Properties

Label 72.22.a.f.1.1
Level $72$
Weight $22$
Character 72.1
Self dual yes
Analytic conductor $201.224$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,22,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(201.223687887\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(57.9766\) of defining polynomial
Character \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.09730e7 q^{5} +9.17385e7 q^{7} +O(q^{10})\) \(q-3.09730e7 q^{5} +9.17385e7 q^{7} -8.72158e10 q^{11} -2.36850e11 q^{13} -7.42283e12 q^{17} +4.68680e9 q^{19} -3.33703e14 q^{23} +4.82489e14 q^{25} -3.23982e15 q^{29} +6.40473e15 q^{31} -2.84142e15 q^{35} -1.61009e16 q^{37} +5.77168e16 q^{41} -2.01468e17 q^{43} -6.62056e17 q^{47} -5.50130e17 q^{49} -4.62651e17 q^{53} +2.70133e18 q^{55} -7.39502e18 q^{59} -5.50188e18 q^{61} +7.33594e18 q^{65} +6.03520e18 q^{67} +4.43147e19 q^{71} -2.48622e19 q^{73} -8.00105e18 q^{77} +5.70948e19 q^{79} +1.31820e20 q^{83} +2.29907e20 q^{85} -3.97023e20 q^{89} -2.17282e19 q^{91} -1.45164e17 q^{95} -9.80402e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 24111774 q^{5} + 295988280 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 24111774 q^{5} + 295988280 q^{7} + 40335108684 q^{11} + 133734425946 q^{13} - 7797732274422 q^{17} + 35788199781996 q^{19} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 32\!\cdots\!42 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.09730e7 −1.41840 −0.709199 0.705008i \(-0.750943\pi\)
−0.709199 + 0.705008i \(0.750943\pi\)
\(6\) 0 0
\(7\) 9.17385e7 0.122750 0.0613751 0.998115i \(-0.480451\pi\)
0.0613751 + 0.998115i \(0.480451\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.72158e10 −1.01385 −0.506923 0.861991i \(-0.669217\pi\)
−0.506923 + 0.861991i \(0.669217\pi\)
\(12\) 0 0
\(13\) −2.36850e11 −0.476505 −0.238253 0.971203i \(-0.576575\pi\)
−0.238253 + 0.971203i \(0.576575\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.42283e12 −0.893009 −0.446505 0.894781i \(-0.647331\pi\)
−0.446505 + 0.894781i \(0.647331\pi\)
\(18\) 0 0
\(19\) 4.68680e9 0.000175373 0 8.76867e−5 1.00000i \(-0.499972\pi\)
8.76867e−5 1.00000i \(0.499972\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.33703e14 −1.67965 −0.839823 0.542860i \(-0.817341\pi\)
−0.839823 + 0.542860i \(0.817341\pi\)
\(24\) 0 0
\(25\) 4.82489e14 1.01185
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.23982e15 −1.43002 −0.715010 0.699114i \(-0.753578\pi\)
−0.715010 + 0.699114i \(0.753578\pi\)
\(30\) 0 0
\(31\) 6.40473e15 1.40347 0.701735 0.712438i \(-0.252409\pi\)
0.701735 + 0.712438i \(0.252409\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.84142e15 −0.174109
\(36\) 0 0
\(37\) −1.61009e16 −0.550467 −0.275234 0.961377i \(-0.588755\pi\)
−0.275234 + 0.961377i \(0.588755\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.77168e16 0.671540 0.335770 0.941944i \(-0.391003\pi\)
0.335770 + 0.941944i \(0.391003\pi\)
\(42\) 0 0
\(43\) −2.01468e17 −1.42163 −0.710817 0.703377i \(-0.751675\pi\)
−0.710817 + 0.703377i \(0.751675\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.62056e17 −1.83598 −0.917989 0.396607i \(-0.870188\pi\)
−0.917989 + 0.396607i \(0.870188\pi\)
\(48\) 0 0
\(49\) −5.50130e17 −0.984932
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.62651e17 −0.363376 −0.181688 0.983356i \(-0.558156\pi\)
−0.181688 + 0.983356i \(0.558156\pi\)
\(54\) 0 0
\(55\) 2.70133e18 1.43804
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.39502e18 −1.88362 −0.941809 0.336148i \(-0.890876\pi\)
−0.941809 + 0.336148i \(0.890876\pi\)
\(60\) 0 0
\(61\) −5.50188e18 −0.987524 −0.493762 0.869597i \(-0.664379\pi\)
−0.493762 + 0.869597i \(0.664379\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.33594e18 0.675874
\(66\) 0 0
\(67\) 6.03520e18 0.404489 0.202244 0.979335i \(-0.435176\pi\)
0.202244 + 0.979335i \(0.435176\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.43147e19 1.61561 0.807803 0.589453i \(-0.200657\pi\)
0.807803 + 0.589453i \(0.200657\pi\)
\(72\) 0 0
\(73\) −2.48622e19 −0.677095 −0.338548 0.940949i \(-0.609936\pi\)
