Properties

Label 192.8.a.r.1.1
Level $192$
Weight $8$
Character 192.1
Self dual yes
Analytic conductor $59.978$
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] = [192,8,Mod(1,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 192.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: \(59.9779248930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{235}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 235 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-15.3297\) of defining polynomial
Character \(\chi\) \(=\) 192.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -335.275 q^{5} -710.377 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} -335.275 q^{5} -710.377 q^{7} +729.000 q^{9} -2891.65 q^{11} -10131.6 q^{13} +9052.43 q^{15} -14241.5 q^{17} -33189.3 q^{19} +19180.2 q^{21} -75372.9 q^{23} +34284.6 q^{25} -19683.0 q^{27} -150880. q^{29} +55120.5 q^{31} +78074.6 q^{33} +238172. q^{35} -510853. q^{37} +273552. q^{39} +604684. q^{41} +479815. q^{43} -244416. q^{45} +159323. q^{47} -318908. q^{49} +384520. q^{51} -319003. q^{53} +969500. q^{55} +896110. q^{57} -1.81794e6 q^{59} -3.00891e6 q^{61} -517865. q^{63} +3.39686e6 q^{65} +3.12562e6 q^{67} +2.03507e6 q^{69} +4.15152e6 q^{71} +1.69454e6 q^{73} -925683. q^{75} +2.05416e6 q^{77} -4.26998e6 q^{79} +531441. q^{81} +7.78317e6 q^{83} +4.77482e6 q^{85} +4.07375e6 q^{87} -9.58244e6 q^{89} +7.19723e6 q^{91} -1.48825e6 q^{93} +1.11275e7 q^{95} -1.01226e7 q^{97} -2.10801e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{3} - 180 q^{5} + 1032 q^{7} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{3} - 180 q^{5} + 1032 q^{7} + 1458 q^{9} - 2840 q^{11} + 340 q^{13} + 4860 q^{15} + 9780 q^{17} - 32040 q^{19} - 27864 q^{21} + 11136 q^{23} - 19730 q^{25} - 39366 q^{27} - 304212 q^{29} - 77640 q^{31} + 76680 q^{33} + 508720 q^{35} - 1015820 q^{37} - 9180 q^{39} + 704100 q^{41} + 395496 q^{43} - 131220 q^{45} + 1157488 q^{47} + 1893426 q^{49} - 264060 q^{51} - 1568580 q^{53} + 977520 q^{55} + 865080 q^{57} - 139240 q^{59} - 2603580 q^{61} + 752328 q^{63} + 5022840 q^{65} + 5289768 q^{67} - 300672 q^{69} + 5721760 q^{71} - 1190700 q^{73} + 532710 q^{75} + 2144160 q^{77} - 398280 q^{79} + 1062882 q^{81} + 6986616 q^{83} + 8504760 q^{85} + 8213724 q^{87} - 8166732 q^{89} + 25442640 q^{91} + 2096280 q^{93} + 11306000 q^{95} - 10361500 q^{97} - 2070360 q^{99}+O(q^{100}) \) 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
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −335.275 −1.19952 −0.599759 0.800181i \(-0.704737\pi\)
−0.599759 + 0.800181i \(0.704737\pi\)
\(6\) 0 0
\(7\) −710.377 −0.782791 −0.391395 0.920223i \(-0.628007\pi\)
−0.391395 + 0.920223i \(0.628007\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −2891.65 −0.655046 −0.327523 0.944843i \(-0.606214\pi\)
−0.327523 + 0.944843i \(0.606214\pi\)
\(12\) 0 0
\(13\) −10131.6 −1.27901 −0.639506 0.768786i \(-0.720861\pi\)
−0.639506 + 0.768786i \(0.720861\pi\)
\(14\) 0 0
\(15\) 9052.43 0.692542
\(16\) 0 0
\(17\) −14241.5 −0.703046 −0.351523 0.936179i \(-0.614336\pi\)
−0.351523 + 0.936179i \(0.614336\pi\)
\(18\) 0 0
\(19\) −33189.3 −1.11010 −0.555048 0.831819i \(-0.687300\pi\)
−0.555048 + 0.831819i \(0.687300\pi\)
\(20\) 0 0
\(21\) 19180.2 0.451944
\(22\) 0 0
\(23\) −75372.9 −1.29172 −0.645859 0.763457i \(-0.723501\pi\)
−0.645859 + 0.763457i \(0.723501\pi\)
\(24\) 0 0
\(25\) 34284.6 0.438842
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −150880. −1.14878 −0.574391 0.818581i \(-0.694761\pi\)
−0.574391 + 0.818581i \(0.694761\pi\)
\(30\) 0 0
\(31\) 55120.5 0.332313 0.166156 0.986099i \(-0.446864\pi\)
0.166156 + 0.986099i \(0.446864\pi\)
\(32\) 0 0
\(33\) 78074.6 0.378191
\(34\) 0 0
\(35\) 238172. 0.938971
