Properties

Label 64.22.a.l.1.2
Level $64$
Weight $22$
Character 64.1
Self dual yes
Analytic conductor $178.866$
Analytic rank $1$
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] = [64,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
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: \(178.865500344\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(57.9766\) 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)
 
\(f(q)\) \(=\) \(q+7983.67 q^{3} -3.09730e7 q^{5} +9.17385e7 q^{7} -1.03966e10 q^{9} +O(q^{10})\) \(q+7983.67 q^{3} -3.09730e7 q^{5} +9.17385e7 q^{7} -1.03966e10 q^{9} -8.72158e10 q^{11} +2.36850e11 q^{13} -2.47278e11 q^{15} +7.42283e12 q^{17} -4.68680e9 q^{19} +7.32410e11 q^{21} +3.33703e14 q^{23} +4.82489e14 q^{25} -1.66515e14 q^{27} -3.23982e15 q^{29} +6.40473e15 q^{31} -6.96303e14 q^{33} -2.84142e15 q^{35} +1.61009e16 q^{37} +1.89093e15 q^{39} -5.77168e16 q^{41} +2.01468e17 q^{43} +3.22014e17 q^{45} +6.62056e17 q^{47} -5.50130e17 q^{49} +5.92615e16 q^{51} -4.62651e17 q^{53} +2.70133e18 q^{55} -3.74179e13 q^{57} -7.39502e18 q^{59} +5.50188e18 q^{61} -9.53770e17 q^{63} -7.33594e18 q^{65} -6.03520e18 q^{67} +2.66418e18 q^{69} -4.43147e19 q^{71} -2.48622e19 q^{73} +3.85204e18 q^{75} -8.00105e18 q^{77} +5.70948e19 q^{79} +1.07423e20 q^{81} +1.31820e20 q^{83} -2.29907e20 q^{85} -2.58657e19 q^{87} +3.97023e20 q^{89} +2.17282e19 q^{91} +5.11333e19 q^{93} +1.45164e17 q^{95} -9.80402e20 q^{97} +9.06749e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 96764 q^{3} + 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 96764 q^{3} + 24111774 q^{5} + 295988280 q^{7} + 18844697239 q^{9} + 40335108684 q^{11} - 133734425946 q^{13} - 1223136458200 q^{15} + 7797732274422 q^{17} - 35788199781996 q^{19} - 198539224853088 q^{21} + 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots + 94\!\cdots\!60 q^{99}+O(q^{100}) \) 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
\(3\) 7983.67 0.0780602 0.0390301 0.999238i \(-0.487573\pi\)
0.0390301 + 0.999238i \(0.487573\pi\)
\(4\) 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
\(9\) −1.03966e10 −0.993907
\(10\) 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 \(-0.423425\pi\)
0.238253 + 0.971203i \(0.423425\pi\)
\(14\) 0 0
\(15\) −2.47278e11 −0.110720
\(16\) 0 0
\(17\) 7.42283e12 0.893009 0.446505 0.894781i \(-0.352669\pi\)
0.446505 + 0.894781i \(0.352669\pi\)
\(18\) 0 0
\(19\) −4.68680e9 −0.000175373 0 −8.76867e−5 1.00000i \(-0.500028\pi\)
−8.76867e−5 1.00000i \(0.500028\pi\)
\(20\) 0 0
\(21\) 7.32410e11 0.00958190
\(22\) 0 0
\(23\) 3.33703e14 1.67965 0.839823 0.542860i \(-0.182659\pi\)
0.839823 + 0.542860i \(0.182659\pi\)
\(24\) 0 0
\(25\) 4.82489e14 1.01185
\(26\) 0 0
\(27\) −1.66515e14 −0.155645
\(28\) 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
\(33\) −6.96303e14 −0.0791410
\(34\) 0 0
\(35\) −2.84142e15 −0.174109
\(36\) 0 0
\(37\) 1.61009e16 0.550467 0.275234 0.961377i \(-0.411245\pi\)
0.275234 + 0.961377i \(0.411245\pi\)
\(38\) 0 0
\(39\) 1.89093e15 0.0371961
\(40\) 0 0
\(41\) −5.77168e16 −0.671540 −0.335770 0.941944i \(-0.608997\pi\)
−0.335770 + 0.941944i \(0.608997\pi\)
\(42\) 0 0
\(43\) 2.01468e17 1.42163 0.710817 0.703377i \(-0.248325\pi\)
0.710817 + 0.703377i \(0.248325\pi\)
\(44\) 0 0
\(45\) 3.22014e17 1.40976
\(46\) 0 0
\(47\) 6.62056e17 1.83598 0.917989 0.396607i \(-0.129812\pi\)
0.917989 + 0.396607i \(0.129812\pi\)
\(48\) 0 0
\(49\) −5.50130e17 −0.984932
\(50\) 0 0
\(51\) 5.92615e16 0.0697085
\(52\) 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