−0.338548 + 0.940949i \(0.609936\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00105e18 −0.124450
\(78\) 0 0
\(79\) 5.70948e19 0.678441 0.339220 0.940707i \(-0.389837\pi\)
0.339220 + 0.940707i \(0.389837\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.31820e20 0.932528 0.466264 0.884646i \(-0.345600\pi\)
0.466264 + 0.884646i \(0.345600\pi\)
\(84\) 0 0
\(85\) 2.29907e20 1.26664
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.97023e20 −1.34965 −0.674823 0.737979i \(-0.735780\pi\)
−0.674823 + 0.737979i \(0.735780\pi\)
\(90\) 0 0
\(91\) −2.17282e19 −0.0584911
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.45164e17 −0.000248749 0
\(96\) 0 0
\(97\) −9.80402e20 −1.34990 −0.674949 0.737864i \(-0.735835\pi\)
−0.674949 + 0.737864i \(0.735835\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.46191e19 −0.0582085 −0.0291042 0.999576i \(-0.509265\pi\)
−0.0291042 + 0.999576i \(0.509265\pi\)
\(102\) 0 0
\(103\) −7.39481e19 −0.0542171 −0.0271085 0.999632i \(-0.508630\pi\)
−0.0271085 + 0.999632i \(0.508630\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.69439e20 −0.378133 −0.189066 0.981964i \(-0.560546\pi\)
−0.189066 + 0.981964i \(0.560546\pi\)
\(108\) 0 0
\(109\) −3.27500e21 −1.32505 −0.662526 0.749039i \(-0.730516\pi\)
−0.662526 + 0.749039i \(0.730516\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.53293e20 0.0701938 0.0350969 0.999384i \(-0.488826\pi\)
0.0350969 + 0.999384i \(0.488826\pi\)
\(114\) 0 0
\(115\) 1.03358e22 2.38241
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.80959e20 −0.109617
\(120\) 0 0
\(121\) 2.06345e20 0.0278836
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.75063e20 −0.0168128
\(126\) 0 0
\(127\) −3.02098e21 −0.245589 −0.122795 0.992432i \(-0.539186\pi\)
−0.122795 + 0.992432i \(0.539186\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.70668e22 1.00185 0.500925 0.865491i \(-0.332993\pi\)
0.500925 + 0.865491i \(0.332993\pi\)
\(132\) 0 0
\(133\) 4.29960e17 2.15271e−5 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.34811e22 0.861296 0.430648 0.902520i \(-0.358285\pi\)
0.430648 + 0.902520i \(0.358285\pi\)
\(138\) 0 0
\(139\) −4.53132e22 −1.42748 −0.713738 0.700413i \(-0.752999\pi\)
−0.713738 + 0.700413i \(0.752999\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.06570e22 0.483103
\(144\) 0 0
\(145\) 1.00347e23 2.02834
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.98352e22 −1.06076 −0.530381 0.847759i \(-0.677951\pi\)
−0.530381 + 0.847759i \(0.677951\pi\)
\(150\) 0 0
\(151\) −4.73560e22 −0.625341 −0.312671 0.949862i \(-0.601224\pi\)
−0.312671 + 0.949862i \(0.601224\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.98374e23 −1.99068
\(156\) 0 0
\(157\) −2.22945e21 −0.0195547 −0.00977733 0.999952i \(-0.503112\pi\)
−0.00977733 + 0.999952i \(0.503112\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.06134e22 −0.206177
\(162\) 0 0
\(163\) 1.48202e23 0.876768 0.438384 0.898788i \(-0.355551\pi\)
0.438384 + 0.898788i \(0.355551\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.25766e23 1.03547 0.517733 0.855542i \(-0.326776\pi\)
0.517733 + 0.855542i \(0.326776\pi\)
\(168\) 0 0
\(169\) −1.90967e23 −0.772943
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.37523e23 1.70182 0.850909 0.525313i \(-0.176052\pi\)
0.850909 + 0.525313i \(0.176052\pi\)
\(174\) 0 0
\(175\) 4.42628e22 0.124205
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.13222e23 −0.250597 −0.125299 0.992119i \(-0.539989\pi\)
−0.125299 + 0.992119i \(0.539989\pi\)
\(180\) 0 0
\(181\) 2.55968e23 0.504151 0.252075 0.967708i \(-0.418887\pi\)
0.252075 + 0.967708i \(0.418887\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.98692e23 0.780781
\(186\) 0 0
\(187\) 6.47388e23 0.905374
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.48387e23 0.838062 0.419031 0.907972i \(-0.362370\pi\)
0.419031 + 0.907972i \(0.362370\pi\)
\(192\) 0 0
\(193\) 1.77062e24 1.77735 0.888674 0.458539i \(-0.151627\pi\)
0.888674 + 0.458539i \(0.151627\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.04387e24 −1.65409 −0.827044 0.562137i \(-0.809979\pi\)
−0.827044 + 0.562137i \(0.809979\pi\)