\(36\) 0 0
\(37\) −510853. −1.65802 −0.829011 0.559233i \(-0.811096\pi\)
−0.829011 + 0.559233i \(0.811096\pi\)
\(38\) 0 0
\(39\) 273552. 0.738438
\(40\) 0 0
\(41\) 604684. 1.37020 0.685101 0.728448i \(-0.259758\pi\)
0.685101 + 0.728448i \(0.259758\pi\)
\(42\) 0 0
\(43\) 479815. 0.920310 0.460155 0.887839i \(-0.347794\pi\)
0.460155 + 0.887839i \(0.347794\pi\)
\(44\) 0 0
\(45\) −244416. −0.399839
\(46\) 0 0
\(47\) 159323. 0.223839 0.111920 0.993717i \(-0.464300\pi\)
0.111920 + 0.993717i \(0.464300\pi\)
\(48\) 0 0
\(49\) −318908. −0.387239
\(50\) 0 0
\(51\) 384520. 0.405904
\(52\) 0 0
\(53\) −319003. −0.294326 −0.147163 0.989112i \(-0.547014\pi\)
−0.147163 + 0.989112i \(0.547014\pi\)
\(54\) 0 0
\(55\) 969500. 0.785739
\(56\) 0 0
\(57\) 896110. 0.640914
\(58\) 0 0
\(59\) −1.81794e6 −1.15239 −0.576193 0.817313i \(-0.695462\pi\)
−0.576193 + 0.817313i \(0.695462\pi\)
\(60\) 0 0
\(61\) −3.00891e6 −1.69728 −0.848642 0.528968i \(-0.822579\pi\)
−0.848642 + 0.528968i \(0.822579\pi\)
\(62\) 0 0
\(63\) −517865. −0.260930
\(64\) 0 0
\(65\) 3.39686e6 1.53420
\(66\) 0 0
\(67\) 3.12562e6 1.26962 0.634812 0.772667i \(-0.281078\pi\)
0.634812 + 0.772667i \(0.281078\pi\)
\(68\) 0 0
\(69\) 2.03507e6 0.745774
\(70\) 0 0
\(71\) 4.15152e6 1.37658 0.688292 0.725434i \(-0.258361\pi\)
0.688292 + 0.725434i \(0.258361\pi\)
\(72\) 0 0
\(73\) 1.69454e6 0.509826 0.254913 0.966964i \(-0.417953\pi\)
0.254913 + 0.966964i \(0.417953\pi\)
\(74\) 0 0
\(75\) −925683. −0.253366
\(76\) 0 0
\(77\) 2.05416e6 0.512764
\(78\) 0 0
\(79\) −4.26998e6 −0.974385 −0.487192 0.873295i \(-0.661979\pi\)
−0.487192 + 0.873295i \(0.661979\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 7.78317e6 1.49411 0.747056 0.664761i \(-0.231466\pi\)
0.747056 + 0.664761i \(0.231466\pi\)
\(84\) 0 0
\(85\) 4.77482e6 0.843316
\(86\) 0 0
\(87\) 4.07375e6 0.663250
\(88\) 0 0
\(89\) −9.58244e6 −1.44082 −0.720412 0.693546i \(-0.756047\pi\)
−0.720412 + 0.693546i \(0.756047\pi\)
\(90\) 0 0
\(91\) 7.19723e6 1.00120
\(92\) 0 0
\(93\) −1.48825e6 −0.191861
\(94\) 0 0
\(95\) 1.11275e7 1.33158
\(96\) 0 0
\(97\) −1.01226e7 −1.12613 −0.563066 0.826412i \(-0.690378\pi\)
−0.563066 + 0.826412i \(0.690378\pi\)
\(98\) 0 0
\(99\) −2.10801e6 −0.218349
\(100\) 0 0
\(101\) 4.00358e6 0.386655 0.193327 0.981134i \(-0.438072\pi\)
0.193327 + 0.981134i \(0.438072\pi\)
\(102\) 0 0
\(103\) 1.25940e7 1.13562 0.567809 0.823160i \(-0.307791\pi\)
0.567809 + 0.823160i \(0.307791\pi\)
\(104\) 0 0
\(105\) −6.43064e6 −0.542115
\(106\) 0 0
\(107\) 1.44326e7 1.13894 0.569470 0.822012i \(-0.307149\pi\)
0.569470 + 0.822012i \(0.307149\pi\)
\(108\) 0 0
\(109\) 6.17092e6 0.456412 0.228206 0.973613i \(-0.426714\pi\)
0.228206 + 0.973613i \(0.426714\pi\)
\(110\) 0 0
\(111\) 1.37930e7 0.957259
\(112\) 0 0
\(113\) 4.83221e6 0.315044 0.157522 0.987515i \(-0.449649\pi\)
0.157522 + 0.987515i \(0.449649\pi\)
\(114\) 0 0
\(115\) 2.52707e7 1.54944
\(116\) 0 0
\(117\) −7.38591e6 −0.426338
\(118\) 0 0
\(119\) 1.01168e7 0.550338
\(120\) 0 0
\(121\) −1.11255e7 −0.570915
\(122\) 0 0
\(123\) −1.63265e7 −0.791087
\(124\) 0 0
\(125\) 1.46986e7 0.673118
\(126\) 0 0
\(127\) −2.67682e7 −1.15959 −0.579796 0.814762i \(-0.696868\pi\)
−0.579796 + 0.814762i \(0.696868\pi\)
\(128\) 0 0
\(129\) −1.29550e7 −0.531341
\(130\) 0 0
\(131\) −3.86901e7 −1.50366 −0.751832 0.659355i \(-0.770829\pi\)
−0.751832 + 0.659355i \(0.770829\pi\)
\(132\) 0 0
\(133\) 2.35769e7 0.868972
\(134\) 0 0
\(135\) 6.59922e6 0.230847
\(136\) 0 0
\(137\) −1.95562e7 −0.649773 −0.324887 0.945753i \(-0.605326\pi\)
−0.324887 + 0.945753i \(0.605326\pi\)
\(138\) 0 0
\(139\) 4.88761e7 1.54363 0.771817 0.635844i \(-0.219348\pi\)
0.771817 + 0.635844i \(0.219348\pi\)
\(140\) 0 0
\(141\) −4.30172e6 −0.129234
\(142\) 0 0