\(57\) −3.74179e13 −1.36897e−5 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 \(-0.335621\pi\)
0.493762 + 0.869597i \(0.335621\pi\)
\(62\) 0 0
\(63\) −9.53770e17 −0.122002
\(64\) 0 0
\(65\) −7.33594e18 −0.675874
\(66\) 0 0
\(67\) −6.03520e18 −0.404489 −0.202244 0.979335i \(-0.564824\pi\)
−0.202244 + 0.979335i \(0.564824\pi\)
\(68\) 0 0
\(69\) 2.66418e18 0.131113
\(70\) 0 0
\(71\) −4.43147e19 −1.61561 −0.807803 0.589453i \(-0.799343\pi\)
−0.807803 + 0.589453i \(0.799343\pi\)
\(72\) 0 0
\(73\) −2.48622e19 −0.677095 −0.338548 0.940949i \(-0.609936\pi\)
−0.338548 + 0.940949i \(0.609936\pi\)
\(74\) 0 0
\(75\) 3.85204e18 0.0789855
\(76\) 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
\(81\) 1.07423e20 0.981757
\(82\) 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
\(87\) −2.58657e19 −0.111628
\(88\) 0 0
\(89\) 3.97023e20 1.34965 0.674823 0.737979i \(-0.264220\pi\)
0.674823 + 0.737979i \(0.264220\pi\)
\(90\) 0 0
\(91\) 2.17282e19 0.0584911
\(92\) 0 0
\(93\) 5.11333e19 0.109555
\(94\) 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
\(99\) 9.06749e20 1.00767
\(100\) 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
\(105\) −2.26849e19 −0.0135910
\(106\) 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 \(-0.269484\pi\)
0.662526 + 0.749039i \(0.269484\pi\)
\(110\) 0 0
\(111\) 1.28544e20 0.0429696
\(112\) 0 0
\(113\) −2.53293e20 −0.0701938 −0.0350969 0.999384i \(-0.511174\pi\)
−0.0350969 + 0.999384i \(0.511174\pi\)
\(114\) 0 0
\(115\) −1.03358e22 −2.38241
\(116\) 0 0
\(117\) −2.46243e21 −0.473602
\(118\) 0 0
\(119\) 6.80959e20 0.109617
\(120\) 0 0
\(121\) 2.06345e20 0.0278836
\(122\) 0 0
\(123\) −4.60792e20 −0.0524206
\(124\) 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
\(129\) 1.60846e21 0.110973
\(130\) 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
\(135\) 5.15748e21 0.220766
\(136\) 0 0
\(137\) −2.34811e22 −0.861296 −0.430648 0.902520i \(-0.641715\pi\)
−0.430648 + 0.902520i \(0.641715\pi\)
\(138\) 0 0
\(139\) 4.53132e22 1.42748 0.713738 0.700413i \(-0.247001\pi\)
0.713738 + 0.700413i \(0.247001\pi\)
\(140\) 0 0
\(141\) 5.28564e21 0.143317
\(142\) 0 0
\(143\) −2.06570e22 −0.483103
\(144\) 0 0
\(145\) 1.00347e23 2.02834
\(146\) 0 0
\(147\) −4.39206e21 −0.0768840
\(148\) 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
\(153\) −7.71723e22 −0.887568
\(154\) 0 0
\(155\) −1.98374e23 −1.99068
\(156\) 0 0
\(157\) 2.22945e21 0.0195547 0.00977733 0.999952i \(-0.496888\pi\)
0.00977733 + 0.999952i \(0.496888\pi\)
\(158\) 0 0
\(159\) −3.69365e21 −0.0283652
\(160\) 0 0
\(161\) 3.06134e22 0.206177
\(162\) 0 0
\(163\) −1.48202e23 −0.876768 −0.438384 0.898788i \(-0.644449\pi\)
−0.438384 + 0.898788i \(0.644449\pi\)
\(164\) 0 0
\(165\) 2.15666e22 0.112253
\(166\) 0 0
\(167\) −2.25766e23 −1.03547 −0.517733 0.855542i \(-0.673224\pi\)
−0.517733 + 0.855542i \(0.673224\pi\)
\(168\) 0 0
\(169\) −1.90967e23 −0.772943
\(170\) 0 0
\(171\) 4.87269e19 0.000174305 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
\(177\) −5.90394e22 −0.147036
\(178\) 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 \(-0.581113\pi\)
−0.252075 + 0.967708i \(0.581113\pi\)
\(182\) 0 0
\(183\) 4.39252e22 0.0770863
\(184\) 0 0
\(185\) −4.98692e23 −0.780781
\(186\) 0 0
\(187\) −6.47388e23 −0.905374
\(188\) 0 0
\(189\) −1.52759e22 −0.0191054
\(190\) 0 0
\(191\) −7.48387e23 −0.838062 −0.419031 0.907972i \(-0.637630\pi\)
−0.419031 + 0.907972i \(0.637630\pi\)
\(192\) 0 0
