Properties

Label 64.24.a.e.1.2
Level $64$
Weight $24$
Character 64.1
Self dual yes
Analytic conductor $214.531$
Analytic rank $0$
Dimension $2$
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] = [64,24,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(214.530583901\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 618312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 4)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-785.828\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+216690. q^{3} -1.16920e8 q^{5} -5.39865e8 q^{7} -4.71886e10 q^{9} +O(q^{10})\) \(q+216690. q^{3} -1.16920e8 q^{5} -5.39865e8 q^{7} -4.71886e10 q^{9} -2.77240e11 q^{11} +1.06685e13 q^{13} -2.53354e13 q^{15} -2.32199e14 q^{17} +4.66455e14 q^{19} -1.16983e14 q^{21} +6.60775e15 q^{23} +1.74937e15 q^{25} -3.06252e16 q^{27} +4.34130e14 q^{29} -2.65725e17 q^{31} -6.00751e16 q^{33} +6.31211e16 q^{35} -1.71922e17 q^{37} +2.31175e18 q^{39} +5.88204e17 q^{41} -4.52759e18 q^{43} +5.51729e18 q^{45} -2.22092e19 q^{47} -2.70773e19 q^{49} -5.03153e19 q^{51} +1.08925e19 q^{53} +3.24149e19 q^{55} +1.01076e20 q^{57} -3.00557e20 q^{59} +4.24437e20 q^{61} +2.54755e19 q^{63} -1.24736e21 q^{65} -6.81182e20 q^{67} +1.43183e21 q^{69} +2.50119e21 q^{71} +2.31511e21 q^{73} +3.79070e20 q^{75} +1.49672e20 q^{77} +4.71022e21 q^{79} -2.19369e21 q^{81} +7.97680e21 q^{83} +2.71488e22 q^{85} +9.40716e19 q^{87} +2.30111e22 q^{89} -5.75954e21 q^{91} -5.75800e22 q^{93} -5.45379e22 q^{95} +9.70590e22 q^{97} +1.30826e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 170520 q^{3} + 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 170520 q^{3} + 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9} + 1247174695800 q^{11} + 7460299980980 q^{13} - 106334360092080 q^{15} - 374897347903260 q^{17} + 840360279212552 q^{19} - 400401295079232 q^{21} + 64\!\cdots\!20 q^{23}+ \cdots + 98\!\cdots\!00 q^{99}+O(q^{100}) \) 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
\(3\) 216690. 0.706227 0.353114 0.935580i \(-0.385123\pi\)
0.353114 + 0.935580i \(0.385123\pi\)
\(4\) 0 0
\(5\) −1.16920e8 −1.07086 −0.535431 0.844579i \(-0.679851\pi\)
−0.535431 + 0.844579i \(0.679851\pi\)
\(6\) 0 0
\(7\) −5.39865e8 −0.103195 −0.0515974 0.998668i \(-0.516431\pi\)
−0.0515974 + 0.998668i \(0.516431\pi\)
\(8\) 0 0
\(9\) −4.71886e10 −0.501243
\(10\) 0 0
\(11\) −2.77240e11 −0.292981 −0.146491 0.989212i \(-0.546798\pi\)
−0.146491 + 0.989212i \(0.546798\pi\)
\(12\) 0 0
\(13\) 1.06685e13 1.65103 0.825513 0.564383i \(-0.190886\pi\)
0.825513 + 0.564383i \(0.190886\pi\)
\(14\) 0 0
\(15\) −2.53354e13 −0.756273
\(16\) 0 0
\(17\) −2.32199e14 −1.64323 −0.821615 0.570043i \(-0.806927\pi\)
−0.821615 + 0.570043i \(0.806927\pi\)
\(18\) 0 0
\(19\) 4.66455e14 0.918635 0.459318 0.888272i \(-0.348094\pi\)
0.459318 + 0.888272i \(0.348094\pi\)
\(20\) 0 0
\(21\) −1.16983e14 −0.0728790
\(22\) 0 0
\(23\) 6.60775e15 1.44605 0.723026 0.690821i \(-0.242751\pi\)
0.723026 + 0.690821i \(0.242751\pi\)
\(24\) 0 0
\(25\) 1.74937e15 0.146747
\(26\) 0 0
\(27\) −3.06252e16 −1.06022
\(28\) 0 0
\(29\) 4.34130e14 0.00660758 0.00330379 0.999995i \(-0.498948\pi\)
0.00330379 + 0.999995i \(0.498948\pi\)
\(30\) 0 0
\(31\) −2.65725e17 −1.87833 −0.939167 0.343460i \(-0.888401\pi\)
−0.939167 + 0.343460i \(0.888401\pi\)
\(32\) 0 0
\(33\) −6.00751e16 −0.206911
\(34\) 0 0
\(35\) 6.31211e16 0.110507
\(36\) 0 0
\(37\) −1.71922e17 −0.158859 −0.0794294 0.996840i \(-0.525310\pi\)
−0.0794294 + 0.996840i \(0.525310\pi\)
\(38\) 0 0
\(39\) 2.31175e18 1.16600
\(40\) 0 0
\(41\) 5.88204e17 0.166922 0.0834610 0.996511i \(-0.473403\pi\)
0.0834610 + 0.996511i \(0.473403\pi\)
\(42\) 0 0
\(43\) −4.52759e18 −0.742985 −0.371492 0.928436i \(-0.621154\pi\)
−0.371492 + 0.928436i \(0.621154\pi\)
\(44\) 0 0
\(45\) 5.51729e18 0.536762
\(46\) 0 0
\(47\) −2.22092e19 −1.31041 −0.655205 0.755451i \(-0.727418\pi\)
−0.655205 + 0.755451i \(0.727418\pi\)
\(48\) 0 0
\(49\) −2.70773e19 −0.989351
\(50\) 0 0
\(51\) −5.03153e19 −1.16049
\(52\) 0 0
\(53\) 1.08925e19 0.161420 0.0807099 0.996738i \(-0.474281\pi\)
0.0807099 + 0.996738i \(0.474281\pi\)
\(54\) 0 0
\(55\) 3.24149e19 0.313743
\(56\) 0 0
\(57\) 1.01076e20 0.648765
\(58\) 0 0
\(59\) −3.00557e20 −1.29756 −0.648782 0.760974i \(-0.724721\pi\)
−0.648782 + 0.760974i \(0.724721\pi\)
\(60\) 0 0
\(61\) 4.24437e20 1.24888 0.624440 0.781073i \(-0.285327\pi\)
0.624440 + 0.781073i \(0.285327\pi\)
