0 0
\(861\) −2.74995e39 −0.655817
\(862\) 0 0
\(863\) −4.96688e39 −1.14799 −0.573994 0.818859i \(-0.694607\pi\)
−0.573994 + 0.818859i \(0.694607\pi\)
\(864\) 0 0
\(865\) 6.09682e39 1.36580
\(866\) 0 0
\(867\) −2.35081e39 −0.510458
\(868\) 0 0
\(869\) 1.51444e39 0.318776
\(870\) 0 0
\(871\) −6.04196e38 −0.123291
\(872\) 0 0
\(873\) −1.55405e39 −0.307449
\(874\) 0 0
\(875\) −4.62633e39 −0.887417
\(876\) 0 0
\(877\) 8.89158e39 1.65381 0.826903 0.562344i \(-0.190100\pi\)
0.826903 + 0.562344i \(0.190100\pi\)
\(878\) 0 0
\(879\) 5.80159e39 1.04640
\(880\) 0 0
\(881\) 8.48498e39 1.48414 0.742072 0.670320i \(-0.233843\pi\)
0.742072 + 0.670320i \(0.233843\pi\)
\(882\) 0 0
\(883\) 8.65208e39 1.46775 0.733873 0.679287i \(-0.237711\pi\)
0.733873 + 0.679287i \(0.237711\pi\)
\(884\) 0 0
\(885\) −3.82793e39 −0.629839
\(886\) 0 0
\(887\) 8.11389e38 0.129497 0.0647486 0.997902i \(-0.479375\pi\)
0.0647486 + 0.997902i \(0.479375\pi\)
\(888\) 0 0
\(889\) 5.54070e39 0.857809
\(890\) 0 0
\(891\) −9.77909e38 −0.146875
\(892\) 0 0
\(893\) 3.64821e38 0.0531601
\(894\) 0 0
\(895\) 8.37931e39 1.18467
\(896\) 0 0
\(897\) −6.05923e38 −0.0831227
\(898\) 0 0
\(899\) −8.62907e39 −1.14870
\(900\) 0 0
\(901\) −4.07846e39 −0.526880
\(902\) 0 0
\(903\) −3.78668e39 −0.474760
\(904\) 0 0
\(905\) 1.20587e40 1.46738
\(906\) 0 0
\(907\) 4.26511e39 0.503769 0.251884 0.967757i \(-0.418950\pi\)
0.251884 + 0.967757i \(0.418950\pi\)
\(908\) 0 0
\(909\) 8.99574e38 0.103139
\(910\) 0 0
\(911\) −1.06535e40 −1.18576 −0.592878 0.805292i \(-0.702009\pi\)
−0.592878 + 0.805292i \(0.702009\pi\)
\(912\) 0 0
\(913\) −1.41296e40 −1.52678
\(914\) 0 0
\(915\) −2.63078e39 −0.275994
\(916\) 0 0
\(917\) −1.04622e40 −1.06570
\(918\) 0 0
\(919\) 1.28657e40 1.27254 0.636271 0.771465i \(-0.280476\pi\)
0.636271 + 0.771465i \(0.280476\pi\)
\(920\) 0 0
\(921\) 5.51952e39 0.530147
\(922\) 0 0
\(923\) 3.42579e38 0.0319550
\(924\) 0 0
\(925\) 1.71748e39 0.155589
\(926\) 0 0
\(927\) 7.28658e39 0.641132
\(928\) 0 0
\(929\) 3.77009e39 0.322211 0.161105 0.986937i \(-0.448494\pi\)
0.161105 + 0.986937i \(0.448494\pi\)
\(930\) 0 0
\(931\) 3.02850e38 0.0251424
\(932\) 0 0
\(933\) −1.16681e40 −0.941020
\(934\) 0 0
\(935\) 6.20457e39 0.486133
\(936\) 0 0
\(937\) 2.50990e40 1.91061 0.955307 0.295617i \(-0.0955251\pi\)
0.955307 + 0.295617i \(0.0955251\pi\)
\(938\) 0 0
\(939\) 9.59578e38 0.0709733
\(940\) 0 0
\(941\) 1.48642e39 0.106827 0.0534136 0.998572i \(-0.482990\pi\)
0.0534136 + 0.998572i \(0.482990\pi\)
\(942\) 0 0
\(943\) −2.79335e40 −1.95082
\(944\) 0 0
\(945\) 3.02203e39 0.205102
\(946\) 0 0
\(947\) −1.95808e40 −1.29153 −0.645766 0.763536i \(-0.723462\pi\)
−0.645766 + 0.763536i \(0.723462\pi\)
\(948\) 0 0
\(949\) 1.17878e38 0.00755679
\(950\) 0 0
\(951\) −8.23575e39 −0.513176
\(952\) 0 0
\(953\) 2.88838e40 1.74945 0.874723 0.484623i \(-0.161043\pi\)
0.874723 + 0.484623i \(0.161043\pi\)
\(954\) 0 0
\(955\) 6.44996e39 0.379762
\(956\) 0 0
\(957\) −1.04325e40 −0.597143
\(958\) 0 0
\(959\) −3.32761e40 −1.85174
\(960\) 0 0
\(961\) 2.13545e40 1.15537
\(962\) 0 0
\(963\) −9.58992e39 −0.504499
\(964\) 0 0
\(965\) −2.94238e40 −1.50515
\(966\) 0 0
\(967\) −2.69517e40 −1.34069 −0.670346 0.742049i \(-0.733854\pi\)
−0.670346 + 0.742049i \(0.733854\pi\)
\(968\) 0 0
\(969\) −3.78252e39 −0.182983
\(970\) 0 0
\(971\) 5.85142e39 0.275297 0.137649 0.990481i \(-0.456046\pi\)
0.137649 + 0.990481i \(0.456046\pi\)
\(972\) 0 0
\(973\) −3.31286e40 −1.51593
\(974\) 0 0
\(975\) 1.84463e38 0.00821006
\(976\) 0 0
\(977\) 6.94583e39 0.300710 0.150355 0.988632i \(-0.451958\pi\)
0.150355 + 0.988632i \(0.451958\pi\)
\(978\) 0 0
\(979\) −4.58394e39 −0.193051
\(980\) 0 0
\(981\) 3.04094e39 0.124588
\(982\) 0 0
\(983\) 1.80358e40 0.718892 0.359446 0.933166i \(-0.382966\pi\)
0.359446 + 0.933166i \(0.382966\pi\)
\(984\) 0 0
\(985\) −8.54233e39 −0.331274
\(986\) 0 0
\(987\) 8.61705e38 0.0325145
\(988\) 0 0
\(989\) −3.84644e40 −1.41224
\(990\) 0 0
\(991\) −4.92955e40 −1.76122 −0.880608 0.473846i \(-0.842865\pi\)
−0.880608 + 0.473846i \(0.842865\pi\)
\(992\) 0 0
\(993\) −2.77258e40 −0.963983
\(994\) 0 0
\(995\) 2.00871e40 0.679681
\(996\) 0 0
\(997\) 8.75195e39 0.288217 0.144109 0.989562i \(-0.453968\pi\)
0.144109 + 0.989562i \(0.453968\pi\)
\(998\) 0 0
\(999\) 5.58307e39 0.178953
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 24.28.a.d.1.1 4
4.3 odd 2 48.28.a.l.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.28.a.d.1.1 4 1.1 even 1 trivial
48.28.a.l.1.1 4 4.3 odd 2