\(193\) 1.77062e24 1.77735 0.888674 0.458539i \(-0.151627\pi\)
0.888674 + 0.458539i \(0.151627\pi\)
\(194\) 0 0
\(195\) −5.85678e22 −0.0527589
\(196\) 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
\(201\) −4.81831e22 −0.0315745
\(202\) 0 0
\(203\) −2.97216e23 −0.175535
\(204\) 0 0
\(205\) 1.78766e24 0.952511
\(206\) 0 0
\(207\) −3.46938e24 −1.66941
\(208\) 0 0
\(209\) 4.08763e20 0.000177802 0
\(210\) 0 0
\(211\) −3.52722e24 −1.38825 −0.694123 0.719856i \(-0.744208\pi\)
−0.694123 + 0.719856i \(0.744208\pi\)
\(212\) 0 0
\(213\) −3.53794e23 −0.126114
\(214\) 0 0
\(215\) −6.24008e24 −2.01644
\(216\) 0 0
\(217\) 5.87560e23 0.172276
\(218\) 0 0
\(219\) −1.98492e23 −0.0528542
\(220\) 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
\(225\) −5.01625e24 −1.00569
\(226\) 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 \(-0.583377\pi\)
−0.258952 + 0.965890i \(0.583377\pi\)
\(230\) 0 0
\(231\) −6.38777e22 −0.00971457
\(232\) 0 0
\(233\) −3.90742e24 −0.542817 −0.271408 0.962464i \(-0.587489\pi\)
−0.271408 + 0.962464i \(0.587489\pi\)
\(234\) 0 0
\(235\) −2.05059e25 −2.60415
\(236\) 0 0
\(237\) 4.55826e23 0.0529592
\(238\) 0 0
\(239\) 2.38784e23 0.0253997 0.0126998 0.999919i \(-0.495957\pi\)
0.0126998 + 0.999919i \(0.495957\pi\)
\(240\) 0 0
\(241\) 1.14005e25 1.11107 0.555537 0.831492i \(-0.312513\pi\)
0.555537 + 0.831492i \(0.312513\pi\)
\(242\) 0 0
\(243\) 2.59944e24 0.232281
\(244\) 0 0
\(245\) 1.70392e25 1.39703
\(246\) 0 0
\(247\) −1.11007e21 −8.35663e−5 0
\(248\) 0 0
\(249\) 1.05241e24 0.0727934
\(250\) 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
\(255\) −1.83551e24 −0.0988744
\(256\) 0 0
\(257\) −2.21415e25 −1.09878 −0.549388 0.835567i \(-0.685139\pi\)
−0.549388 + 0.835567i \(0.685139\pi\)
\(258\) 0 0
\(259\) 1.47707e24 0.0675699
\(260\) 0 0
\(261\) 3.36832e25 1.42131
\(262\) 0 0
\(263\) −3.74763e25 −1.45956 −0.729779 0.683683i \(-0.760377\pi\)
−0.729779 + 0.683683i \(0.760377\pi\)
\(264\) 0 0
\(265\) 1.43297e25 0.515412
\(266\) 0 0
\(267\) 3.16970e24 0.105354
\(268\) 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
\(273\) 1.73471e23 0.00456583
\(274\) 0 0
\(275\) −4.20807e25 −1.02586
\(276\) 0 0
\(277\) −5.26085e25 −1.18855 −0.594277 0.804261i \(-0.702562\pi\)
−0.594277 + 0.804261i \(0.702562\pi\)
\(278\) 0 0
\(279\) −6.65875e25 −1.39492
\(280\) 0 0
\(281\) 6.45378e25 1.25429 0.627145 0.778903i \(-0.284223\pi\)
0.627145 + 0.778903i \(0.284223\pi\)
\(282\) 0 0
\(283\) −4.56042e25 −0.822711 −0.411355 0.911475i \(-0.634944\pi\)
−0.411355 + 0.911475i \(0.634944\pi\)
\(284\) 0 0
\(285\) 1.15894e21 1.94174e−5 0
\(286\) 0 0
\(287\) −5.29485e24 −0.0824317
\(288\) 0 0
\(289\) −1.39935e25 −0.202535
\(290\) 0 0
\(291\) −7.82721e24 −0.105373
\(292\) 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
\(297\) 1.45228e25 0.157800
\(298\) 0 0
\(299\) 7.90374e25 0.800360
\(300\) 0 0
\(301\) 1.84824e25 0.174506
\(302\) 0 0
\(303\) −5.15898e23 −0.00454377
\(304\) 0 0
\(305\) −1.70410e26 −1.40070
\(306\) 0 0
\(307\) 1.65443e26 1.26969 0.634843 0.772641i \(-0.281065\pi\)
0.634843 + 0.772641i \(0.281065\pi\)
\(308\) 0 0
\(309\) −5.90377e23 −0.00423220
\(310\) 0 0
\(311\) −7.32700e24 −0.0490842 −0.0245421 0.999699i \(-0.507813\pi\)
−0.0245421 + 0.999699i \(0.507813\pi\)
\(312\) 0 0
\(313\) −7.64901e25 −0.479059 −0.239530 0.970889i \(-0.576993\pi\)
−0.239530 + 0.970889i \(0.576993\pi\)
\(314\) 0 0
\(315\) 2.95411e25 0.173048
\(316\) 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
\(321\) −6.14295e24 −0.0295171
\(322\) 0 0
\(323\) −3.47893e22 −0.000156610 0