\(198\) 0 0
\(199\) −8.13486e22 −0.0592097 −0.0296048 0.999562i \(-0.509425\pi\)
−0.0296048 + 0.999562i \(0.509425\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.97216e23 −0.175535
\(204\) 0 0
\(205\) −1.78766e24 −0.952511
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.08763e20 −0.000177802 0
\(210\) 0 0
\(211\) 3.52722e24 1.38825 0.694123 0.719856i \(-0.255792\pi\)
0.694123 + 0.719856i \(0.255792\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.24008e24 2.01644
\(216\) 0 0
\(217\) 5.87560e23 0.172276
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.75809e24 0.425523
\(222\) 0 0
\(223\) −2.37573e24 −0.523114 −0.261557 0.965188i \(-0.584236\pi\)
−0.261557 + 0.965188i \(0.584236\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.71337e24 −1.22650 −0.613252 0.789887i \(-0.710139\pi\)
−0.613252 + 0.789887i \(0.710139\pi\)
\(228\) 0 0
\(229\) 3.10829e24 0.517903 0.258952 0.965890i \(-0.416623\pi\)
0.258952 + 0.965890i \(0.416623\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.90742e24 0.542817 0.271408 0.962464i \(-0.412511\pi\)
0.271408 + 0.962464i \(0.412511\pi\)
\(234\) 0 0
\(235\) 2.05059e25 2.60415
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.38784e23 −0.0253997 −0.0126998 0.999919i \(-0.504043\pi\)
−0.0126998 + 0.999919i \(0.504043\pi\)
\(240\) 0 0
\(241\) 1.14005e25 1.11107 0.555537 0.831492i \(-0.312513\pi\)
0.555537 + 0.831492i \(0.312513\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.70392e25 1.39703
\(246\) 0 0
\(247\) −1.11007e21 −8.35663e−5 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.80122e24 0.368931 0.184466 0.982839i \(-0.440945\pi\)
0.184466 + 0.982839i \(0.440945\pi\)
\(252\) 0 0
\(253\) 2.91042e25 1.70290
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.21415e25 1.09878 0.549388 0.835567i \(-0.314861\pi\)
0.549388 + 0.835567i \(0.314861\pi\)
\(258\) 0 0
\(259\) −1.47707e24 −0.0675699
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.74763e25 1.45956 0.729779 0.683683i \(-0.239623\pi\)
0.729779 + 0.683683i \(0.239623\pi\)
\(264\) 0 0
\(265\) 1.43297e25 0.515412
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.64824e25 −0.506549 −0.253274 0.967394i \(-0.581508\pi\)
−0.253274 + 0.967394i \(0.581508\pi\)
\(270\) 0 0
\(271\) −2.86168e25 −0.813660 −0.406830 0.913504i \(-0.633366\pi\)
−0.406830 + 0.913504i \(0.633366\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.20807e25 −1.02586
\(276\) 0 0
\(277\) 5.26085e25 1.18855 0.594277 0.804261i \(-0.297438\pi\)
0.594277 + 0.804261i \(0.297438\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.45378e25 −1.25429 −0.627145 0.778903i \(-0.715777\pi\)
−0.627145 + 0.778903i \(0.715777\pi\)
\(282\) 0 0
\(283\) 4.56042e25 0.822711 0.411355 0.911475i \(-0.365056\pi\)
0.411355 + 0.911475i \(0.365056\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.29485e24 0.0824317
\(288\) 0 0
\(289\) −1.39935e25 −0.202535
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.87863e25 0.736489 0.368244 0.929729i \(-0.379959\pi\)
0.368244 + 0.929729i \(0.379959\pi\)
\(294\) 0 0
\(295\) 2.29046e26 2.67172
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.90374e25 0.800360
\(300\) 0 0
\(301\) −1.84824e25 −0.174506
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.70410e26 1.40070
\(306\) 0 0
\(307\) −1.65443e26 −1.26969 −0.634843 0.772641i \(-0.718935\pi\)
−0.634843 + 0.772641i \(0.718935\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.32700e24 0.0490842 0.0245421 0.999699i \(-0.492187\pi\)
0.0245421 + 0.999699i \(0.492187\pi\)
\(312\) 0 0
\(313\) −7.64901e25 −0.479059 −0.239530 0.970889i \(-0.576993\pi\)
−0.239530 + 0.970889i \(0.576993\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.82997e25 0.538803 0.269402 0.963028i \(-0.413174\pi\)
0.269402 + 0.963028i \(0.413174\pi\)
\(318\) 0 0
\(319\) 2.82564e26 1.44982
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.47893e22 −0.000156610 0
\(324\) 0 0
\(325\) −1.14277e26 −0.482153
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.07360e25 −0.225367
\(330\) 0 0