\(62\) 0 0
\(63\) 2.54755e19 0.0517256
\(64\) 0 0
\(65\) −1.24736e21 −1.76802
\(66\) 0 0
\(67\) −6.81182e20 −0.681402 −0.340701 0.940172i \(-0.610664\pi\)
−0.340701 + 0.940172i \(0.610664\pi\)
\(68\) 0 0
\(69\) 1.43183e21 1.02124
\(70\) 0 0
\(71\) 2.50119e21 1.28433 0.642163 0.766568i \(-0.278037\pi\)
0.642163 + 0.766568i \(0.278037\pi\)
\(72\) 0 0
\(73\) 2.31511e21 0.863693 0.431847 0.901947i \(-0.357862\pi\)
0.431847 + 0.901947i \(0.357862\pi\)
\(74\) 0 0
\(75\) 3.79070e20 0.103637
\(76\) 0 0
\(77\) 1.49672e20 0.0302342
\(78\) 0 0
\(79\) 4.71022e21 0.708483 0.354241 0.935154i \(-0.384739\pi\)
0.354241 + 0.935154i \(0.384739\pi\)
\(80\) 0 0
\(81\) −2.19369e21 −0.247513
\(82\) 0 0
\(83\) 7.97680e21 0.679878 0.339939 0.940448i \(-0.389594\pi\)
0.339939 + 0.940448i \(0.389594\pi\)
\(84\) 0 0
\(85\) 2.71488e22 1.75967
\(86\) 0 0
\(87\) 9.40716e19 0.00466646
\(88\) 0 0
\(89\) 2.30111e22 0.878926 0.439463 0.898261i \(-0.355169\pi\)
0.439463 + 0.898261i \(0.355169\pi\)
\(90\) 0 0
\(91\) −5.75954e21 −0.170377
\(92\) 0 0
\(93\) −5.75800e22 −1.32653
\(94\) 0 0
\(95\) −5.45379e22 −0.983732
\(96\) 0 0
\(97\) 9.70590e22 1.37772 0.688860 0.724894i \(-0.258111\pi\)
0.688860 + 0.724894i \(0.258111\pi\)
\(98\) 0 0
\(99\) 1.30826e22 0.146855
\(100\) 0 0
\(101\) 5.15267e22 0.459554 0.229777 0.973243i \(-0.426200\pi\)
0.229777 + 0.973243i \(0.426200\pi\)
\(102\) 0 0
\(103\) −1.07411e23 −0.764578 −0.382289 0.924043i \(-0.624864\pi\)
−0.382289 + 0.924043i \(0.624864\pi\)
\(104\) 0 0
\(105\) 1.36777e22 0.0780434
\(106\) 0 0
\(107\) 2.13016e23 0.978359 0.489180 0.872183i \(-0.337296\pi\)
0.489180 + 0.872183i \(0.337296\pi\)
\(108\) 0 0
\(109\) 2.91111e22 0.108057 0.0540286 0.998539i \(-0.482794\pi\)
0.0540286 + 0.998539i \(0.482794\pi\)
\(110\) 0 0
\(111\) −3.72538e22 −0.112190
\(112\) 0 0
\(113\) 8.27983e22 0.203058 0.101529 0.994833i \(-0.467627\pi\)
0.101529 + 0.994833i \(0.467627\pi\)
\(114\) 0 0
\(115\) −7.72579e23 −1.54852
\(116\) 0 0
\(117\) −5.03431e23 −0.827565
\(118\) 0 0
\(119\) 1.25356e23 0.169573
\(120\) 0 0
\(121\) −8.18568e23 −0.914162
\(122\) 0 0
\(123\) 1.27458e23 0.117885
\(124\) 0 0
\(125\) 1.18926e24 0.913717
\(126\) 0 0
\(127\) 1.61404e24 1.03317 0.516583 0.856237i \(-0.327204\pi\)
0.516583 + 0.856237i \(0.327204\pi\)
\(128\) 0 0
\(129\) −9.81084e23 −0.524716
\(130\) 0 0
\(131\) 3.84884e24 1.72468 0.862342 0.506326i \(-0.168997\pi\)
0.862342 + 0.506326i \(0.168997\pi\)
\(132\) 0 0
\(133\) −2.51823e23 −0.0947984
\(134\) 0 0
\(135\) 3.58070e24 1.13535
\(136\) 0 0
\(137\) −3.12427e24 −0.836492 −0.418246 0.908334i \(-0.637355\pi\)
−0.418246 + 0.908334i \(0.637355\pi\)
\(138\) 0 0
\(139\) −7.53219e24 −1.70707 −0.853535 0.521036i \(-0.825546\pi\)
−0.853535 + 0.521036i \(0.825546\pi\)
\(140\) 0 0
\(141\) −4.81251e24 −0.925448
\(142\) 0 0
\(143\) −2.95773e24 −0.483720
\(144\) 0 0
\(145\) −5.07585e22 −0.00707582
\(146\) 0 0
\(147\) −5.86738e24 −0.698707
\(148\) 0 0
\(149\) 9.61982e24 0.980674 0.490337 0.871533i \(-0.336874\pi\)
0.490337 + 0.871533i \(0.336874\pi\)
\(150\) 0 0
\(151\) −1.64397e25 −1.43767 −0.718835 0.695180i \(-0.755324\pi\)
−0.718835 + 0.695180i \(0.755324\pi\)
\(152\) 0 0
\(153\) 1.09572e25 0.823657
\(154\) 0 0
\(155\) 3.10686e25 2.01144
\(156\) 0 0
\(157\) −2.55113e25 −1.42524 −0.712619 0.701551i \(-0.752491\pi\)
−0.712619 + 0.701551i \(0.752491\pi\)
\(158\) 0 0
\(159\) 2.36031e24 0.113999
\(160\) 0 0
\(161\) −3.56730e24 −0.149225
\(162\) 0 0
\(163\) −9.60360e24 −0.348559 −0.174280 0.984696i \(-0.555760\pi\)
−0.174280 + 0.984696i \(0.555760\pi\)
\(164\) 0 0
\(165\) 7.02399e24 0.221574
\(166\) 0 0
\(167\) 1.23180e25 0.338300 0.169150 0.985590i \(-0.445898\pi\)
0.169150 + 0.985590i \(0.445898\pi\)
\(168\) 0 0
\(169\) 7.20626e25 1.72589
\(170\) 0 0
\(171\) −2.20113e25 −0.460459
\(172\) 0 0
\(173\) 4.66836e25 0.854347 0.427174 0.904170i \(-0.359509\pi\)
0.427174 + 0.904170i \(0.359509\pi\)
\(174\) 0 0
\(175\) −9.44421e23 −0.0151436
\(176\) 0 0
\(177\) −6.51278e25 −0.916376
\(178\) 0 0
\(179\) 1.17980e26 1.45881 0.729405 0.684082i \(-0.239797\pi\)
0.729405 + 0.684082i \(0.239797\pi\)
\(180\) 0 0
\(181\) 3.32252e25 0.361546 0.180773 0.983525i \(-0.442140\pi\)
0.180773 + 0.983525i \(0.442140\pi\)
\(182\) 0 0
\(183\) 9.19714e25 0.881993
\(184\) 0 0
\(185\) 2.01011e25 0.170116
\(186\) 0 0
\(187\) 6.43749e25 0.481436
\(188\) 0 0
\(189\) 1.65335e25 0.109409
\(190\) 0 0