\(324\) 0 0
\(325\) 1.14277e26 0.482153
\(326\) 0 0
\(327\) 2.61465e25 0.103434
\(328\) 0 0
\(329\) 6.07360e25 0.225367
\(330\) 0 0
\(331\) 2.46189e26 0.857187 0.428593 0.903498i \(-0.359009\pi\)
0.428593 + 0.903498i \(0.359009\pi\)
\(332\) 0 0
\(333\) −1.67395e26 −0.547113
\(334\) 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
\(339\) −2.02221e24 −0.00547934
\(340\) 0 0
\(341\) −5.58594e26 −1.42290
\(342\) 0 0
\(343\) −1.01708e26 −0.243651
\(344\) 0 0
\(345\) −8.25175e25 −0.185971
\(346\) 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 \(-0.484240\pi\)
0.0494911 + 0.998775i \(0.484240\pi\)
\(350\) 0 0
\(351\) −3.94391e25 −0.0741655
\(352\) 0 0
\(353\) 1.88459e26 0.333873 0.166937 0.985968i \(-0.446613\pi\)
0.166937 + 0.985968i \(0.446613\pi\)
\(354\) 0 0
\(355\) 1.37256e27 2.29157
\(356\) 0 0
\(357\) 5.43656e24 0.00855673
\(358\) 0 0
\(359\) 5.21557e26 0.774123 0.387062 0.922054i \(-0.373490\pi\)
0.387062 + 0.922054i \(0.373490\pi\)
\(360\) 0 0
\(361\) −7.14209e26 −1.00000
\(362\) 0 0
\(363\) 1.64740e24 0.00217660
\(364\) 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
\(369\) 6.00060e26 0.667448
\(370\) 0 0
\(371\) −4.24429e25 −0.0446045
\(372\) 0 0
\(373\) −1.92501e25 −0.0191201 −0.00956004 0.999954i \(-0.503043\pi\)
−0.00956004 + 0.999954i \(0.503043\pi\)
\(374\) 0 0
\(375\) −1.39765e24 −0.00131241
\(376\) 0 0
\(377\) −7.67351e26 −0.681412
\(378\) 0 0
\(379\) 2.04774e27 1.72014 0.860070 0.510175i \(-0.170419\pi\)
0.860070 + 0.510175i \(0.170419\pi\)
\(380\) 0 0
\(381\) −2.41186e25 −0.0191707
\(382\) 0 0
\(383\) 1.75914e27 1.32347 0.661733 0.749740i \(-0.269821\pi\)
0.661733 + 0.749740i \(0.269821\pi\)
\(384\) 0 0
\(385\) 2.47816e26 0.176519
\(386\) 0 0
\(387\) −2.09459e27 −1.41297
\(388\) 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
\(393\) 1.36256e26 0.0782046
\(394\) 0 0
\(395\) −1.76840e27 −0.962299
\(396\) 0 0
\(397\) 1.21691e27 0.627998 0.313999 0.949423i \(-0.398331\pi\)
0.313999 + 0.949423i \(0.398331\pi\)
\(398\) 0 0
\(399\) −3.43266e21 −1.68041e−6 0
\(400\) 0 0
\(401\) −1.15061e27 −0.534455 −0.267228 0.963633i \(-0.586108\pi\)
−0.267228 + 0.963633i \(0.586108\pi\)
\(402\) 0 0
\(403\) 1.51696e27 0.668761
\(404\) 0 0
\(405\) −3.32721e27 −1.39252
\(406\) 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
\(411\) −1.87465e26 −0.0672330
\(412\) 0 0
\(413\) −6.78408e26 −0.231215
\(414\) 0 0
\(415\) −4.08287e27 −1.32270
\(416\) 0 0
\(417\) 3.61766e26 0.111429
\(418\) 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 \(-0.266307\pi\)
0.669969 + 0.742389i \(0.266307\pi\)
\(422\) 0 0
\(423\) −6.88314e27 −1.82479
\(424\) 0 0
\(425\) 3.58144e27 0.903594
\(426\) 0 0
\(427\) 5.04734e26 0.121219
\(428\) 0 0
\(429\) −1.64919e26 −0.0377111
\(430\) 0 0
\(431\) −4.51446e27 −0.983092 −0.491546 0.870852i \(-0.663568\pi\)
−0.491546 + 0.870852i \(0.663568\pi\)
\(432\) 0 0
\(433\) −4.04253e27 −0.838552 −0.419276 0.907859i \(-0.637716\pi\)
−0.419276 + 0.907859i \(0.637716\pi\)
\(434\) 0 0
\(435\) 8.01138e26 0.158332
\(436\) 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
\(441\) 5.71949e27 0.978931
\(442\) 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
\(447\) −5.57541e26 −0.0828034
\(448\) 0 0
\(449\) −4.22534e27 −0.598790 −0.299395 0.954129i \(-0.596785\pi\)
−0.299395 + 0.954129i \(0.596785\pi\)
\(450\) 0 0
\(451\) 5.03382e27 0.680838
\(452\) 0 0
\(453\) −3.78075e26 −0.0488143
\(454\) 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
\(459\) −1.23601e27 −0.138992