\(331\) −2.46189e26 −0.857187 −0.428593 0.903498i \(-0.640991\pi\)
−0.428593 + 0.903498i \(0.640991\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.86928e26 −0.573726
\(336\) 0 0
\(337\) 4.89205e26 1.41051 0.705257 0.708952i \(-0.250832\pi\)
0.705257 + 0.708952i \(0.250832\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.58594e26 −1.42290
\(342\) 0 0
\(343\) −1.01708e26 −0.243651
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.00972e26 −0.214161 −0.107080 0.994250i \(-0.534150\pi\)
−0.107080 + 0.994250i \(0.534150\pi\)
\(348\) 0 0
\(349\) −4.95706e25 −0.0989822 −0.0494911 0.998775i \(-0.515760\pi\)
−0.0494911 + 0.998775i \(0.515760\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.88459e26 −0.333873 −0.166937 0.985968i \(-0.553387\pi\)
−0.166937 + 0.985968i \(0.553387\pi\)
\(354\) 0 0
\(355\) −1.37256e27 −2.29157
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.21557e26 −0.774123 −0.387062 0.922054i \(-0.626510\pi\)
−0.387062 + 0.922054i \(0.626510\pi\)
\(360\) 0 0
\(361\) −7.14209e26 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.70057e26 0.960390
\(366\) 0 0
\(367\) −4.38478e25 −0.0516362 −0.0258181 0.999667i \(-0.508219\pi\)
−0.0258181 + 0.999667i \(0.508219\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.24429e25 −0.0446045
\(372\) 0 0
\(373\) 1.92501e25 0.0191201 0.00956004 0.999954i \(-0.496957\pi\)
0.00956004 + 0.999954i \(0.496957\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.67351e26 0.681412
\(378\) 0 0
\(379\) −2.04774e27 −1.72014 −0.860070 0.510175i \(-0.829581\pi\)
−0.860070 + 0.510175i \(0.829581\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.75914e27 −1.32347 −0.661733 0.749740i \(-0.730179\pi\)
−0.661733 + 0.749740i \(0.730179\pi\)
\(384\) 0 0
\(385\) 2.47816e26 0.176519
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.73236e26 0.174609 0.0873046 0.996182i \(-0.472175\pi\)
0.0873046 + 0.996182i \(0.472175\pi\)
\(390\) 0 0
\(391\) 2.47702e27 1.49994
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.76840e27 −0.962299
\(396\) 0 0
\(397\) −1.21691e27 −0.627998 −0.313999 0.949423i \(-0.601669\pi\)
−0.313999 + 0.949423i \(0.601669\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.15061e27 0.534455 0.267228 0.963633i \(-0.413892\pi\)
0.267228 + 0.963633i \(0.413892\pi\)
\(402\) 0 0
\(403\) −1.51696e27 −0.668761
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.40425e27 0.558089
\(408\) 0 0
\(409\) −1.54628e26 −0.0583705 −0.0291852 0.999574i \(-0.509291\pi\)
−0.0291852 + 0.999574i \(0.509291\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.78408e26 −0.231215
\(414\) 0 0
\(415\) −4.08287e27 −1.32270
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.10695e27 −1.20302 −0.601510 0.798866i \(-0.705434\pi\)
−0.601510 + 0.798866i \(0.705434\pi\)
\(420\) 0 0
\(421\) −4.80891e27 −1.33994 −0.669969 0.742389i \(-0.733693\pi\)
−0.669969 + 0.742389i \(0.733693\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.58144e27 −0.903594
\(426\) 0 0
\(427\) −5.04734e26 −0.121219
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.51446e27 0.983092 0.491546 0.870852i \(-0.336432\pi\)
0.491546 + 0.870852i \(0.336432\pi\)
\(432\) 0 0
\(433\) −4.04253e27 −0.838552 −0.419276 0.907859i \(-0.637716\pi\)
−0.419276 + 0.907859i \(0.637716\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.56400e24 −0.000294565 0
\(438\) 0 0
\(439\) 3.73290e26 0.0670145 0.0335072 0.999438i \(-0.489332\pi\)
0.0335072 + 0.999438i \(0.489332\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.67248e27 −1.41548 −0.707741 0.706472i \(-0.750286\pi\)
−0.707741 + 0.706472i \(0.750286\pi\)
\(444\) 0 0
\(445\) 1.22970e28 1.91434
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.22534e27 0.598790 0.299395 0.954129i \(-0.403215\pi\)
0.299395 + 0.954129i \(0.403215\pi\)
\(450\) 0 0
\(451\) −5.03382e27 −0.680838
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.72988e26 0.0829637
\(456\) 0 0
\(457\) 8.06069e27 0.948969 0.474484 0.880264i \(-0.342635\pi\)
0.474484 + 0.880264i \(0.342635\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.88705e27 1.06220 0.531101 0.847309i \(-0.321779\pi\)