\(191\) 2.29936e26 1.34810 0.674051 0.738684i \(-0.264553\pi\)
0.674051 + 0.738684i \(0.264553\pi\)
\(192\) 0 0
\(193\) −8.72143e25 −0.453606 −0.226803 0.973941i \(-0.572827\pi\)
−0.226803 + 0.973941i \(0.572827\pi\)
\(194\) 0 0
\(195\) −2.70290e26 −1.24863
\(196\) 0 0
\(197\) −1.13225e25 −0.0465138 −0.0232569 0.999730i \(-0.507404\pi\)
−0.0232569 + 0.999730i \(0.507404\pi\)
\(198\) 0 0
\(199\) 4.25068e25 0.155470 0.0777352 0.996974i \(-0.475231\pi\)
0.0777352 + 0.996974i \(0.475231\pi\)
\(200\) 0 0
\(201\) −1.47605e26 −0.481225
\(202\) 0 0
\(203\) −2.34372e23 −0.000681868 0
\(204\) 0 0
\(205\) −6.87728e25 −0.178751
\(206\) 0 0
\(207\) −3.11811e26 −0.724823
\(208\) 0 0
\(209\) −1.29320e26 −0.269143
\(210\) 0 0
\(211\) 6.05293e26 1.12906 0.564530 0.825412i \(-0.309057\pi\)
0.564530 + 0.825412i \(0.309057\pi\)
\(212\) 0 0
\(213\) 5.41982e26 0.907026
\(214\) 0 0
\(215\) 5.29366e26 0.795635
\(216\) 0 0
\(217\) 1.43456e26 0.193834
\(218\) 0 0
\(219\) 5.01662e26 0.609964
\(220\) 0 0
\(221\) −2.47721e27 −2.71302
\(222\) 0 0
\(223\) 3.90751e26 0.385828 0.192914 0.981216i \(-0.438206\pi\)
0.192914 + 0.981216i \(0.438206\pi\)
\(224\) 0 0
\(225\) −8.25501e25 −0.0735561
\(226\) 0 0
\(227\) 9.06311e26 0.729424 0.364712 0.931120i \(-0.381167\pi\)
0.364712 + 0.931120i \(0.381167\pi\)
\(228\) 0 0
\(229\) 2.76026e26 0.200836 0.100418 0.994945i \(-0.467982\pi\)
0.100418 + 0.994945i \(0.467982\pi\)
\(230\) 0 0
\(231\) 3.24325e25 0.0213522
\(232\) 0 0
\(233\) 2.39900e27 1.43034 0.715168 0.698953i \(-0.246350\pi\)
0.715168 + 0.698953i \(0.246350\pi\)
\(234\) 0 0
\(235\) 2.59670e27 1.40327
\(236\) 0 0
\(237\) 1.02066e27 0.500350
\(238\) 0 0
\(239\) 2.04793e27 0.911465 0.455732 0.890117i \(-0.349377\pi\)
0.455732 + 0.890117i \(0.349377\pi\)
\(240\) 0 0
\(241\) −3.44324e27 −1.39242 −0.696211 0.717837i \(-0.745132\pi\)
−0.696211 + 0.717837i \(0.745132\pi\)
\(242\) 0 0
\(243\) 2.40780e27 0.885418
\(244\) 0 0
\(245\) 3.16588e27 1.05946
\(246\) 0 0
\(247\) 4.97636e27 1.51669
\(248\) 0 0
\(249\) 1.72849e27 0.480148
\(250\) 0 0
\(251\) 1.69682e27 0.429921 0.214960 0.976623i \(-0.431038\pi\)
0.214960 + 0.976623i \(0.431038\pi\)
\(252\) 0 0
\(253\) −1.83193e27 −0.423666
\(254\) 0 0
\(255\) 5.88287e27 1.24273
\(256\) 0 0
\(257\) 2.16634e27 0.418308 0.209154 0.977883i \(-0.432929\pi\)
0.209154 + 0.977883i \(0.432929\pi\)
\(258\) 0 0
\(259\) 9.28146e25 0.0163934
\(260\) 0 0
\(261\) −2.04860e25 −0.00331200
\(262\) 0 0
\(263\) 3.76273e27 0.557201 0.278601 0.960407i \(-0.410129\pi\)
0.278601 + 0.960407i \(0.410129\pi\)
\(264\) 0 0
\(265\) −1.27356e27 −0.172858
\(266\) 0 0
\(267\) 4.98628e27 0.620722
\(268\) 0 0
\(269\) 1.38207e28 1.57898 0.789491 0.613762i \(-0.210344\pi\)
0.789491 + 0.613762i \(0.210344\pi\)
\(270\) 0 0
\(271\) −1.13487e27 −0.119069 −0.0595343 0.998226i \(-0.518962\pi\)
−0.0595343 + 0.998226i \(0.518962\pi\)
\(272\) 0 0
\(273\) −1.24804e27 −0.120325
\(274\) 0 0
\(275\) −4.84994e26 −0.0429942
\(276\) 0 0
\(277\) −9.82908e26 −0.0801670 −0.0400835 0.999196i \(-0.512762\pi\)
−0.0400835 + 0.999196i \(0.512762\pi\)
\(278\) 0 0
\(279\) 1.25392e28 0.941502
\(280\) 0 0
\(281\) −1.82184e28 −1.26005 −0.630024 0.776576i \(-0.716955\pi\)
−0.630024 + 0.776576i \(0.716955\pi\)
\(282\) 0 0
\(283\) 7.32526e27 0.466960 0.233480 0.972362i \(-0.424989\pi\)
0.233480 + 0.972362i \(0.424989\pi\)
\(284\) 0 0
\(285\) −1.18178e28 −0.694739
\(286\) 0 0
\(287\) −3.17551e26 −0.0172255
\(288\) 0 0
\(289\) 3.39490e28 1.70021
\(290\) 0 0
\(291\) 2.10317e28 0.972984
\(292\) 0 0
\(293\) −3.19438e28 −1.36587 −0.682934 0.730480i \(-0.739296\pi\)
−0.682934 + 0.730480i \(0.739296\pi\)
\(294\) 0 0
\(295\) 3.51412e28 1.38951
\(296\) 0 0
\(297\) 8.49053e27 0.310624
\(298\) 0 0
\(299\) 7.04947e28 2.38747
\(300\) 0 0
\(301\) 2.44429e27 0.0766722
\(302\) 0 0
\(303\) 1.11653e28 0.324550
\(304\) 0 0
\(305\) −4.96252e28 −1.33738
\(306\) 0 0
\(307\) 3.20565e28 0.801355 0.400678 0.916219i \(-0.368775\pi\)
0.400678 + 0.916219i \(0.368775\pi\)
\(308\) 0 0
\(309\) −2.32750e28 −0.539966
\(310\) 0 0
\(311\) 1.76164e27 0.0379465 0.0189733 0.999820i \(-0.493960\pi\)
0.0189733 + 0.999820i \(0.493960\pi\)
\(312\) 0 0
\(313\) −5.11434e27 −0.102336 −0.0511682 0.998690i \(-0.516294\pi\)
−0.0511682 + 0.998690i \(0.516294\pi\)
\(314\) 0 0
\(315\) −2.97859e27 −0.0553911
\(316\) 0 0
\(317\) 2.90574e28 0.502430 0.251215 0.967931i \(-0.419170\pi\)
0.251215 + 0.967931i \(0.419170\pi\)
\(318\) 0 0