\(460\) 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
\(465\) −1.58375e27 −0.155393
\(466\) 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
\(471\) 1.77992e25 0.00152644
\(472\) 0 0
\(473\) −1.75712e28 −1.44132
\(474\) 0 0
\(475\) −2.26133e24 −0.000177452 0
\(476\) 0 0
\(477\) 4.81000e27 0.361162
\(478\) 0 0
\(479\) −9.14679e27 −0.657273 −0.328636 0.944457i \(-0.606589\pi\)
−0.328636 + 0.944457i \(0.606589\pi\)
\(480\) 0 0
\(481\) 3.81349e27 0.262300
\(482\) 0 0
\(483\) 2.44407e26 0.0160942
\(484\) 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
\(489\) −1.18320e27 −0.0684407
\(490\) 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
\(495\) −2.80847e28 −1.42927
\(496\) 0 0
\(497\) −4.06537e27 −0.198316
\(498\) 0 0
\(499\) −2.04342e28 −0.955657 −0.477829 0.878453i \(-0.658576\pi\)
−0.477829 + 0.878453i \(0.658576\pi\)
\(500\) 0 0
\(501\) −1.80244e27 −0.0808288
\(502\) 0 0
\(503\) 1.67038e28 0.718375 0.359188 0.933265i \(-0.383054\pi\)
0.359188 + 0.933265i \(0.383054\pi\)
\(504\) 0 0
\(505\) 2.00145e27 0.0825628
\(506\) 0 0
\(507\) −1.52462e27 −0.0603361
\(508\) 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
\(513\) 7.80424e23 2.72960e−5 0
\(514\) 0 0
\(515\) 2.29039e27 0.0769014
\(516\) 0 0
\(517\) −5.77418e28 −1.86140
\(518\) 0 0
\(519\) 4.29141e27 0.132844
\(520\) 0 0
\(521\) 1.37414e28 0.408541 0.204271 0.978914i \(-0.434518\pi\)
0.204271 + 0.978914i \(0.434518\pi\)
\(522\) 0 0
\(523\) 3.33678e28 0.952927 0.476463 0.879194i \(-0.341919\pi\)
0.476463 + 0.879194i \(0.341919\pi\)
\(524\) 0 0
\(525\) 3.53380e26 0.00969548
\(526\) 0 0
\(527\) 4.75412e28 1.25331
\(528\) 0 0
\(529\) 7.18860e28 1.82121
\(530\) 0 0
\(531\) 7.68831e28 1.87214
\(532\) 0 0
\(533\) −1.36702e28 −0.319992
\(534\) 0 0
\(535\) 2.38318e28 0.536343
\(536\) 0 0
\(537\) −9.03932e26 −0.0195617
\(538\) 0 0
\(539\) 4.79800e28 0.998570
\(540\) 0 0
\(541\) 9.34113e28 1.86994 0.934971 0.354723i \(-0.115425\pi\)
0.934971 + 0.354723i \(0.115425\pi\)
\(542\) 0 0
\(543\) −2.04356e27 −0.0393541
\(544\) 0 0
\(545\) −1.01437e29 −1.87945
\(546\) 0 0
\(547\) −1.90655e28 −0.339923 −0.169961 0.985451i \(-0.554364\pi\)
−0.169961 + 0.985451i \(0.554364\pi\)
\(548\) 0 0
\(549\) −5.72009e28 −0.981507
\(550\) 0 0
\(551\) 1.51844e25 0.000250788 0
\(552\) 0 0
\(553\) 5.23779e27 0.0832787
\(554\) 0 0
\(555\) −3.98140e27 −0.0609480
\(556\) 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
\(561\) −5.16854e27 −0.0706736
\(562\) 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
\(567\) 9.85481e27 0.120511
\(568\) 0 0
\(569\) −8.25601e28 −0.972952 −0.486476 0.873694i \(-0.661718\pi\)
−0.486476 + 0.873694i \(0.661718\pi\)
\(570\) 0 0
\(571\) 3.00348e28 0.341150 0.170575 0.985345i \(-0.445437\pi\)
0.170575 + 0.985345i \(0.445437\pi\)
\(572\) 0 0
\(573\) −5.97488e27 −0.0654193
\(574\) 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
\(579\) 1.41360e28 0.138740
\(580\) 0 0
\(581\) 1.20930e28 0.114468
\(582\) 0 0
\(583\) 4.03504e28 0.368408
\(584\) 0 0
\(585\) 7.62690e28 0.671756
\(586\) 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
\(591\) −1.63176e28 −0.129118
\(592\) 0 0
\(593\) 3.89818e28 0.297706 0.148853 0.988859i \(-0.452442\pi\)
0.148853 + 0.988859i \(0.452442\pi\)
\(594\) 0 0
\(595\) −2.10913e28 −0.155481
\(596\) 0 0
\(597\) −6.49461e26 −0.00462192
\(598\) 0 0
\(599\) 1.28684e29 0.884186 0.442093 0.896969i \(-0.354236\pi\)
0.442093 + 0.896969i \(0.354236\pi\)
\(600\) 0 0
\(601\) −1.81454e29 −1.20388 −0.601942 0.798540i \(-0.705606\pi\)