\(143\) 2.92970e7 0.837812
\(144\) 0 0
\(145\) 5.05862e7 1.37798
\(146\) 0 0
\(147\) 8.61051e6 0.223572
\(148\) 0 0
\(149\) −1.88273e7 −0.466270 −0.233135 0.972444i \(-0.574898\pi\)
−0.233135 + 0.972444i \(0.574898\pi\)
\(150\) 0 0
\(151\) −5.29908e7 −1.25251 −0.626255 0.779618i \(-0.715413\pi\)
−0.626255 + 0.779618i \(0.715413\pi\)
\(152\) 0 0
\(153\) −1.03820e7 −0.234349
\(154\) 0 0
\(155\) −1.84805e7 −0.398615
\(156\) 0 0
\(157\) −6.61106e7 −1.36340 −0.681699 0.731633i \(-0.738759\pi\)
−0.681699 + 0.731633i \(0.738759\pi\)
\(158\) 0 0
\(159\) 8.61307e6 0.169929
\(160\) 0 0
\(161\) 5.35431e7 1.01114
\(162\) 0 0
\(163\) 4.13135e7 0.747197 0.373599 0.927590i \(-0.378124\pi\)
0.373599 + 0.927590i \(0.378124\pi\)
\(164\) 0 0
\(165\) −2.61765e7 −0.453647
\(166\) 0 0
\(167\) −203790. −0.00338590 −0.00169295 0.999999i \(-0.500539\pi\)
−0.00169295 + 0.999999i \(0.500539\pi\)
\(168\) 0 0
\(169\) 3.99001e7 0.635873
\(170\) 0 0
\(171\) −2.41950e7 −0.370032
\(172\) 0 0
\(173\) −2.42803e7 −0.356528 −0.178264 0.983983i \(-0.557048\pi\)
−0.178264 + 0.983983i \(0.557048\pi\)
\(174\) 0 0
\(175\) −2.43550e7 −0.343522
\(176\) 0 0
\(177\) 4.90845e7 0.665331
\(178\) 0 0
\(179\) −1.03121e8 −1.34388 −0.671941 0.740604i \(-0.734539\pi\)
−0.671941 + 0.740604i \(0.734539\pi\)
\(180\) 0 0
\(181\) 6.74847e6 0.0845922 0.0422961 0.999105i \(-0.486533\pi\)
0.0422961 + 0.999105i \(0.486533\pi\)
\(182\) 0 0
\(183\) 8.12405e7 0.979927
\(184\) 0 0
\(185\) 1.71277e8 1.98883
\(186\) 0 0
\(187\) 4.11814e7 0.460527
\(188\) 0 0
\(189\) 1.39823e7 0.150648
\(190\) 0 0
\(191\) 1.92545e7 0.199947 0.0999735 0.994990i \(-0.468124\pi\)
0.0999735 + 0.994990i \(0.468124\pi\)
\(192\) 0 0
\(193\) −1.68684e8 −1.68897 −0.844486 0.535577i \(-0.820094\pi\)
−0.844486 + 0.535577i \(0.820094\pi\)
\(194\) 0 0
\(195\) −9.17153e7 −0.885770
\(196\) 0 0
\(197\) 1.57329e8 1.46615 0.733073 0.680149i \(-0.238085\pi\)
0.733073 + 0.680149i \(0.238085\pi\)
\(198\) 0 0
\(199\) −1.33321e8 −1.19926 −0.599630 0.800277i \(-0.704686\pi\)
−0.599630 + 0.800277i \(0.704686\pi\)
\(200\) 0 0
\(201\) −8.43918e7 −0.733017
\(202\) 0 0
\(203\) 1.07181e8 0.899256
\(204\) 0 0
\(205\) −2.02736e8 −1.64358
\(206\) 0 0
\(207\) −5.49468e7 −0.430573
\(208\) 0 0
\(209\) 9.59718e7 0.727163
\(210\) 0 0
\(211\) −2.66491e8 −1.95296 −0.976480 0.215609i \(-0.930826\pi\)
−0.976480 + 0.215609i \(0.930826\pi\)
\(212\) 0 0
\(213\) −1.12091e8 −0.794771
\(214\) 0 0
\(215\) −1.60870e8 −1.10393
\(216\) 0 0
\(217\) −3.91563e7 −0.260131
\(218\) 0 0
\(219\) −4.57526e7 −0.294348
\(220\) 0 0
\(221\) 1.44288e8 0.899205
\(222\) 0 0
\(223\) −2.59409e8 −1.56645 −0.783226 0.621737i \(-0.786427\pi\)
−0.783226 + 0.621737i \(0.786427\pi\)
\(224\) 0 0
\(225\) 2.49934e7 0.146281
\(226\) 0 0
\(227\) −1.14021e8 −0.646987 −0.323494 0.946230i \(-0.604857\pi\)
−0.323494 + 0.946230i \(0.604857\pi\)
\(228\) 0 0
\(229\) −9.63317e7 −0.530084 −0.265042 0.964237i \(-0.585386\pi\)
−0.265042 + 0.964237i \(0.585386\pi\)
\(230\) 0 0
\(231\) −5.54624e7 −0.296044
\(232\) 0 0
\(233\) −1.31106e8 −0.679009 −0.339504 0.940604i \(-0.610259\pi\)
−0.339504 + 0.940604i \(0.610259\pi\)
\(234\) 0 0
\(235\) −5.34171e7 −0.268499
\(236\) 0 0
\(237\) 1.15289e8 0.562561
\(238\) 0 0
\(239\) −1.92716e8 −0.913116 −0.456558 0.889694i \(-0.650918\pi\)
−0.456558 + 0.889694i \(0.650918\pi\)
\(240\) 0 0
\(241\) −3.36481e8 −1.54846 −0.774231 0.632904i \(-0.781863\pi\)
−0.774231 + 0.632904i \(0.781863\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) 1.06922e8 0.464500
\(246\) 0 0
\(247\) 3.36259e8 1.41983
\(248\) 0 0
\(249\) −2.10146e8 −0.862626
\(250\) 0 0
\(251\) −2.10771e8 −0.841306 −0.420653 0.907222i \(-0.638199\pi\)
−0.420653 + 0.907222i \(0.638199\pi\)
\(252\) 0 0
\(253\) 2.17952e8 0.846134