0.531101 + 0.847309i \(0.321779\pi\)
\(462\) 0 0
\(463\) 1.00097e28 1.02759 0.513794 0.857913i \(-0.328239\pi\)
0.513794 + 0.857913i \(0.328239\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.60212e28 −1.50268 −0.751342 0.659913i \(-0.770593\pi\)
−0.751342 + 0.659913i \(0.770593\pi\)
\(468\) 0 0
\(469\) 5.53660e26 0.0496511
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.75712e28 1.44132
\(474\) 0 0
\(475\) 2.26133e24 0.000177452 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.14679e27 0.657273 0.328636 0.944457i \(-0.393411\pi\)
0.328636 + 0.944457i \(0.393411\pi\)
\(480\) 0 0
\(481\) 3.81349e27 0.262300
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.03660e28 1.91469
\(486\) 0 0
\(487\) −1.63338e28 −0.986357 −0.493179 0.869928i \(-0.664165\pi\)
−0.493179 + 0.869928i \(0.664165\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.10020e28 −0.609702 −0.304851 0.952400i \(-0.598607\pi\)
−0.304851 + 0.952400i \(0.598607\pi\)
\(492\) 0 0
\(493\) 2.40487e28 1.27702
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.06537e27 0.198316
\(498\) 0 0
\(499\) 2.04342e28 0.955657 0.477829 0.878453i \(-0.341424\pi\)
0.477829 + 0.878453i \(0.341424\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.67038e28 −0.718375 −0.359188 0.933265i \(-0.616946\pi\)
−0.359188 + 0.933265i \(0.616946\pi\)
\(504\) 0 0
\(505\) 2.00145e27 0.0825628
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.42146e27 0.205864 0.102932 0.994688i \(-0.467178\pi\)
0.102932 + 0.994688i \(0.467178\pi\)
\(510\) 0 0
\(511\) −2.28082e27 −0.0831136
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.29039e27 0.0769014
\(516\) 0 0
\(517\) 5.77418e28 1.86140
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.37414e28 −0.408541 −0.204271 0.978914i \(-0.565482\pi\)
−0.204271 + 0.978914i \(0.565482\pi\)
\(522\) 0 0
\(523\) −3.33678e28 −0.952927 −0.476463 0.879194i \(-0.658081\pi\)
−0.476463 + 0.879194i \(0.658081\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.75412e28 −1.25331
\(528\) 0 0
\(529\) 7.18860e28 1.82121
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.36702e28 −0.319992
\(534\) 0 0
\(535\) 2.38318e28 0.536343
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.79800e28 0.998570
\(540\) 0 0
\(541\) −9.34113e28 −1.86994 −0.934971 0.354723i \(-0.884575\pi\)
−0.934971 + 0.354723i \(0.884575\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.01437e29 1.87945
\(546\) 0 0
\(547\) 1.90655e28 0.339923 0.169961 0.985451i \(-0.445636\pi\)
0.169961 + 0.985451i \(0.445636\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.51844e25 −0.000250788 0
\(552\) 0 0
\(553\) 5.23779e27 0.0832787
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.85895e28 0.716247 0.358124 0.933674i \(-0.383417\pi\)
0.358124 + 0.933674i \(0.383417\pi\)
\(558\) 0 0
\(559\) 4.77177e28 0.677416
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.82251e28 0.635236 0.317618 0.948219i \(-0.397117\pi\)
0.317618 + 0.948219i \(0.397117\pi\)
\(564\) 0 0
\(565\) −7.84523e27 −0.0995628
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.25601e28 0.972952 0.486476 0.873694i \(-0.338282\pi\)
0.486476 + 0.873694i \(0.338282\pi\)
\(570\) 0 0
\(571\) −3.00348e28 −0.341150 −0.170575 0.985345i \(-0.554563\pi\)
−0.170575 + 0.985345i \(0.554563\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.61008e29 −1.69956
\(576\) 0 0
\(577\) 7.52665e28 0.766047 0.383024 0.923739i \(-0.374883\pi\)
0.383024 + 0.923739i \(0.374883\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.20930e28 0.114468
\(582\) 0 0
\(583\) 4.03504e28 0.368408
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.52918e29 −1.29945 −0.649723 0.760171i \(-0.725115\pi\)
−0.649723 + 0.760171i \(0.725115\pi\)
\(588\) 0 0
\(589\) 3.00177e25 0.000246131 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.89818e28 −0.297706 −0.148853 0.988859i \(-0.547558\pi\)
−0.148853 + 0.988859i \(0.547558\pi\)
\(594\) 0 0
\(595\) 2.10913e28 0.155481
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.28684e29 −0.884186 −0.442093 0.896969i \(-0.645764\pi\)