\(319\) −1.20358e26 −0.00193590
\(320\) 0 0
\(321\) 4.61584e28 0.690944
\(322\) 0 0
\(323\) −1.08310e29 −1.50953
\(324\) 0 0
\(325\) 1.86631e28 0.242284
\(326\) 0 0
\(327\) 6.30809e27 0.0763130
\(328\) 0 0
\(329\) 1.19900e28 0.135228
\(330\) 0 0
\(331\) −1.94295e28 −0.204380 −0.102190 0.994765i \(-0.532585\pi\)
−0.102190 + 0.994765i \(0.532585\pi\)
\(332\) 0 0
\(333\) 8.11275e27 0.0796268
\(334\) 0 0
\(335\) 7.96439e28 0.729688
\(336\) 0 0
\(337\) −7.14896e28 −0.611644 −0.305822 0.952089i \(-0.598931\pi\)
−0.305822 + 0.952089i \(0.598931\pi\)
\(338\) 0 0
\(339\) 1.79416e28 0.143405
\(340\) 0 0
\(341\) 7.36696e28 0.550317
\(342\) 0 0
\(343\) 2.93935e28 0.205291
\(344\) 0 0
\(345\) −1.67410e29 −1.09361
\(346\) 0 0
\(347\) 6.32031e28 0.386322 0.193161 0.981167i \(-0.438126\pi\)
0.193161 + 0.981167i \(0.438126\pi\)
\(348\) 0 0
\(349\) 1.65356e29 0.946080 0.473040 0.881041i \(-0.343157\pi\)
0.473040 + 0.881041i \(0.343157\pi\)
\(350\) 0 0
\(351\) −3.26724e29 −1.75045
\(352\) 0 0
\(353\) 3.87413e29 1.94431 0.972153 0.234346i \(-0.0752950\pi\)
0.972153 + 0.234346i \(0.0752950\pi\)
\(354\) 0 0
\(355\) −2.92439e29 −1.37534
\(356\) 0 0
\(357\) 2.71635e28 0.119757
\(358\) 0 0
\(359\) 1.97727e29 0.817484 0.408742 0.912650i \(-0.365968\pi\)
0.408742 + 0.912650i \(0.365968\pi\)
\(360\) 0 0
\(361\) −4.02496e28 −0.156109
\(362\) 0 0
\(363\) −1.77376e29 −0.645606
\(364\) 0 0
\(365\) −2.70683e29 −0.924897
\(366\) 0 0
\(367\) −4.25634e29 −1.36577 −0.682883 0.730528i \(-0.739274\pi\)
−0.682883 + 0.730528i \(0.739274\pi\)
\(368\) 0 0
\(369\) −2.77565e28 −0.0836685
\(370\) 0 0
\(371\) −5.88050e27 −0.0166577
\(372\) 0 0
\(373\) −2.45786e29 −0.654495 −0.327247 0.944939i \(-0.606121\pi\)
−0.327247 + 0.944939i \(0.606121\pi\)
\(374\) 0 0
\(375\) 2.57701e29 0.645292
\(376\) 0 0
\(377\) 4.63151e27 0.0109093
\(378\) 0 0
\(379\) 5.19016e29 1.15035 0.575175 0.818031i \(-0.304934\pi\)
0.575175 + 0.818031i \(0.304934\pi\)
\(380\) 0 0
\(381\) 3.49745e29 0.729651
\(382\) 0 0
\(383\) −2.42152e29 −0.475665 −0.237833 0.971306i \(-0.576437\pi\)
−0.237833 + 0.971306i \(0.576437\pi\)
\(384\) 0 0
\(385\) −1.74997e28 −0.0323766
\(386\) 0 0
\(387\) 2.13651e29 0.372416
\(388\) 0 0
\(389\) −2.56541e29 −0.421441 −0.210721 0.977546i \(-0.567581\pi\)
−0.210721 + 0.977546i \(0.567581\pi\)
\(390\) 0 0
\(391\) −1.53432e30 −2.37620
\(392\) 0 0
\(393\) 8.34005e29 1.21802
\(394\) 0 0
\(395\) −5.50719e29 −0.758688
\(396\) 0 0
\(397\) −9.53634e29 −1.23963 −0.619813 0.784749i \(-0.712792\pi\)
−0.619813 + 0.784749i \(0.712792\pi\)
\(398\) 0 0
\(399\) −5.45675e28 −0.0669492
\(400\) 0 0
\(401\) −1.61868e30 −1.87500 −0.937501 0.347982i \(-0.886867\pi\)
−0.937501 + 0.347982i \(0.886867\pi\)
\(402\) 0 0
\(403\) −2.83488e30 −3.10118
\(404\) 0 0
\(405\) 2.56486e29 0.265052
\(406\) 0 0
\(407\) 4.76636e28 0.0465427
\(408\) 0 0
\(409\) 4.84424e29 0.447103 0.223551 0.974692i \(-0.428235\pi\)
0.223551 + 0.974692i \(0.428235\pi\)
\(410\) 0 0
\(411\) −6.76998e29 −0.590754
\(412\) 0 0
\(413\) 1.62260e29 0.133902
\(414\) 0 0
\(415\) −9.32648e29 −0.728056
\(416\) 0 0
\(417\) −1.63215e30 −1.20558
\(418\) 0 0
\(419\) 2.19166e29 0.153219 0.0766095 0.997061i \(-0.475591\pi\)
0.0766095 + 0.997061i \(0.475591\pi\)
\(420\) 0 0
\(421\) 1.70067e30 1.12558 0.562789 0.826600i \(-0.309728\pi\)
0.562789 + 0.826600i \(0.309728\pi\)
\(422\) 0 0
\(423\) 1.04802e30 0.656834
\(424\) 0 0
\(425\) −4.06202e29 −0.241140
\(426\) 0 0
\(427\) −2.29139e29 −0.128878
\(428\) 0 0
\(429\) −6.40911e29 −0.341616
\(430\) 0 0
\(431\) −2.28932e30 −1.15669 −0.578346 0.815791i \(-0.696302\pi\)
−0.578346 + 0.815791i \(0.696302\pi\)
\(432\) 0 0
\(433\) 2.06239e30 0.988009 0.494004 0.869459i \(-0.335533\pi\)
0.494004 + 0.869459i \(0.335533\pi\)
\(434\) 0 0
\(435\) −1.09989e28 −0.00499714
\(436\) 0 0
\(437\) 3.08222e30 1.32839
\(438\) 0 0
\(439\) −1.57142e29 −0.0642613 −0.0321306 0.999484i \(-0.510229\pi\)
−0.0321306 + 0.999484i \(0.510229\pi\)
\(440\) 0 0
\(441\) 1.27774e30 0.495905
\(442\) 0 0
\(443\) −2.88129e30 −1.06156 −0.530779 0.847511i \(-0.678100\pi\)
−0.530779 + 0.847511i \(0.678100\pi\)
\(444\) 0 0
\(445\) −2.69046e30 −0.941209
\(446\) 0 0
\(447\) 2.08452e30 0.692579
\(448\) 0 0
\(449\) 7.97062e28 0.0251570 0.0125785 0.999921i \(-0.495996\pi\)
0.0125785 + 0.999921i \(0.495996\pi\)
\(450\) 0 0
\(451\) −1.63074e29 −0.0489050
\(452\) 0 0
\(453\) −3.56232e30 −1.01532
\(454\) 0 0
\(455\) 6.73406e29 0.182451
\(456\) 0 0