\(254\) 0 0
\(255\) −1.28920e8 −0.486889
\(256\) 0 0
\(257\) −1.23129e7 −0.0452476 −0.0226238 0.999744i \(-0.507202\pi\)
−0.0226238 + 0.999744i \(0.507202\pi\)
\(258\) 0 0
\(259\) 3.62898e8 1.29788
\(260\) 0 0
\(261\) −1.09991e8 −0.382927
\(262\) 0 0
\(263\) 4.33617e8 1.46981 0.734905 0.678170i \(-0.237227\pi\)
0.734905 + 0.678170i \(0.237227\pi\)
\(264\) 0 0
\(265\) 1.06954e8 0.353049
\(266\) 0 0
\(267\) 2.58726e8 0.831860
\(268\) 0 0
\(269\) 5.84288e8 1.83018 0.915090 0.403250i \(-0.132119\pi\)
0.915090 + 0.403250i \(0.132119\pi\)
\(270\) 0 0
\(271\) −2.27111e7 −0.0693180 −0.0346590 0.999399i \(-0.511035\pi\)
−0.0346590 + 0.999399i \(0.511035\pi\)
\(272\) 0 0
\(273\) −1.94325e8 −0.578043
\(274\) 0 0
\(275\) −9.91390e7 −0.287462
\(276\) 0 0
\(277\) 2.89498e8 0.818400 0.409200 0.912445i \(-0.365808\pi\)
0.409200 + 0.912445i \(0.365808\pi\)
\(278\) 0 0
\(279\) 4.01828e7 0.110771
\(280\) 0 0
\(281\) 3.03041e8 0.814758 0.407379 0.913259i \(-0.366443\pi\)
0.407379 + 0.913259i \(0.366443\pi\)
\(282\) 0 0
\(283\) −5.31646e8 −1.39434 −0.697172 0.716903i \(-0.745559\pi\)
−0.697172 + 0.716903i \(0.745559\pi\)
\(284\) 0 0
\(285\) −3.00444e8 −0.768787
\(286\) 0 0
\(287\) −4.29553e8 −1.07258
\(288\) 0 0
\(289\) −2.07519e8 −0.505726
\(290\) 0 0
\(291\) 2.73309e8 0.650172
\(292\) 0 0
\(293\) −1.69254e8 −0.393099 −0.196550 0.980494i \(-0.562974\pi\)
−0.196550 + 0.980494i \(0.562974\pi\)
\(294\) 0 0
\(295\) 6.09511e8 1.38231
\(296\) 0 0
\(297\) 5.69164e7 0.126064
\(298\) 0 0
\(299\) 7.63645e8 1.65212
\(300\) 0 0
\(301\) −3.40849e8 −0.720410
\(302\) 0 0
\(303\) −1.08097e8 −0.223235
\(304\) 0 0
\(305\) 1.00881e9 2.03592
\(306\) 0 0
\(307\) −7.89572e8 −1.55743 −0.778713 0.627380i \(-0.784127\pi\)
−0.778713 + 0.627380i \(0.784127\pi\)
\(308\) 0 0
\(309\) −3.40037e8 −0.655650
\(310\) 0 0
\(311\) 1.66047e8 0.313018 0.156509 0.987677i \(-0.449976\pi\)
0.156509 + 0.987677i \(0.449976\pi\)
\(312\) 0 0
\(313\) 4.18218e8 0.770899 0.385450 0.922729i \(-0.374046\pi\)
0.385450 + 0.922729i \(0.374046\pi\)
\(314\) 0 0
\(315\) 1.73627e8 0.312990
\(316\) 0 0
\(317\) 5.75019e8 1.01385 0.506926 0.861990i \(-0.330782\pi\)
0.506926 + 0.861990i \(0.330782\pi\)
\(318\) 0 0
\(319\) 4.36291e8 0.752505
\(320\) 0 0
\(321\) −3.89679e8 −0.657567
\(322\) 0 0
\(323\) 4.72664e8 0.780448
\(324\) 0 0
\(325\) −3.47356e8 −0.561285
\(326\) 0 0
\(327\) −1.66615e8 −0.263510
\(328\) 0 0
\(329\) −1.13179e8 −0.175219
\(330\) 0 0
\(331\) 1.14042e9 1.72850 0.864249 0.503065i \(-0.167794\pi\)
0.864249 + 0.503065i \(0.167794\pi\)
\(332\) 0 0
\(333\) −3.72412e8 −0.552674
\(334\) 0 0
\(335\) −1.04794e9 −1.52294
\(336\) 0 0
\(337\) 1.16512e8 0.165832 0.0829159 0.996557i \(-0.473577\pi\)
0.0829159 + 0.996557i \(0.473577\pi\)
\(338\) 0 0
\(339\) −1.30470e8 −0.181891
\(340\) 0 0
\(341\) −1.59389e8 −0.217680
\(342\) 0 0
\(343\) 8.11571e8 1.08592
\(344\) 0 0
\(345\) −6.82308e8 −0.894568
\(346\) 0 0
\(347\) 6.08010e8 0.781192 0.390596 0.920562i \(-0.372269\pi\)
0.390596 + 0.920562i \(0.372269\pi\)
\(348\) 0 0
\(349\) −8.63094e8 −1.08685 −0.543424 0.839458i \(-0.682872\pi\)
−0.543424 + 0.839458i \(0.682872\pi\)
\(350\) 0 0
\(351\) 1.99420e8 0.246146
\(352\) 0 0
\(353\) 6.33693e8 0.766774 0.383387 0.923588i \(-0.374757\pi\)
0.383387 + 0.923588i \(0.374757\pi\)
\(354\) 0 0
\(355\) −1.39190e9 −1.65124
\(356\) 0 0
\(357\) −2.73154e8 −0.317738
\(358\) 0 0
\(359\) 9.24215e8 1.05425 0.527123 0.849789i \(-0.323271\pi\)
0.527123 + 0.849789i \(0.323271\pi\)
\(360\) 0 0
\(361\) 2.07656e8 0.232311
\(362\) 0 0
\(363\) 3.00389e8 0.329618
\(364\) 0 0
\(365\) −5.68138e8 −0.611545
\(366\) 0 0
\(367\) 1.07990e9 1.14039 0.570193 0.821511i \(-0.306868\pi\)
0.570193 + 0.821511i \(0.306868\pi\)
\(368\) 0 0
\(369\) 4.40814e8 0.456734