−0.601942 + 0.798540i \(0.705606\pi\)
\(602\) 0 0
\(603\) 6.27457e28 0.402024
\(604\) 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
\(609\) −2.37288e27 −0.0137023
\(610\) 0 0
\(611\) 1.56808e29 0.874853
\(612\) 0 0
\(613\) −8.09843e28 −0.436582 −0.218291 0.975884i \(-0.570048\pi\)
−0.218291 + 0.975884i \(0.570048\pi\)
\(614\) 0 0
\(615\) 1.42721e28 0.0743532
\(616\) 0 0
\(617\) 3.47454e29 1.74946 0.874728 0.484614i \(-0.161040\pi\)
0.874728 + 0.484614i \(0.161040\pi\)
\(618\) 0 0
\(619\) 2.90076e29 1.41176 0.705878 0.708333i \(-0.250553\pi\)
0.705878 + 0.708333i \(0.250553\pi\)
\(620\) 0 0
\(621\) −5.55666e28 −0.261428
\(622\) 0 0
\(623\) 3.64223e28 0.165669
\(624\) 0 0
\(625\) −2.24647e29 −0.988006
\(626\) 0 0
\(627\) 3.26343e24 1.38792e−5 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
\(633\) −2.81602e28 −0.108367
\(634\) 0 0
\(635\) 9.35689e28 0.348343
\(636\) 0 0
\(637\) −1.30298e29 −0.469325
\(638\) 0 0
\(639\) 4.60723e29 1.60576
\(640\) 0 0
\(641\) −3.43690e29 −1.15920 −0.579598 0.814902i \(-0.696791\pi\)
−0.579598 + 0.814902i \(0.696791\pi\)
\(642\) 0 0
\(643\) −1.89646e29 −0.619055 −0.309527 0.950891i \(-0.600171\pi\)
−0.309527 + 0.950891i \(0.600171\pi\)
\(644\) 0 0
\(645\) −4.98187e28 −0.157404
\(646\) 0 0
\(647\) 3.06810e29 0.938370 0.469185 0.883100i \(-0.344548\pi\)
0.469185 + 0.883100i \(0.344548\pi\)
\(648\) 0 0
\(649\) 6.44962e29 1.90970
\(650\) 0 0
\(651\) 4.69089e27 0.0134479
\(652\) 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
\(657\) 2.58483e29 0.672969
\(658\) 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 \(-0.339384\pi\)
0.483448 + 0.875373i \(0.339384\pi\)
\(662\) 0 0
\(663\) 1.40361e28 0.0332164
\(664\) 0 0
\(665\) 1.33172e25 3.05340e−5 0
\(666\) 0 0
\(667\) −1.08114e30 −2.40193
\(668\) 0 0
\(669\) −1.89671e28 −0.0408343
\(670\) 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
\(675\) −8.03418e28 −0.157490
\(676\) 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
\(681\) −5.35973e28 −0.0957411
\(682\) 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
\(687\) −2.48156e28 −0.0404276
\(688\) 0 0
\(689\) −1.09579e29 −0.173151
\(690\) 0 0
\(691\) −2.46471e29 −0.377787 −0.188893 0.981998i \(-0.560490\pi\)
−0.188893 + 0.981998i \(0.560490\pi\)
\(692\) 0 0
\(693\) 8.31838e28 0.123691
\(694\) 0 0
\(695\) −1.40348e30 −2.02473
\(696\) 0 0
\(697\) −4.28422e29 −0.599691
\(698\) 0 0
\(699\) −3.11956e28 −0.0423724
\(700\) 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
\(705\) −1.63712e29 −0.203280
\(706\) 0 0
\(707\) −5.92806e27 −0.00714510
\(708\) 0 0
\(709\) −7.52975e29 −0.881039 −0.440520 0.897743i \(-0.645206\pi\)
−0.440520 + 0.897743i \(0.645206\pi\)
\(710\) 0 0
\(711\) −5.93592e29 −0.674307
\(712\) 0 0
\(713\) 2.13728e30 2.35733
\(714\) 0 0
\(715\) 6.39810e29 0.685232
\(716\) 0 0
\(717\) 1.90638e27 0.00198270
\(718\) 0 0
\(719\) 4.85535e29 0.490419 0.245210 0.969470i \(-0.421143\pi\)
0.245210 + 0.969470i \(0.421143\pi\)
\(720\) 0 0
\(721\) −6.78389e27 −0.00665516
\(722\) 0 0
\(723\) 9.10175e28 0.0867307
\(724\) 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
\(729\) −1.10293e30 −0.963625
\(730\) 0 0
\(731\) 1.49546e30 1.26953
\(732\) 0 0
\(733\) 3.73906e29 0.308440 0.154220 0.988037i \(-0.450714\pi\)
0.154220 + 0.988037i \(0.450714\pi\)
\(734\) 0 0
\(735\) 1.36035e29 0.109052
\(736\) 0 0
\(737\) 5.26365e29 0.410089
\(738\) 0 0
\(739\) −8.38257e29 −0.634762 −0.317381 0.948298i \(-0.602803\pi\)
−0.317381 + 0.948298i \(0.602803\pi\)
\(740\) 0 0
\(741\) −8.86242e24 −6.52321e−6 0