\(457\) −2.35544e30 −0.606786 −0.303393 0.952866i \(-0.598119\pi\)
−0.303393 + 0.952866i \(0.598119\pi\)
\(458\) 0 0
\(459\) 7.11115e30 1.74218
\(460\) 0 0
\(461\) 5.69014e30 1.32606 0.663029 0.748594i \(-0.269271\pi\)
0.663029 + 0.748594i \(0.269271\pi\)
\(462\) 0 0
\(463\) 1.47214e30 0.326413 0.163206 0.986592i \(-0.447816\pi\)
0.163206 + 0.986592i \(0.447816\pi\)
\(464\) 0 0
\(465\) 6.73225e30 1.42053
\(466\) 0 0
\(467\) 6.23957e30 1.25317 0.626585 0.779353i \(-0.284452\pi\)
0.626585 + 0.779353i \(0.284452\pi\)
\(468\) 0 0
\(469\) 3.67747e29 0.0703171
\(470\) 0 0
\(471\) −5.52806e30 −1.00654
\(472\) 0 0
\(473\) 1.25523e30 0.217681
\(474\) 0 0
\(475\) 8.16000e29 0.134807
\(476\) 0 0
\(477\) −5.14004e29 −0.0809105
\(478\) 0 0
\(479\) 5.00921e30 0.751468 0.375734 0.926727i \(-0.377391\pi\)
0.375734 + 0.926727i \(0.377391\pi\)
\(480\) 0 0
\(481\) −1.83415e30 −0.262280
\(482\) 0 0
\(483\) −7.72998e29 −0.105387
\(484\) 0 0
\(485\) −1.13481e31 −1.47535
\(486\) 0 0
\(487\) −3.34402e30 −0.414654 −0.207327 0.978272i \(-0.566476\pi\)
−0.207327 + 0.978272i \(0.566476\pi\)
\(488\) 0 0
\(489\) −2.08101e30 −0.246162
\(490\) 0 0
\(491\) −1.19218e31 −1.34556 −0.672782 0.739841i \(-0.734901\pi\)
−0.672782 + 0.739841i \(0.734901\pi\)
\(492\) 0 0
\(493\) −1.00805e29 −0.0108578
\(494\) 0 0
\(495\) −1.52961e30 −0.157261
\(496\) 0 0
\(497\) −1.35030e30 −0.132536
\(498\) 0 0
\(499\) 2.82630e30 0.264888 0.132444 0.991190i \(-0.457718\pi\)
0.132444 + 0.991190i \(0.457718\pi\)
\(500\) 0 0
\(501\) 2.66919e30 0.238916
\(502\) 0 0
\(503\) 2.24263e31 1.91746 0.958729 0.284321i \(-0.0917680\pi\)
0.958729 + 0.284321i \(0.0917680\pi\)
\(504\) 0 0
\(505\) −6.02450e30 −0.492119
\(506\) 0 0
\(507\) 1.56152e31 1.21887
\(508\) 0 0
\(509\) 7.05286e30 0.526152 0.263076 0.964775i \(-0.415263\pi\)
0.263076 + 0.964775i \(0.415263\pi\)
\(510\) 0 0
\(511\) −1.24985e30 −0.0891286
\(512\) 0 0
\(513\) −1.42853e31 −0.973954
\(514\) 0 0
\(515\) 1.25585e31 0.818758
\(516\) 0 0
\(517\) 6.15727e30 0.383926
\(518\) 0 0
\(519\) 1.01159e31 0.603363
\(520\) 0 0
\(521\) −1.98048e31 −1.13015 −0.565075 0.825040i \(-0.691153\pi\)
−0.565075 + 0.825040i \(0.691153\pi\)
\(522\) 0 0
\(523\) −1.73584e30 −0.0947852 −0.0473926 0.998876i \(-0.515091\pi\)
−0.0473926 + 0.998876i \(0.515091\pi\)
\(524\) 0 0
\(525\) −2.04647e29 −0.0106948
\(526\) 0 0
\(527\) 6.17012e31 3.08654
\(528\) 0 0
\(529\) 2.27820e31 1.09107
\(530\) 0 0
\(531\) 1.41829e31 0.650395
\(532\) 0 0
\(533\) 6.27524e30 0.275593
\(534\) 0 0
\(535\) −2.49058e31 −1.04769
\(536\) 0 0
\(537\) 2.55651e31 1.03025
\(538\) 0 0
\(539\) 7.50691e30 0.289861
\(540\) 0 0
\(541\) −2.16112e31 −0.799669 −0.399834 0.916587i \(-0.630932\pi\)
−0.399834 + 0.916587i \(0.630932\pi\)
\(542\) 0 0
\(543\) 7.19957e30 0.255334
\(544\) 0 0
\(545\) −3.40367e30 −0.115715
\(546\) 0 0
\(547\) 2.49009e31 0.811634 0.405817 0.913954i \(-0.366987\pi\)
0.405817 + 0.913954i \(0.366987\pi\)
\(548\) 0 0
\(549\) −2.00286e31 −0.625992
\(550\) 0 0
\(551\) 2.02502e29 0.00606996
\(552\) 0 0
\(553\) −2.54288e30 −0.0731117
\(554\) 0 0
\(555\) 4.35571e30 0.120141
\(556\) 0 0
\(557\) 3.55158e31 0.939910 0.469955 0.882690i \(-0.344270\pi\)
0.469955 + 0.882690i \(0.344270\pi\)
\(558\) 0 0
\(559\) −4.83025e31 −1.22669
\(560\) 0 0
\(561\) 1.39494e31 0.340003
\(562\) 0 0
\(563\) 3.99841e31 0.935493 0.467747 0.883863i \(-0.345066\pi\)
0.467747 + 0.883863i \(0.345066\pi\)
\(564\) 0 0
\(565\) −9.68077e30 −0.217447
\(566\) 0 0
\(567\) 1.18430e30 0.0255420
\(568\) 0 0
\(569\) −1.04446e31 −0.216323 −0.108161 0.994133i \(-0.534496\pi\)
−0.108161 + 0.994133i \(0.534496\pi\)
\(570\) 0 0
\(571\) −4.71952e31 −0.938820 −0.469410 0.882980i \(-0.655533\pi\)
−0.469410 + 0.882980i \(0.655533\pi\)
\(572\) 0 0
\(573\) 4.98248e31 0.952067
\(574\) 0 0
\(575\) 1.15594e31 0.212204
\(576\) 0 0
\(577\) 7.18074e31 1.26662 0.633312 0.773897i \(-0.281695\pi\)
0.633312 + 0.773897i \(0.281695\pi\)
\(578\) 0 0
\(579\) −1.88985e31 −0.320349
\(580\) 0 0
\(581\) −4.30640e30 −0.0701598
\(582\) 0 0
\(583\) −3.01985e30 −0.0472930
\(584\) 0 0
\(585\) 5.88611e31 0.886209
\(586\) 0 0
\(587\) −1.52916e31 −0.221368 −0.110684 0.993856i \(-0.535304\pi\)
−0.110684 + 0.993856i \(0.535304\pi\)
\(588\) 0 0
\(589\) −1.23949e32 −1.72550
\(590\) 0 0
\(591\) −2.45348e30 −0.0328493
\(592\) 0 0
\(593\) −1.17874e32 −1.51805 −0.759027 0.651059i \(-0.774325\pi\)
−0.759027 + 0.651059i \(0.774325\pi\)
\(594\) 0 0