\(370\) 0 0
\(371\) 2.26612e8 0.230396
\(372\) 0 0
\(373\) −3.27025e8 −0.326287 −0.163143 0.986602i \(-0.552163\pi\)
−0.163143 + 0.986602i \(0.552163\pi\)
\(374\) 0 0
\(375\) −3.96863e8 −0.388625
\(376\) 0 0
\(377\) 1.52865e9 1.46931
\(378\) 0 0
\(379\) 9.75385e8 0.920320 0.460160 0.887836i \(-0.347792\pi\)
0.460160 + 0.887836i \(0.347792\pi\)
\(380\) 0 0
\(381\) 7.22740e8 0.669491
\(382\) 0 0
\(383\) 9.77181e8 0.888750 0.444375 0.895841i \(-0.353426\pi\)
0.444375 + 0.895841i \(0.353426\pi\)
\(384\) 0 0
\(385\) −6.88710e8 −0.615069
\(386\) 0 0
\(387\) 3.49785e8 0.306770
\(388\) 0 0
\(389\) 1.13465e9 0.977327 0.488663 0.872472i \(-0.337485\pi\)
0.488663 + 0.872472i \(0.337485\pi\)
\(390\) 0 0
\(391\) 1.07342e9 0.908137
\(392\) 0 0
\(393\) 1.04463e9 0.868141
\(394\) 0 0
\(395\) 1.43162e9 1.16879
\(396\) 0 0
\(397\) −9.26214e8 −0.742924 −0.371462 0.928448i \(-0.621143\pi\)
−0.371462 + 0.928448i \(0.621143\pi\)
\(398\) 0 0
\(399\) −6.36576e8 −0.501701
\(400\) 0 0
\(401\) −1.30250e9 −1.00872 −0.504362 0.863492i \(-0.668272\pi\)
−0.504362 + 0.863492i \(0.668272\pi\)
\(402\) 0 0
\(403\) −5.58457e8 −0.425032
\(404\) 0 0
\(405\) −1.78179e8 −0.133280
\(406\) 0 0
\(407\) 1.47721e9 1.08608
\(408\) 0 0
\(409\) −2.43668e9 −1.76103 −0.880515 0.474018i \(-0.842803\pi\)
−0.880515 + 0.474018i \(0.842803\pi\)
\(410\) 0 0
\(411\) 5.28016e8 0.375147
\(412\) 0 0
\(413\) 1.29142e9 0.902077
\(414\) 0 0
\(415\) −2.60951e9 −1.79221
\(416\) 0 0
\(417\) −1.31965e9 −0.891218
\(418\) 0 0
\(419\) −1.61835e9 −1.07479 −0.537393 0.843332i \(-0.680591\pi\)
−0.537393 + 0.843332i \(0.680591\pi\)
\(420\) 0 0
\(421\) 2.08462e8 0.136157 0.0680786 0.997680i \(-0.478313\pi\)
0.0680786 + 0.997680i \(0.478313\pi\)
\(422\) 0 0
\(423\) 1.16147e8 0.0746131
\(424\) 0 0
\(425\) −4.88263e8 −0.308526
\(426\) 0 0
\(427\) 2.13746e9 1.32862
\(428\) 0 0
\(429\) −7.91018e8 −0.483711
\(430\) 0 0
\(431\) −9.30757e8 −0.559971 −0.279986 0.960004i \(-0.590330\pi\)
−0.279986 + 0.960004i \(0.590330\pi\)
\(432\) 0 0
\(433\) −1.38986e9 −0.822741 −0.411371 0.911468i \(-0.634950\pi\)
−0.411371 + 0.911468i \(0.634950\pi\)
\(434\) 0 0
\(435\) −1.36583e9 −0.795580
\(436\) 0 0
\(437\) 2.50157e9 1.43393
\(438\) 0 0
\(439\) −7.74625e8 −0.436984 −0.218492 0.975839i \(-0.570114\pi\)
−0.218492 + 0.975839i \(0.570114\pi\)
\(440\) 0 0
\(441\) −2.32484e8 −0.129080
\(442\) 0 0
\(443\) −1.87372e9 −1.02398 −0.511991 0.858991i \(-0.671092\pi\)
−0.511991 + 0.858991i \(0.671092\pi\)
\(444\) 0 0
\(445\) 3.21276e9 1.72829
\(446\) 0 0
\(447\) 5.08338e8 0.269201
\(448\) 0 0
\(449\) 2.43736e9 1.27074 0.635372 0.772207i \(-0.280847\pi\)
0.635372 + 0.772207i \(0.280847\pi\)
\(450\) 0 0
\(451\) −1.74853e9 −0.897545
\(452\) 0 0
\(453\) 1.43075e9 0.723137
\(454\) 0 0
\(455\) −2.41305e9 −1.20096
\(456\) 0 0
\(457\) −4.41576e8 −0.216421 −0.108210 0.994128i \(-0.534512\pi\)
−0.108210 + 0.994128i \(0.534512\pi\)
\(458\) 0 0
\(459\) 2.80315e8 0.135301
\(460\) 0 0
\(461\) −1.77394e9 −0.843309 −0.421654 0.906757i \(-0.638550\pi\)
−0.421654 + 0.906757i \(0.638550\pi\)
\(462\) 0 0
\(463\) −2.81182e9 −1.31660 −0.658300 0.752756i \(-0.728724\pi\)
−0.658300 + 0.752756i \(0.728724\pi\)
\(464\) 0 0
\(465\) 4.98974e8 0.230140
\(466\) 0 0
\(467\) −1.67741e9 −0.762131 −0.381066 0.924548i \(-0.624443\pi\)
−0.381066 + 0.924548i \(0.624443\pi\)
\(468\) 0 0
\(469\) −2.22037e9 −0.993849
\(470\) 0 0
\(471\) 1.78499e9 0.787158
\(472\) 0 0
\(473\) −1.38746e9 −0.602845
\(474\) 0 0
\(475\) −1.13788e9 −0.487157
\(476\) 0 0
\(477\) −2.32553e8 −0.0981087
\(478\) 0 0
\(479\) −2.38838e9 −0.992953 −0.496477 0.868050i \(-0.665373\pi\)
−0.496477 + 0.868050i \(0.665373\pi\)
\(480\) 0 0
\(481\) 5.17574e9 2.12063
\(482\) 0 0
\(483\) −1.44566e9 −0.583785
\(484\) 0 0