−0.442093 + 0.896969i \(0.645764\pi\)
\(600\) 0 0
\(601\) −1.81454e29 −1.20388 −0.601942 0.798540i \(-0.705606\pi\)
−0.601942 + 0.798540i \(0.705606\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.39114e27 −0.0395500
\(606\) 0 0
\(607\) −2.81972e29 −1.68549 −0.842743 0.538316i \(-0.819061\pi\)
−0.842743 + 0.538316i \(0.819061\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.56808e29 0.874853
\(612\) 0 0
\(613\) 8.09843e28 0.436582 0.218291 0.975884i \(-0.429952\pi\)
0.218291 + 0.975884i \(0.429952\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.47454e29 −1.74946 −0.874728 0.484614i \(-0.838960\pi\)
−0.874728 + 0.484614i \(0.838960\pi\)
\(618\) 0 0
\(619\) −2.90076e29 −1.41176 −0.705878 0.708333i \(-0.749447\pi\)
−0.705878 + 0.708333i \(0.749447\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.64223e28 −0.165669
\(624\) 0 0
\(625\) −2.24647e29 −0.988006
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.19514e29 0.491572
\(630\) 0 0
\(631\) −4.23594e29 −1.68516 −0.842581 0.538570i \(-0.818965\pi\)
−0.842581 + 0.538570i \(0.818965\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.35689e28 0.348343
\(636\) 0 0
\(637\) 1.30298e29 0.469325
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.43690e29 1.15920 0.579598 0.814902i \(-0.303209\pi\)
0.579598 + 0.814902i \(0.303209\pi\)
\(642\) 0 0
\(643\) 1.89646e29 0.619055 0.309527 0.950891i \(-0.399829\pi\)
0.309527 + 0.950891i \(0.399829\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.06810e29 −0.938370 −0.469185 0.883100i \(-0.655452\pi\)
−0.469185 + 0.883100i \(0.655452\pi\)
\(648\) 0 0
\(649\) 6.44962e29 1.90970
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.17693e29 −0.326710 −0.163355 0.986567i \(-0.552232\pi\)
−0.163355 + 0.986567i \(0.552232\pi\)
\(654\) 0 0
\(655\) −5.28609e29 −1.42102
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3.42901e29 −0.864712 −0.432356 0.901703i \(-0.642318\pi\)
−0.432356 + 0.901703i \(0.642318\pi\)
\(660\) 0 0
\(661\) −3.95818e29 −0.966897 −0.483448 0.875373i \(-0.660616\pi\)
−0.483448 + 0.875373i \(0.660616\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.33172e25 −3.05340e−5 0
\(666\) 0 0
\(667\) 1.08114e30 2.40193
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.79851e29 1.00120
\(672\) 0 0
\(673\) −9.58339e29 −1.93803 −0.969017 0.246996i \(-0.920557\pi\)
−0.969017 + 0.246996i \(0.920557\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.12797e29 −0.974461 −0.487230 0.873273i \(-0.661993\pi\)
−0.487230 + 0.873273i \(0.661993\pi\)
\(678\) 0 0
\(679\) −8.99406e28 −0.165700
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5.75298e29 −0.996495 −0.498248 0.867035i \(-0.666023\pi\)
−0.498248 + 0.867035i \(0.666023\pi\)
\(684\) 0 0
\(685\) −7.27280e29 −1.22166
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.09579e29 0.173151
\(690\) 0 0
\(691\) 2.46471e29 0.377787 0.188893 0.981998i \(-0.439510\pi\)
0.188893 + 0.981998i \(0.439510\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.40348e30 2.02473
\(696\) 0 0
\(697\) −4.28422e29 −0.599691
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.94607e29 1.17922 0.589608 0.807690i \(-0.299282\pi\)
0.589608 + 0.807690i \(0.299282\pi\)
\(702\) 0 0
\(703\) −7.54616e25 −9.65373e−5 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.92806e27 −0.00714510
\(708\) 0 0
\(709\) 7.52975e29 0.881039 0.440520 0.897743i \(-0.354794\pi\)
0.440520 + 0.897743i \(0.354794\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.13728e30 −2.35733
\(714\) 0 0
\(715\) −6.39810e29 −0.685232
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.85535e29 −0.490419 −0.245210 0.969470i \(-0.578857\pi\)
−0.245210 + 0.969470i \(0.578857\pi\)
\(720\) 0 0
\(721\) −6.78389e27 −0.00665516
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.56318e30 −1.44697
\(726\) 0 0
\(727\) 1.86351e30 1.67579 0.837895 0.545831i \(-0.183786\pi\)
0.837895 + 0.545831i \(0.183786\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.49546e30 1.26953
\(732\) 0 0
\(733\) −3.73906e29 −0.308440 −0.154220 0.988037i \(-0.549286\pi\)