\(742\) 0 0
\(743\) −7.60139e29 −0.543889 −0.271945 0.962313i \(-0.587667\pi\)
−0.271945 + 0.962313i \(0.587667\pi\)
\(744\) 0 0
\(745\) 2.16300e30 1.50458
\(746\) 0 0
\(747\) −1.37048e30 −0.926846
\(748\) 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
\(753\) 4.63151e28 0.0287988
\(754\) 0 0
\(755\) 1.46676e30 0.886983
\(756\) 0 0
\(757\) −3.06224e30 −1.80108 −0.900540 0.434774i \(-0.856828\pi\)
−0.900540 + 0.434774i \(0.856828\pi\)
\(758\) 0 0
\(759\) −2.32358e29 −0.132929
\(760\) 0 0
\(761\) −3.46912e30 −1.93055 −0.965274 0.261241i \(-0.915868\pi\)
−0.965274 + 0.261241i \(0.915868\pi\)
\(762\) 0 0
\(763\) 3.00444e29 0.162650
\(764\) 0 0
\(765\) 2.39026e30 1.25892
\(766\) 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
\(771\) −1.76770e29 −0.0857707
\(772\) 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
\(777\) 1.17924e28 0.00527452
\(778\) 0 0
\(779\) 2.70507e26 0.000117770 0
\(780\) 0 0
\(781\) 3.86494e30 1.63798
\(782\) 0 0
\(783\) 5.39480e29 0.222575
\(784\) 0 0
\(785\) −6.90526e28 −0.0277363
\(786\) 0 0
\(787\) −1.22284e30 −0.478227 −0.239114 0.970992i \(-0.576857\pi\)
−0.239114 + 0.970992i \(0.576857\pi\)
\(788\) 0 0
\(789\) −2.99199e29 −0.113933
\(790\) 0 0
\(791\) −2.32367e28 −0.00861630
\(792\) 0 0
\(793\) 1.30312e30 0.470560
\(794\) 0 0
\(795\) 1.14403e29 0.0402332
\(796\) 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
\(801\) −4.12769e30 −1.34142
\(802\) 0 0
\(803\) 2.16838e30 0.686470
\(804\) 0 0
\(805\) −9.48189e29 −0.292441
\(806\) 0 0
\(807\) −1.31590e29 −0.0395413
\(808\) 0 0
\(809\) 5.54086e30 1.62225 0.811124 0.584874i \(-0.198856\pi\)
0.811124 + 0.584874i \(0.198856\pi\)
\(810\) 0 0
\(811\) 4.51703e30 1.28865 0.644323 0.764753i \(-0.277139\pi\)
0.644323 + 0.764753i \(0.277139\pi\)
\(812\) 0 0
\(813\) −2.28467e29 −0.0635145
\(814\) 0 0
\(815\) 4.59026e30 1.24361
\(816\) 0 0
\(817\) −9.44242e26 −0.000249317 0
\(818\) 0 0
\(819\) −2.25900e29 −0.0581347
\(820\) 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
\(825\) −3.35959e29 −0.0800791
\(826\) 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 \(-0.388717\pi\)
0.342526 + 0.939508i \(0.388717\pi\)
\(830\) 0 0
\(831\) −4.20009e29 −0.0927787
\(832\) 0 0
\(833\) −4.08352e30 −0.879554
\(834\) 0 0
\(835\) 6.99266e30 1.46870
\(836\) 0 0
\(837\) −1.06649e30 −0.218443
\(838\) 0 0
\(839\) −6.48734e30 −1.29588 −0.647942 0.761690i \(-0.724370\pi\)
−0.647942 + 0.761690i \(0.724370\pi\)
\(840\) 0 0
\(841\) 5.36360e30 1.04496
\(842\) 0 0
\(843\) 5.15249e29 0.0979101
\(844\) 0 0
\(845\) 5.91481e30 1.09634
\(846\) 0 0
\(847\) 1.89298e28 0.00342272
\(848\) 0 0
\(849\) −3.64089e29 −0.0642210
\(850\) 0 0
\(851\) 5.37291e30 0.924590
\(852\) 0 0
\(853\) −1.83172e30 −0.307535 −0.153767 0.988107i \(-0.549141\pi\)
−0.153767 + 0.988107i \(0.549141\pi\)
\(854\) 0 0
\(855\) −1.50922e27 −0.000247234 0
\(856\) 0 0
\(857\) 1.04074e31 1.66358 0.831792 0.555088i \(-0.187315\pi\)
0.831792 + 0.555088i \(0.187315\pi\)
\(858\) 0 0
\(859\) 8.40237e29 0.131061 0.0655306 0.997851i \(-0.479126\pi\)
0.0655306 + 0.997851i \(0.479126\pi\)
\(860\) 0 0
\(861\) −4.22724e28 −0.00643463
\(862\) 0 0
\(863\) −6.66021e30 −0.989406 −0.494703 0.869062i \(-0.664723\pi\)
−0.494703 + 0.869062i \(0.664723\pi\)
\(864\) 0 0
\(865\) −1.66487e31 −2.41386
\(866\) 0 0
\(867\) −1.11720e29 −0.0158099
\(868\) 0 0
\(869\) −4.97957e30 −0.687835
\(870\) 0 0
\(871\) −1.42944e30 −0.192741
\(872\) 0 0
\(873\) 1.01929e31 1.34167
\(874\) 0 0
\(875\) −1.60600e28 −0.00206377
\(876\) 0 0