\(485\) 3.39384e9 1.35081
\(486\) 0 0
\(487\) 6.54404e8 0.256741 0.128370 0.991726i \(-0.459025\pi\)
0.128370 + 0.991726i \(0.459025\pi\)
\(488\) 0 0
\(489\) −1.11546e9 −0.431395
\(490\) 0 0
\(491\) 8.01260e8 0.305484 0.152742 0.988266i \(-0.451190\pi\)
0.152742 + 0.988266i \(0.451190\pi\)
\(492\) 0 0
\(493\) 2.14875e9 0.807647
\(494\) 0 0
\(495\) 7.06765e8 0.261913
\(496\) 0 0
\(497\) −2.94914e9 −1.07758
\(498\) 0 0
\(499\) 2.30647e9 0.830989 0.415494 0.909596i \(-0.363609\pi\)
0.415494 + 0.909596i \(0.363609\pi\)
\(500\) 0 0
\(501\) 5.50232e6 0.00195485
\(502\) 0 0
\(503\) 1.82507e9 0.639426 0.319713 0.947514i \(-0.396413\pi\)
0.319713 + 0.947514i \(0.396413\pi\)
\(504\) 0 0
\(505\) −1.34230e9 −0.463799
\(506\) 0 0
\(507\) −1.07730e9 −0.367121
\(508\) 0 0
\(509\) −5.36457e9 −1.80311 −0.901556 0.432663i \(-0.857574\pi\)
−0.901556 + 0.432663i \(0.857574\pi\)
\(510\) 0 0
\(511\) −1.20376e9 −0.399087
\(512\) 0 0
\(513\) 6.53264e8 0.213638
\(514\) 0 0
\(515\) −4.22245e9 −1.36219
\(516\) 0 0
\(517\) −4.60707e8 −0.146625
\(518\) 0 0
\(519\) 6.55569e8 0.205841
\(520\) 0 0
\(521\) −1.11907e9 −0.346677 −0.173339 0.984862i \(-0.555455\pi\)
−0.173339 + 0.984862i \(0.555455\pi\)
\(522\) 0 0
\(523\) 1.51072e8 0.0461772 0.0230886 0.999733i \(-0.492650\pi\)
0.0230886 + 0.999733i \(0.492650\pi\)
\(524\) 0 0
\(525\) 6.57584e8 0.198332
\(526\) 0 0
\(527\) −7.84997e8 −0.233631
\(528\) 0 0
\(529\) 2.27624e9 0.668535
\(530\) 0 0
\(531\) −1.32528e9 −0.384129
\(532\) 0 0
\(533\) −6.12639e9 −1.75251
\(534\) 0 0
\(535\) −4.83889e9 −1.36618
\(536\) 0 0
\(537\) 2.78427e9 0.775891
\(538\) 0 0
\(539\) 9.22171e8 0.253659
\(540\) 0 0
\(541\) 1.71582e9 0.465888 0.232944 0.972490i \(-0.425164\pi\)
0.232944 + 0.972490i \(0.425164\pi\)
\(542\) 0 0
\(543\) −1.82209e8 −0.0488393
\(544\) 0 0
\(545\) −2.06896e9 −0.547475
\(546\) 0 0
\(547\) 3.72932e9 0.974259 0.487129 0.873330i \(-0.338044\pi\)
0.487129 + 0.873330i \(0.338044\pi\)
\(548\) 0 0
\(549\) −2.19349e9 −0.565761
\(550\) 0 0
\(551\) 5.00759e9 1.27526
\(552\) 0 0
\(553\) 3.03329e9 0.762739
\(554\) 0 0
\(555\) −4.62447e9 −1.14825
\(556\) 0 0
\(557\) −3.74622e9 −0.918544 −0.459272 0.888296i \(-0.651890\pi\)
−0.459272 + 0.888296i \(0.651890\pi\)
\(558\) 0 0
\(559\) −4.86127e9 −1.17709
\(560\) 0 0
\(561\) −1.11190e9 −0.265886
\(562\) 0 0
\(563\) −6.71873e8 −0.158675 −0.0793374 0.996848i \(-0.525280\pi\)
−0.0793374 + 0.996848i \(0.525280\pi\)
\(564\) 0 0
\(565\) −1.62012e9 −0.377901
\(566\) 0 0
\(567\) −3.77523e8 −0.0869767
\(568\) 0 0
\(569\) 1.38230e9 0.314564 0.157282 0.987554i \(-0.449727\pi\)
0.157282 + 0.987554i \(0.449727\pi\)
\(570\) 0 0
\(571\) 4.22076e9 0.948778 0.474389 0.880315i \(-0.342669\pi\)
0.474389 + 0.880315i \(0.342669\pi\)
\(572\) 0 0
\(573\) −5.19871e8 −0.115439
\(574\) 0 0
\(575\) −2.58413e9 −0.566861
\(576\) 0 0
\(577\) 3.00405e9 0.651017 0.325508 0.945539i \(-0.394465\pi\)
0.325508 + 0.945539i \(0.394465\pi\)
\(578\) 0 0
\(579\) 4.55446e9 0.975129
\(580\) 0 0
\(581\) −5.52899e9 −1.16958
\(582\) 0 0
\(583\) 9.22445e8 0.192797
\(584\) 0 0
\(585\) 2.47631e9 0.511399
\(586\) 0 0
\(587\) −4.90415e9 −1.00076 −0.500380 0.865806i \(-0.666806\pi\)
−0.500380 + 0.865806i \(0.666806\pi\)
\(588\) 0 0
\(589\) −1.82941e9 −0.368899
\(590\) 0 0
\(591\) −4.24789e9 −0.846480
\(592\) 0 0
\(593\) 5.24962e9 1.03380 0.516900 0.856046i \(-0.327086\pi\)
0.516900 + 0.856046i \(0.327086\pi\)
\(594\) 0 0
\(595\) −3.39192e9 −0.660140
\(596\) 0 0
\(597\) 3.59967e9 0.692393
\(598\) 0 0
\(599\) 2.43405e9 0.462739 0.231369 0.972866i \(-0.425679\pi\)
0.231369 + 0.972866i \(0.425679\pi\)
\(600\) 0 0
\(601\) 2.91489e9 0.547723 0.273862 0.961769i \(-0.411699\pi\)
0.273862 + 0.961769i \(0.411699\pi\)
\(602\) 0 0
\(603\) 2.27858e9 0.423208
\(604\) 0 0