\(595\) −1.46567e31 −0.181589
\(596\) 0 0
\(597\) 9.21079e30 0.109797
\(598\) 0 0
\(599\) −1.02527e32 −1.17606 −0.588031 0.808839i \(-0.700097\pi\)
−0.588031 + 0.808839i \(0.700097\pi\)
\(600\) 0 0
\(601\) −2.41324e31 −0.266406 −0.133203 0.991089i \(-0.542526\pi\)
−0.133203 + 0.991089i \(0.542526\pi\)
\(602\) 0 0
\(603\) 3.21440e31 0.341548
\(604\) 0 0
\(605\) 9.57070e31 0.978942
\(606\) 0 0
\(607\) 1.84578e31 0.181765 0.0908823 0.995862i \(-0.471031\pi\)
0.0908823 + 0.995862i \(0.471031\pi\)
\(608\) 0 0
\(609\) −5.07860e28 −0.000481554 0
\(610\) 0 0
\(611\) −2.36938e32 −2.16352
\(612\) 0 0
\(613\) 1.88798e32 1.66036 0.830180 0.557495i \(-0.188238\pi\)
0.830180 + 0.557495i \(0.188238\pi\)
\(614\) 0 0
\(615\) −1.49024e31 −0.126239
\(616\) 0 0
\(617\) −1.33198e32 −1.08697 −0.543487 0.839418i \(-0.682896\pi\)
−0.543487 + 0.839418i \(0.682896\pi\)
\(618\) 0 0
\(619\) 6.61060e30 0.0519755 0.0259877 0.999662i \(-0.491727\pi\)
0.0259877 + 0.999662i \(0.491727\pi\)
\(620\) 0 0
\(621\) −2.02364e32 −1.53313
\(622\) 0 0
\(623\) −1.24229e31 −0.0907006
\(624\) 0 0
\(625\) −1.59902e32 −1.12521
\(626\) 0 0
\(627\) −2.80223e31 −0.190076
\(628\) 0 0
\(629\) 3.99202e31 0.261042
\(630\) 0 0
\(631\) 1.70413e32 1.07440 0.537198 0.843456i \(-0.319483\pi\)
0.537198 + 0.843456i \(0.319483\pi\)
\(632\) 0 0
\(633\) 1.31161e32 0.797373
\(634\) 0 0
\(635\) −1.88713e32 −1.10638
\(636\) 0 0
\(637\) −2.88874e32 −1.63344
\(638\) 0 0
\(639\) −1.18027e32 −0.643759
\(640\) 0 0
\(641\) 1.04888e32 0.551898 0.275949 0.961172i \(-0.411008\pi\)
0.275949 + 0.961172i \(0.411008\pi\)
\(642\) 0 0
\(643\) 2.47552e32 1.25673 0.628364 0.777920i \(-0.283725\pi\)
0.628364 + 0.777920i \(0.283725\pi\)
\(644\) 0 0
\(645\) 1.14708e32 0.561899
\(646\) 0 0
\(647\) −1.42958e31 −0.0675784 −0.0337892 0.999429i \(-0.510757\pi\)
−0.0337892 + 0.999429i \(0.510757\pi\)
\(648\) 0 0
\(649\) 8.33265e31 0.380162
\(650\) 0 0
\(651\) 3.10854e31 0.136891
\(652\) 0 0
\(653\) 1.82338e32 0.775132 0.387566 0.921842i \(-0.373316\pi\)
0.387566 + 0.921842i \(0.373316\pi\)
\(654\) 0 0
\(655\) −4.50006e32 −1.84690
\(656\) 0 0
\(657\) −1.09247e32 −0.432920
\(658\) 0 0
\(659\) −4.48747e32 −1.71719 −0.858597 0.512651i \(-0.828664\pi\)
−0.858597 + 0.512651i \(0.828664\pi\)
\(660\) 0 0
\(661\) −1.17730e32 −0.435082 −0.217541 0.976051i \(-0.569804\pi\)
−0.217541 + 0.976051i \(0.569804\pi\)
\(662\) 0 0
\(663\) −5.36788e32 −1.91601
\(664\) 0 0
\(665\) 2.94431e31 0.101516
\(666\) 0 0
\(667\) 2.86862e30 0.00955491
\(668\) 0 0
\(669\) 8.46718e31 0.272483
\(670\) 0 0
\(671\) −1.17671e32 −0.365898
\(672\) 0 0
\(673\) 2.89151e32 0.868864 0.434432 0.900705i \(-0.356949\pi\)
0.434432 + 0.900705i \(0.356949\pi\)
\(674\) 0 0
\(675\) −5.35746e31 −0.155584
\(676\) 0 0
\(677\) −4.29377e32 −1.20523 −0.602614 0.798033i \(-0.705874\pi\)
−0.602614 + 0.798033i \(0.705874\pi\)
\(678\) 0 0
\(679\) −5.23988e31 −0.142174
\(680\) 0 0
\(681\) 1.96389e32 0.515139
\(682\) 0 0
\(683\) −3.11964e32 −0.791164 −0.395582 0.918431i \(-0.629457\pi\)
−0.395582 + 0.918431i \(0.629457\pi\)
\(684\) 0 0
\(685\) 3.65290e32 0.895769
\(686\) 0 0
\(687\) 5.98121e31 0.141836
\(688\) 0 0
\(689\) 1.16207e32 0.266508
\(690\) 0 0
\(691\) 7.49648e31 0.166288 0.0831438 0.996538i \(-0.473504\pi\)
0.0831438 + 0.996538i \(0.473504\pi\)
\(692\) 0 0
\(693\) −7.06282e30 −0.0151546
\(694\) 0 0
\(695\) 8.80664e32 1.82804
\(696\) 0 0
\(697\) −1.36581e32 −0.274291
\(698\) 0 0
\(699\) 5.19840e32 1.01014
\(700\) 0 0
\(701\) 2.70909e32 0.509408 0.254704 0.967019i \(-0.418022\pi\)
0.254704 + 0.967019i \(0.418022\pi\)
\(702\) 0 0
\(703\) −8.01938e31 −0.145933
\(704\) 0 0
\(705\) 5.62679e32 0.991028
\(706\) 0 0
\(707\) −2.78175e31 −0.0474236
\(708\) 0 0
\(709\) 1.59384e32 0.263034 0.131517 0.991314i \(-0.458015\pi\)
0.131517 + 0.991314i \(0.458015\pi\)
\(710\) 0 0
\(711\) −2.22268e32 −0.355122
\(712\) 0 0
\(713\) −1.75585e33 −2.71617
\(714\) 0 0
\(715\) 3.45818e32 0.517998
\(716\) 0 0
\(717\) 4.43767e32 0.643701
\(718\) 0 0
\(719\) 1.86352e32 0.261789 0.130895 0.991396i \(-0.458215\pi\)
0.130895 + 0.991396i \(0.458215\pi\)
\(720\) 0 0
\(721\) 5.79876e31 0.0789005
\(722\) 0 0
\(723\) −7.46115e32 −0.983367
\(724\) 0 0
\(725\) 7.59452e29 0.000969646 0
\(726\) 0 0
\(727\) 6.67960e32 0.826237 0.413119 0.910677i \(-0.364439\pi\)
0.413119 + 0.910677i \(0.364439\pi\)
\(728\) 0 0
\(729\) 7.28268e32 0.872820
\(730\) 0 0
\(731\) 1.05130e33 1.22090
\(732\) 0 0