\(877\) 1.99416e30 0.250187 0.125093 0.992145i \(-0.460077\pi\)
0.125093 + 0.992145i \(0.460077\pi\)
\(878\) 0 0
\(879\) 4.69331e29 0.0574905
\(880\) 0 0
\(881\) −7.25457e30 −0.867690 −0.433845 0.900987i \(-0.642843\pi\)
−0.433845 + 0.900987i \(0.642843\pi\)
\(882\) 0 0
\(883\) −7.43622e30 −0.868491 −0.434245 0.900795i \(-0.642985\pi\)
−0.434245 + 0.900795i \(0.642985\pi\)
\(884\) 0 0
\(885\) 1.82863e30 0.208555
\(886\) 0 0
\(887\) 5.32005e30 0.592540 0.296270 0.955104i \(-0.404257\pi\)
0.296270 + 0.955104i \(0.404257\pi\)
\(888\) 0 0
\(889\) −2.77140e29 −0.0301461
\(890\) 0 0
\(891\) −9.36897e30 −0.995350
\(892\) 0 0
\(893\) −3.10293e27 −0.000321982 0
\(894\) 0 0
\(895\) 3.50684e30 0.355446
\(896\) 0 0
\(897\) 6.31009e29 0.0624762
\(898\) 0 0
\(899\) −2.07502e31 −2.00699
\(900\) 0 0
\(901\) −3.43418e30 −0.324498
\(902\) 0 0
\(903\) 1.47557e29 0.0136220
\(904\) 0 0
\(905\) 7.92809e30 0.715086
\(906\) 0 0
\(907\) 3.82478e30 0.337077 0.168539 0.985695i \(-0.446095\pi\)
0.168539 + 0.985695i \(0.446095\pi\)
\(908\) 0 0
\(909\) 6.71820e29 0.0578538
\(910\) 0 0
\(911\) −3.98935e30 −0.335706 −0.167853 0.985812i \(-0.553683\pi\)
−0.167853 + 0.985812i \(0.553683\pi\)
\(912\) 0 0
\(913\) −1.14968e31 −0.945440
\(914\) 0 0
\(915\) −1.36050e30 −0.109339
\(916\) 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
\(921\) 1.32085e30 0.0991119
\(922\) 0 0
\(923\) −1.04959e31 −0.769844
\(924\) 0 0
\(925\) 7.76850e30 0.556992
\(926\) 0 0
\(927\) 7.68810e29 0.0538867
\(928\) 0 0
\(929\) −5.26649e30 −0.360875 −0.180437 0.983586i \(-0.557751\pi\)
−0.180437 + 0.983586i \(0.557751\pi\)
\(930\) 0 0
\(931\) 2.57835e27 0.000172731 0
\(932\) 0 0
\(933\) −5.84963e28 −0.00383152
\(934\) 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
\(939\) −6.10672e29 −0.0373955
\(940\) 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
\(945\) 4.73139e29 0.0270991
\(946\) 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
\(951\) 7.84793e29 0.0420591
\(952\) 0 0
\(953\) −4.71750e30 −0.247307 −0.123654 0.992325i \(-0.539461\pi\)
−0.123654 + 0.992325i \(0.539461\pi\)
\(954\) 0 0
\(955\) 2.31798e31 1.18871
\(956\) 0 0
\(957\) 2.25590e30 0.113173
\(958\) 0 0
\(959\) −2.15412e30 −0.105724
\(960\) 0 0
\(961\) 2.01951e31 0.969728
\(962\) 0 0
\(963\) 7.99956e30 0.375829
\(964\) 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
\(969\) −2.77747e26 −1.22250e−5 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
\(975\) 9.12354e29 0.0376370
\(976\) 0 0
\(977\) −1.85934e31 −0.750700 −0.375350 0.926883i \(-0.622477\pi\)
−0.375350 + 0.926883i \(0.622477\pi\)
\(978\) 0 0
\(979\) −3.46266e31 −1.36833
\(980\) 0 0
\(981\) −3.40489e31 −1.31698
\(982\) 0 0
\(983\) −2.18619e31 −0.827704 −0.413852 0.910344i \(-0.635817\pi\)
−0.413852 + 0.910344i \(0.635817\pi\)
\(984\) 0 0
\(985\) 6.33049e31 2.34616
\(986\) 0 0
\(987\) 4.84897e29 0.0175922
\(988\) 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
\(993\) 1.96550e30 0.0669122
\(994\) 0 0
\(995\) 2.51961e30 0.0839829
\(996\) 0 0
\(997\) −1.31572e31 −0.429403 −0.214701 0.976680i \(-0.568878\pi\)
−0.214701 + 0.976680i \(0.568878\pi\)
\(998\) 0 0
\(999\) −2.68104e30 −0.0856773
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 64.22.a.l.1.2 3
4.3 odd 2 64.22.a.m.1.2 3
8.3 odd 2 16.22.a.f.1.2 3
8.5 even 2 8.22.a.b.1.2 3
24.5 odd 2 72.22.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.2 3 8.5 even 2
16.22.a.f.1.2 3 8.3 odd 2
64.22.a.l.1.2 3 1.1 even 1 trivial
64.22.a.m.1.2 3 4.3 odd 2
72.22.a.f.1.1 3 24.5 odd 2