\(605\) 3.73011e9 0.684823
\(606\) 0 0
\(607\) −3.86425e9 −0.701302 −0.350651 0.936506i \(-0.614040\pi\)
−0.350651 + 0.936506i \(0.614040\pi\)
\(608\) 0 0
\(609\) −2.89390e9 −0.519186
\(610\) 0 0
\(611\) −1.61419e9 −0.286293
\(612\) 0 0
\(613\) 5.13751e9 0.900826 0.450413 0.892820i \(-0.351277\pi\)
0.450413 + 0.892820i \(0.351277\pi\)
\(614\) 0 0
\(615\) 5.47386e9 0.948922
\(616\) 0 0
\(617\) −7.30389e9 −1.25186 −0.625930 0.779879i \(-0.715281\pi\)
−0.625930 + 0.779879i \(0.715281\pi\)
\(618\) 0 0
\(619\) −8.08687e9 −1.37045 −0.685225 0.728332i \(-0.740296\pi\)
−0.685225 + 0.728332i \(0.740296\pi\)
\(620\) 0 0
\(621\) 1.48356e9 0.248591
\(622\) 0 0
\(623\) 6.80714e9 1.12786
\(624\) 0 0
\(625\) −7.60657e9 −1.24626
\(626\) 0 0
\(627\) −2.59124e9 −0.419828
\(628\) 0 0
\(629\) 7.27531e9 1.16567
\(630\) 0 0
\(631\) 3.02687e9 0.479613 0.239807 0.970821i \(-0.422916\pi\)
0.239807 + 0.970821i \(0.422916\pi\)
\(632\) 0 0
\(633\) 7.19525e9 1.12754
\(634\) 0 0
\(635\) 8.97470e9 1.39095
\(636\) 0 0
\(637\) 3.23104e9 0.495283
\(638\) 0 0
\(639\) 3.02646e9 0.458861
\(640\) 0 0
\(641\) −8.84536e9 −1.32652 −0.663258 0.748391i \(-0.730827\pi\)
−0.663258 + 0.748391i \(0.730827\pi\)
\(642\) 0 0
\(643\) 6.32368e9 0.938062 0.469031 0.883182i \(-0.344603\pi\)
0.469031 + 0.883182i \(0.344603\pi\)
\(644\) 0 0
\(645\) 4.34349e9 0.637353
\(646\) 0 0
\(647\) −3.92152e9 −0.569233 −0.284616 0.958641i \(-0.591866\pi\)
−0.284616 + 0.958641i \(0.591866\pi\)
\(648\) 0 0
\(649\) 5.25686e9 0.754866
\(650\) 0 0
\(651\) 1.05722e9 0.150187
\(652\) 0 0
\(653\) −7.02434e9 −0.987210 −0.493605 0.869686i \(-0.664321\pi\)
−0.493605 + 0.869686i \(0.664321\pi\)
\(654\) 0 0
\(655\) 1.29718e10 1.80367
\(656\) 0 0
\(657\) 1.23532e9 0.169942
\(658\) 0 0
\(659\) 9.72714e9 1.32399 0.661997 0.749506i \(-0.269709\pi\)
0.661997 + 0.749506i \(0.269709\pi\)
\(660\) 0 0
\(661\) −5.19552e9 −0.699719 −0.349860 0.936802i \(-0.613771\pi\)
−0.349860 + 0.936802i \(0.613771\pi\)
\(662\) 0 0
\(663\) −3.89579e9 −0.519156
\(664\) 0 0
\(665\) −7.90475e9 −1.04235
\(666\) 0 0
\(667\) 1.13722e10 1.48390
\(668\) 0 0
\(669\) 7.00403e9 0.904392
\(670\) 0 0
\(671\) 8.70071e9 1.11180
\(672\) 0 0
\(673\) −2.24492e9 −0.283889 −0.141945 0.989875i \(-0.545335\pi\)
−0.141945 + 0.989875i \(0.545335\pi\)
\(674\) 0 0
\(675\) −6.74823e8 −0.0844553
\(676\) 0 0
\(677\) −1.59249e9 −0.197250 −0.0986251 0.995125i \(-0.531444\pi\)
−0.0986251 + 0.995125i \(0.531444\pi\)
\(678\) 0 0
\(679\) 7.19083e9 0.881525
\(680\) 0 0
\(681\) 3.07858e9 0.373538
\(682\) 0 0
\(683\) 9.68643e9 1.16330 0.581649 0.813440i \(-0.302408\pi\)
0.581649 + 0.813440i \(0.302408\pi\)
\(684\) 0 0
\(685\) 6.55670e9 0.779414
\(686\) 0 0
\(687\) 2.60096e9 0.306044
\(688\) 0 0
\(689\) 3.23200e9 0.376447
\(690\) 0 0
\(691\) −2.97399e9 −0.342899 −0.171449 0.985193i \(-0.554845\pi\)
−0.171449 + 0.985193i \(0.554845\pi\)
\(692\) 0 0
\(693\) 1.49748e9 0.170921
\(694\) 0 0
\(695\) −1.63869e10 −1.85162
\(696\) 0 0
\(697\) −8.61159e9 −0.963315
\(698\) 0 0
\(699\) 3.53985e9 0.392026
\(700\) 0 0
\(701\) 1.90884e9 0.209294 0.104647 0.994509i \(-0.466629\pi\)
0.104647 + 0.994509i \(0.466629\pi\)
\(702\) 0 0
\(703\) 1.69549e10 1.84056
\(704\) 0 0
\(705\) 1.44226e9 0.155018
\(706\) 0 0
\(707\) −2.84405e9 −0.302670
\(708\) 0 0
\(709\) 3.59009e9 0.378306 0.189153 0.981948i \(-0.439426\pi\)
0.189153 + 0.981948i \(0.439426\pi\)
\(710\) 0 0
\(711\) −3.11281e9 −0.324795
\(712\) 0 0
\(713\) −4.15459e9 −0.429254
\(714\) 0 0
\(715\) −9.82255e9 −1.00497
\(716\) 0 0
\(717\) 5.20334e9 0.527188
\(718\) 0 0
\(719\) −2.83832e9 −0.284780 −0.142390 0.989811i \(-0.545479\pi\)
−0.142390 + 0.989811i \(0.545479\pi\)
\(720\) 0 0
\(721\) −8.94646e9 −0.888951
\(722\) 0 0
\(723\) 9.08497e9 0.894004