\(733\) −1.40627e33 −1.58261 −0.791305 0.611422i \(-0.790598\pi\)
−0.791305 + 0.611422i \(0.790598\pi\)
\(734\) 0 0
\(735\) 6.86014e32 0.748219
\(736\) 0 0
\(737\) 1.88851e32 0.199638
\(738\) 0 0
\(739\) 1.57416e31 0.0161301 0.00806506 0.999967i \(-0.497433\pi\)
0.00806506 + 0.999967i \(0.497433\pi\)
\(740\) 0 0
\(741\) 1.07833e33 1.07113
\(742\) 0 0
\(743\) 1.10884e33 1.06782 0.533912 0.845540i \(-0.320721\pi\)
0.533912 + 0.845540i \(0.320721\pi\)
\(744\) 0 0
\(745\) −1.12475e33 −1.05017
\(746\) 0 0
\(747\) −3.76414e32 −0.340784
\(748\) 0 0
\(749\) −1.15000e32 −0.100962
\(750\) 0 0
\(751\) 1.61832e33 1.37786 0.688931 0.724827i \(-0.258080\pi\)
0.688931 + 0.724827i \(0.258080\pi\)
\(752\) 0 0
\(753\) 3.67685e32 0.303622
\(754\) 0 0
\(755\) 1.92213e33 1.53955
\(756\) 0 0
\(757\) −3.85513e31 −0.0299527 −0.0149764 0.999888i \(-0.504767\pi\)
−0.0149764 + 0.999888i \(0.504767\pi\)
\(758\) 0 0
\(759\) −3.96962e32 −0.299205
\(760\) 0 0
\(761\) −3.70529e32 −0.270956 −0.135478 0.990780i \(-0.543257\pi\)
−0.135478 + 0.990780i \(0.543257\pi\)
\(762\) 0 0
\(763\) −1.57161e31 −0.0111509
\(764\) 0 0
\(765\) −1.28111e33 −0.882024
\(766\) 0 0
\(767\) −3.20649e33 −2.14231
\(768\) 0 0
\(769\) −1.16079e33 −0.752663 −0.376331 0.926485i \(-0.622815\pi\)
−0.376331 + 0.926485i \(0.622815\pi\)
\(770\) 0 0
\(771\) 4.69425e32 0.295420
\(772\) 0 0
\(773\) −9.29133e32 −0.567562 −0.283781 0.958889i \(-0.591589\pi\)
−0.283781 + 0.958889i \(0.591589\pi\)
\(774\) 0 0
\(775\) −4.64850e32 −0.275641
\(776\) 0 0
\(777\) 2.01120e31 0.0115775
\(778\) 0 0
\(779\) 2.74371e32 0.153340
\(780\) 0 0
\(781\) −6.93429e32 −0.376283
\(782\) 0 0
\(783\) −1.32953e31 −0.00700548
\(784\) 0 0
\(785\) 2.98279e33 1.52623
\(786\) 0 0
\(787\) 2.04310e33 1.01526 0.507632 0.861574i \(-0.330521\pi\)
0.507632 + 0.861574i \(0.330521\pi\)
\(788\) 0 0
\(789\) 8.15347e32 0.393511
\(790\) 0 0
\(791\) −4.46999e31 −0.0209545
\(792\) 0 0
\(793\) 4.52810e33 2.06193
\(794\) 0 0
\(795\) −2.75967e32 −0.122077
\(796\) 0 0
\(797\) 8.24304e32 0.354255 0.177128 0.984188i \(-0.443319\pi\)
0.177128 + 0.984188i \(0.443319\pi\)
\(798\) 0 0
\(799\) 5.15696e33 2.15331
\(800\) 0 0
\(801\) −1.08586e33 −0.440555
\(802\) 0 0
\(803\) −6.41842e32 −0.253046
\(804\) 0 0
\(805\) 4.17088e32 0.159800
\(806\) 0 0
\(807\) 2.99480e33 1.11512
\(808\) 0 0
\(809\) −4.69127e33 −1.69778 −0.848892 0.528567i \(-0.822730\pi\)
−0.848892 + 0.528567i \(0.822730\pi\)
\(810\) 0 0
\(811\) 5.15175e33 1.81224 0.906119 0.423023i \(-0.139031\pi\)
0.906119 + 0.423023i \(0.139031\pi\)
\(812\) 0 0
\(813\) −2.45914e32 −0.0840896
\(814\) 0 0
\(815\) 1.12285e33 0.373259
\(816\) 0 0
\(817\) −2.11192e33 −0.682532
\(818\) 0 0
\(819\) 2.71785e32 0.0854004
\(820\) 0 0
\(821\) 6.68845e32 0.204352 0.102176 0.994766i \(-0.467419\pi\)
0.102176 + 0.994766i \(0.467419\pi\)
\(822\) 0 0
\(823\) −2.06514e33 −0.613553 −0.306776 0.951782i \(-0.599250\pi\)
−0.306776 + 0.951782i \(0.599250\pi\)
\(824\) 0 0
\(825\) −1.05093e32 −0.0303637
\(826\) 0 0
\(827\) −2.42837e33 −0.682341 −0.341171 0.940001i \(-0.610823\pi\)
−0.341171 + 0.940001i \(0.610823\pi\)
\(828\) 0 0
\(829\) 2.49649e33 0.682265 0.341133 0.940015i \(-0.389189\pi\)
0.341133 + 0.940015i \(0.389189\pi\)
\(830\) 0 0
\(831\) −2.12986e32 −0.0566161
\(832\) 0 0
\(833\) 6.28733e33 1.62573
\(834\) 0 0
\(835\) −1.44022e33 −0.362273
\(836\) 0 0
\(837\) 8.13788e33 1.99145
\(838\) 0 0
\(839\) −3.48206e33 −0.829035 −0.414518 0.910041i \(-0.636050\pi\)
−0.414518 + 0.910041i \(0.636050\pi\)
\(840\) 0 0
\(841\) −4.31653e33 −0.999956
\(842\) 0 0
\(843\) −3.94774e33 −0.889880
\(844\) 0 0
\(845\) −8.42556e33 −1.84819
\(846\) 0 0
\(847\) 4.41916e32 0.0943368
\(848\) 0 0
\(849\) 1.58731e33 0.329780
\(850\) 0 0
\(851\) −1.13602e33 −0.229718
\(852\) 0 0
\(853\) 2.25375e33 0.443600 0.221800 0.975092i \(-0.428807\pi\)
0.221800 + 0.975092i \(0.428807\pi\)
\(854\) 0 0
\(855\) 2.57357e33 0.493089
\(856\) 0 0
\(857\) 4.25946e33 0.794465 0.397233 0.917718i \(-0.369971\pi\)
0.397233 + 0.917718i \(0.369971\pi\)
\(858\) 0 0
\(859\) −1.00049e34 −1.81673 −0.908363 0.418183i \(-0.862667\pi\)
−0.908363 + 0.418183i \(0.862667\pi\)
\(860\) 0 0
\(861\) −6.88101e31 −0.0121651
\(862\) 0 0
\(863\) 2.91159e33 0.501194 0.250597 0.968092i \(-0.419373\pi\)
0.250597 + 0.968092i \(0.419373\pi\)
\(864\) 0 0
\(865\) −5.45825e33 −0.914889
\(866\) 0 0
\(867\) 7.35641e33 1.20073
\(868\) 0 0
\(869\) −1.30586e33 −0.207572