−0.154220 + 0.988037i \(0.549286\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.26365e29 −0.410089
\(738\) 0 0
\(739\) 8.38257e29 0.634762 0.317381 0.948298i \(-0.397197\pi\)
0.317381 + 0.948298i \(0.397197\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.60139e29 0.543889 0.271945 0.962313i \(-0.412333\pi\)
0.271945 + 0.962313i \(0.412333\pi\)
\(744\) 0 0
\(745\) 2.16300e30 1.50458
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7.05871e28 −0.0464159
\(750\) 0 0
\(751\) −1.08974e30 −0.696794 −0.348397 0.937347i \(-0.613274\pi\)
−0.348397 + 0.937347i \(0.613274\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.46676e30 0.886983
\(756\) 0 0
\(757\) 3.06224e30 1.80108 0.900540 0.434774i \(-0.143172\pi\)
0.900540 + 0.434774i \(0.143172\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.46912e30 1.93055 0.965274 0.261241i \(-0.0841318\pi\)
0.965274 + 0.261241i \(0.0841318\pi\)
\(762\) 0 0
\(763\) −3.00444e29 −0.162650
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.75151e30 0.897554
\(768\) 0 0
\(769\) −7.20768e29 −0.359392 −0.179696 0.983722i \(-0.557511\pi\)
−0.179696 + 0.983722i \(0.557511\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.63916e30 0.773990 0.386995 0.922082i \(-0.373513\pi\)
0.386995 + 0.922082i \(0.373513\pi\)
\(774\) 0 0
\(775\) 3.09021e30 1.42011
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.70507e26 0.000117770 0
\(780\) 0 0
\(781\) −3.86494e30 −1.63798
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.90526e28 0.0277363
\(786\) 0 0
\(787\) 1.22284e30 0.478227 0.239114 0.970992i \(-0.423143\pi\)
0.239114 + 0.970992i \(0.423143\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.32367e28 0.00861630
\(792\) 0 0
\(793\) 1.30312e30 0.470560
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.32063e30 −0.452345 −0.226173 0.974087i \(-0.572621\pi\)
−0.226173 + 0.974087i \(0.572621\pi\)
\(798\) 0 0
\(799\) 4.91433e30 1.63954
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.16838e30 0.686470
\(804\) 0 0
\(805\) 9.48189e29 0.292441
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.54086e30 −1.62225 −0.811124 0.584874i \(-0.801144\pi\)
−0.811124 + 0.584874i \(0.801144\pi\)
\(810\) 0 0
\(811\) −4.51703e30 −1.28865 −0.644323 0.764753i \(-0.722861\pi\)
−0.644323 + 0.764753i \(0.722861\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.59026e30 −1.24361
\(816\) 0 0
\(817\) −9.44242e26 −0.000249317 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.49500e30 −1.62921 −0.814603 0.580019i \(-0.803045\pi\)
−0.814603 + 0.580019i \(0.803045\pi\)
\(822\) 0 0
\(823\) 6.58977e30 1.61128 0.805642 0.592402i \(-0.201820\pi\)
0.805642 + 0.592402i \(0.201820\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.99217e29 0.0695309 0.0347655 0.999395i \(-0.488932\pi\)
0.0347655 + 0.999395i \(0.488932\pi\)
\(828\) 0 0
\(829\) −3.02375e30 −0.685053 −0.342526 0.939508i \(-0.611283\pi\)
−0.342526 + 0.939508i \(0.611283\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.08352e30 0.879554
\(834\) 0 0
\(835\) −6.99266e30 −1.46870
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.48734e30 1.29588 0.647942 0.761690i \(-0.275630\pi\)
0.647942 + 0.761690i \(0.275630\pi\)
\(840\) 0 0
\(841\) 5.36360e30 1.04496
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.91481e30 1.09634
\(846\) 0 0
\(847\) 1.89298e28 0.00342272
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.37291e30 0.924590
\(852\) 0 0
\(853\) 1.83172e30 0.307535 0.153767 0.988107i \(-0.450859\pi\)
0.153767 + 0.988107i \(0.450859\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.04074e31 −1.66358 −0.831792 0.555088i \(-0.812685\pi\)
−0.831792 + 0.555088i \(0.812685\pi\)
\(858\) 0 0
\(859\) −8.40237e29 −0.131061 −0.0655306 0.997851i \(-0.520874\pi\)
−0.0655306 + 0.997851i \(0.520874\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.66021e30 0.989406 0.494703 0.869062i \(-0.335277\pi\)
0.494703 + 0.869062i \(0.335277\pi\)
\(864\) 0 0
\(865\) −1.66487e31 −2.41386
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.97957e30 −0.687835
\(870\) 0 0