\(870\) 0 0
\(871\) −7.26718e33 −1.12501
\(872\) 0 0
\(873\) −4.58008e33 −0.690572
\(874\) 0 0
\(875\) −6.42040e32 −0.0942908
\(876\) 0 0
\(877\) −7.84349e33 −1.12205 −0.561027 0.827797i \(-0.689594\pi\)
−0.561027 + 0.827797i \(0.689594\pi\)
\(878\) 0 0
\(879\) −6.92190e33 −0.964613
\(880\) 0 0
\(881\) −4.76220e33 −0.646525 −0.323262 0.946309i \(-0.604780\pi\)
−0.323262 + 0.946309i \(0.604780\pi\)
\(882\) 0 0
\(883\) −4.87573e33 −0.644899 −0.322450 0.946587i \(-0.604506\pi\)
−0.322450 + 0.946587i \(0.604506\pi\)
\(884\) 0 0
\(885\) 7.61474e33 0.981313
\(886\) 0 0
\(887\) −8.40226e33 −1.05505 −0.527527 0.849538i \(-0.676881\pi\)
−0.527527 + 0.849538i \(0.676881\pi\)
\(888\) 0 0
\(889\) −8.71362e32 −0.106617
\(890\) 0 0
\(891\) 6.08179e32 0.0725167
\(892\) 0 0
\(893\) −1.03596e34 −1.20379
\(894\) 0 0
\(895\) −1.37942e34 −1.56219
\(896\) 0 0
\(897\) 1.52755e34 1.68610
\(898\) 0 0
\(899\) −1.15359e32 −0.0124113
\(900\) 0 0
\(901\) −2.52924e33 −0.265250
\(902\) 0 0
\(903\) 5.29653e32 0.0541480
\(904\) 0 0
\(905\) −3.88469e33 −0.387167
\(906\) 0 0
\(907\) −1.45779e34 −1.41648 −0.708242 0.705969i \(-0.750512\pi\)
−0.708242 + 0.705969i \(0.750512\pi\)
\(908\) 0 0
\(909\) −2.43147e33 −0.230348
\(910\) 0 0
\(911\) 2.18817e31 0.00202125 0.00101063 0.999999i \(-0.499678\pi\)
0.00101063 + 0.999999i \(0.499678\pi\)
\(912\) 0 0
\(913\) −2.21149e33 −0.199191
\(914\) 0 0
\(915\) −1.07533e34 −0.944494
\(916\) 0 0
\(917\) −2.07785e33 −0.177978
\(918\) 0 0
\(919\) −1.60404e33 −0.133995 −0.0669974 0.997753i \(-0.521342\pi\)
−0.0669974 + 0.997753i \(0.521342\pi\)
\(920\) 0 0
\(921\) 6.94633e33 0.565939
\(922\) 0 0
\(923\) 2.66839e34 2.12046
\(924\) 0 0
\(925\) −3.00754e32 −0.0233121
\(926\) 0 0
\(927\) 5.06859e33 0.383239
\(928\) 0 0
\(929\) 2.03965e34 1.50444 0.752219 0.658913i \(-0.228984\pi\)
0.752219 + 0.658913i \(0.228984\pi\)
\(930\) 0 0
\(931\) −1.26303e34 −0.908852
\(932\) 0 0
\(933\) 3.81729e32 0.0267989
\(934\) 0 0
\(935\) −7.52672e33 −0.515552
\(936\) 0 0
\(937\) −2.91671e34 −1.94934 −0.974670 0.223649i \(-0.928203\pi\)
−0.974670 + 0.223649i \(0.928203\pi\)
\(938\) 0 0
\(939\) −1.10823e33 −0.0722727
\(940\) 0 0
\(941\) 7.95391e33 0.506174 0.253087 0.967443i \(-0.418554\pi\)
0.253087 + 0.967443i \(0.418554\pi\)
\(942\) 0 0
\(943\) 3.88671e33 0.241378
\(944\) 0 0
\(945\) −1.93309e33 −0.117162
\(946\) 0 0
\(947\) 2.04508e34 1.20972 0.604862 0.796330i \(-0.293228\pi\)
0.604862 + 0.796330i \(0.293228\pi\)
\(948\) 0 0
\(949\) 2.46987e34 1.42598
\(950\) 0 0
\(951\) 6.29646e33 0.354830
\(952\) 0 0
\(953\) −3.16254e34 −1.73967 −0.869837 0.493340i \(-0.835776\pi\)
−0.869837 + 0.493340i \(0.835776\pi\)
\(954\) 0 0
\(955\) −2.68841e34 −1.44363
\(956\) 0 0
\(957\) −2.60804e31 −0.00136718
\(958\) 0 0
\(959\) 1.68668e33 0.0863217
\(960\) 0 0
\(961\) 5.05965e34 2.52814
\(962\) 0 0
\(963\) −1.00519e34 −0.490396
\(964\) 0 0
\(965\) 1.01971e34 0.485750
\(966\) 0 0
\(967\) 2.16807e34 1.00848 0.504242 0.863562i \(-0.331772\pi\)
0.504242 + 0.863562i \(0.331772\pi\)
\(968\) 0 0
\(969\) −2.34698e34 −1.06607
\(970\) 0 0
\(971\) −1.64733e34 −0.730734 −0.365367 0.930864i \(-0.619056\pi\)
−0.365367 + 0.930864i \(0.619056\pi\)
\(972\) 0 0
\(973\) 4.06637e33 0.176161
\(974\) 0 0
\(975\) 4.04410e33 0.171107
\(976\) 0 0
\(977\) 1.91054e34 0.789529 0.394765 0.918782i \(-0.370826\pi\)
0.394765 + 0.918782i \(0.370826\pi\)
\(978\) 0 0
\(979\) −6.37960e33 −0.257509
\(980\) 0 0
\(981\) −1.37371e33 −0.0541629
\(982\) 0 0
\(983\) −2.04129e34 −0.786213 −0.393107 0.919493i \(-0.628600\pi\)
−0.393107 + 0.919493i \(0.628600\pi\)
\(984\) 0 0
\(985\) 1.32383e33 0.0498099
\(986\) 0 0
\(987\) 2.59811e33 0.0955014
\(988\) 0 0
\(989\) −2.99172e34 −1.07439
\(990\) 0 0
\(991\) −2.46345e34 −0.864364 −0.432182 0.901786i \(-0.642256\pi\)
−0.432182 + 0.901786i \(0.642256\pi\)
\(992\) 0 0
\(993\) −4.21017e33 −0.144339
\(994\) 0 0
\(995\) −4.96989e33 −0.166487
\(996\) 0 0
\(997\) −3.06239e33 −0.100246 −0.0501229 0.998743i \(-0.515961\pi\)
−0.0501229 + 0.998743i \(0.515961\pi\)
\(998\) 0 0
\(999\) 5.26514e33 0.168425
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 64.24.a.e.1.2 2
4.3 odd 2 64.24.a.f.1.1 2
8.3 odd 2 16.24.a.c.1.2 2
8.5 even 2 4.24.a.a.1.1 2
24.5 odd 2 36.24.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.24.a.a.1.1 2 8.5 even 2
16.24.a.c.1.2 2 8.3 odd 2
36.24.a.e.1.1 2 24.5 odd 2
64.24.a.e.1.2 2 1.1 even 1 trivial
64.24.a.f.1.1 2 4.3 odd 2