\(871\) −1.42944e30 −0.192741
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.60600e28 −0.00206377
\(876\) 0 0
\(877\) −1.99416e30 −0.250187 −0.125093 0.992145i \(-0.539923\pi\)
−0.125093 + 0.992145i \(0.539923\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.25457e30 0.867690 0.433845 0.900987i \(-0.357157\pi\)
0.433845 + 0.900987i \(0.357157\pi\)
\(882\) 0 0
\(883\) 7.43622e30 0.868491 0.434245 0.900795i \(-0.357015\pi\)
0.434245 + 0.900795i \(0.357015\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.32005e30 −0.592540 −0.296270 0.955104i \(-0.595743\pi\)
−0.296270 + 0.955104i \(0.595743\pi\)
\(888\) 0 0
\(889\) −2.77140e29 −0.0301461
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.10293e27 −0.000321982 0
\(894\) 0 0
\(895\) 3.50684e30 0.355446
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.07502e31 −2.00699
\(900\) 0 0
\(901\) 3.43418e30 0.324498
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.92809e30 −0.715086
\(906\) 0 0
\(907\) −3.82478e30 −0.337077 −0.168539 0.985695i \(-0.553905\pi\)
−0.168539 + 0.985695i \(0.553905\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.98935e30 0.335706 0.167853 0.985812i \(-0.446317\pi\)
0.167853 + 0.985812i \(0.446317\pi\)
\(912\) 0 0
\(913\) −1.14968e31 −0.945440
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.56568e30 0.122977
\(918\) 0 0
\(919\) −2.24335e31 −1.72220 −0.861102 0.508432i \(-0.830225\pi\)
−0.861102 + 0.508432i \(0.830225\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.04959e31 −0.769844
\(924\) 0 0
\(925\) −7.76850e30 −0.556992
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5.26649e30 0.360875 0.180437 0.983586i \(-0.442249\pi\)
0.180437 + 0.983586i \(0.442249\pi\)
\(930\) 0 0
\(931\) −2.57835e27 −0.000172731 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.00515e31 −1.28418
\(936\) 0 0
\(937\) −3.29722e30 −0.206482 −0.103241 0.994656i \(-0.532921\pi\)
−0.103241 + 0.994656i \(0.532921\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.14709e31 1.28576 0.642879 0.765968i \(-0.277740\pi\)
0.642879 + 0.765968i \(0.277740\pi\)
\(942\) 0 0
\(943\) −1.92603e31 −1.12795
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.56465e31 0.876483 0.438242 0.898857i \(-0.355601\pi\)
0.438242 + 0.898857i \(0.355601\pi\)
\(948\) 0 0
\(949\) 5.88861e30 0.322639
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.71750e30 0.247307 0.123654 0.992325i \(-0.460539\pi\)
0.123654 + 0.992325i \(0.460539\pi\)
\(954\) 0 0
\(955\) −2.31798e31 −1.18871
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.15412e30 0.105724
\(960\) 0 0
\(961\) 2.01951e31 0.969728
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5.48413e31 −2.52099
\(966\) 0 0
\(967\) −3.43670e31 −1.54584 −0.772919 0.634505i \(-0.781204\pi\)
−0.772919 + 0.634505i \(0.781204\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.39842e31 −1.89450 −0.947249 0.320497i \(-0.896150\pi\)
−0.947249 + 0.320497i \(0.896150\pi\)
\(972\) 0 0
\(973\) −4.15696e30 −0.175223
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.85934e31 0.750700 0.375350 0.926883i \(-0.377523\pi\)
0.375350 + 0.926883i \(0.377523\pi\)
\(978\) 0 0
\(979\) 3.46266e31 1.36833
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.18619e31 0.827704 0.413852 0.910344i \(-0.364183\pi\)
0.413852 + 0.910344i \(0.364183\pi\)
\(984\) 0 0
\(985\) 6.33049e31 2.34616
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.72306e31 2.38784
\(990\) 0 0
\(991\) −8.68993e30 −0.302164 −0.151082 0.988521i \(-0.548276\pi\)
−0.151082 + 0.988521i \(0.548276\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.51961e30 0.0839829
\(996\) 0 0
\(997\) 1.31572e31 0.429403 0.214701 0.976680i \(-0.431122\pi\)
0.214701 + 0.976680i \(0.431122\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 72.22.a.f.1.1 3
3.2 odd 2 8.22.a.b.1.2 3
12.11 even 2 16.22.a.f.1.2 3
24.5 odd 2 64.22.a.l.1.2 3
24.11 even 2 64.22.a.m.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.2 3 3.2 odd 2
16.22.a.f.1.2 3 12.11 even 2
64.22.a.l.1.2 3 24.5 odd 2
64.22.a.m.1.2 3 24.11 even 2
72.22.a.f.1.1 3 1.1 even 1 trivial