\(724\) 0 0
\(725\) −5.17284e9 −0.504134
\(726\) 0 0
\(727\) −4.47195e6 −0.000431645 0 −0.000215822 1.00000i \(-0.500069\pi\)
−0.000215822 1.00000i \(0.500069\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) −6.83327e9 −0.647020
\(732\) 0 0
\(733\) 8.23619e9 0.772436 0.386218 0.922408i \(-0.373781\pi\)
0.386218 + 0.922408i \(0.373781\pi\)
\(734\) 0 0
\(735\) −2.88689e9 −0.268179
\(736\) 0 0
\(737\) −9.03822e9 −0.831661
\(738\) 0 0
\(739\) −3.33578e9 −0.304048 −0.152024 0.988377i \(-0.548579\pi\)
−0.152024 + 0.988377i \(0.548579\pi\)
\(740\) 0 0
\(741\) −9.07900e9 −0.819737
\(742\) 0 0
\(743\) −5.44577e9 −0.487078 −0.243539 0.969891i \(-0.578308\pi\)
−0.243539 + 0.969891i \(0.578308\pi\)
\(744\) 0 0
\(745\) 6.31234e9 0.559299
\(746\) 0 0
\(747\) 5.67393e9 0.498038
\(748\) 0 0
\(749\) −1.02526e10 −0.891551
\(750\) 0 0
\(751\) 7.09357e6 0.000611118 0 0.000305559 1.00000i \(-0.499903\pi\)
0.000305559 1.00000i \(0.499903\pi\)
\(752\) 0 0
\(753\) 5.69083e9 0.485728
\(754\) 0 0
\(755\) 1.77665e10 1.50241
\(756\) 0 0
\(757\) 4.93300e9 0.413310 0.206655 0.978414i \(-0.433742\pi\)
0.206655 + 0.978414i \(0.433742\pi\)
\(758\) 0 0
\(759\) −5.88471e9 −0.488516
\(760\) 0 0
\(761\) −1.28057e10 −1.05331 −0.526654 0.850079i \(-0.676554\pi\)
−0.526654 + 0.850079i \(0.676554\pi\)
\(762\) 0 0
\(763\) −4.38368e9 −0.357275
\(764\) 0 0
\(765\) 3.48084e9 0.281105
\(766\) 0 0
\(767\) 1.84186e10 1.47392
\(768\) 0 0
\(769\) 1.36621e10 1.08336 0.541681 0.840584i \(-0.317788\pi\)
0.541681 + 0.840584i \(0.317788\pi\)
\(770\) 0 0
\(771\) 3.32449e8 0.0261237
\(772\) 0 0
\(773\) 3.39415e9 0.264304 0.132152 0.991229i \(-0.457811\pi\)
0.132152 + 0.991229i \(0.457811\pi\)
\(774\) 0 0
\(775\) 1.88978e9 0.145833
\(776\) 0 0
\(777\) −9.79825e9 −0.749334
\(778\) 0 0
\(779\) −2.00690e10 −1.52105
\(780\) 0 0
\(781\) −1.20047e10 −0.901726
\(782\) 0 0
\(783\) 2.96976e9 0.221083
\(784\) 0 0
\(785\) 2.21653e10 1.63542
\(786\) 0 0
\(787\) −8.38934e9 −0.613503 −0.306751 0.951790i \(-0.599242\pi\)
−0.306751 + 0.951790i \(0.599242\pi\)
\(788\) 0 0
\(789\) −1.17077e10 −0.848595
\(790\) 0 0
\(791\) −3.43269e9 −0.246613
\(792\) 0 0
\(793\) 3.04849e10 2.17085
\(794\) 0 0
\(795\) −2.88775e9 −0.203833
\(796\) 0 0
\(797\) −1.19751e10 −0.837867 −0.418934 0.908017i \(-0.637596\pi\)
−0.418934 + 0.908017i \(0.637596\pi\)
\(798\) 0 0
\(799\) −2.26900e9 −0.157369
\(800\) 0 0
\(801\) −6.98560e9 −0.480275
\(802\) 0 0
\(803\) −4.90002e9 −0.333959
\(804\) 0 0
\(805\) −1.79517e10 −1.21289
\(806\) 0 0
\(807\) −1.57758e10 −1.05665
\(808\) 0 0
\(809\) 8.77076e9 0.582395 0.291197 0.956663i \(-0.405946\pi\)
0.291197 + 0.956663i \(0.405946\pi\)
\(810\) 0 0
\(811\) −1.19935e10 −0.789535 −0.394767 0.918781i \(-0.629175\pi\)
−0.394767 + 0.918781i \(0.629175\pi\)
\(812\) 0 0
\(813\) 6.13200e8 0.0400207
\(814\) 0 0
\(815\) −1.38514e10 −0.896276
\(816\) 0 0
\(817\) −1.59247e10 −1.02163
\(818\) 0 0
\(819\) 5.24678e9 0.333733
\(820\) 0 0
\(821\) −1.24857e9 −0.0787428 −0.0393714 0.999225i \(-0.512536\pi\)
−0.0393714 + 0.999225i \(0.512536\pi\)
\(822\) 0 0
\(823\) −3.45738e9 −0.216196 −0.108098 0.994140i \(-0.534476\pi\)
−0.108098 + 0.994140i \(0.534476\pi\)
\(824\) 0 0
\(825\) 2.67675e9 0.165966
\(826\) 0 0
\(827\) 1.76043e9 0.108231 0.0541153 0.998535i \(-0.482766\pi\)
0.0541153 + 0.998535i \(0.482766\pi\)
\(828\) 0 0
\(829\) −8.84745e9 −0.539358 −0.269679 0.962950i \(-0.586918\pi\)
−0.269679 + 0.962950i \(0.586918\pi\)
\(830\) 0 0
\(831\) −7.81644e9 −0.472504
\(832\) 0 0
\(833\) 4.54172e9 0.272247
\(834\) 0 0
\(835\) 6.83257e7 0.00406145
\(836\) 0 0
\(837\) −1.08494e9 −0.0639536
\(838\) 0 0
\(839\) 2.90818e10 1.70002 0.850010 0.526767i \(-0.176596\pi\)
0.850010 + 0.526767i \(0.176596\pi\)
\(840\) 0 0
\(841\) 5.51478e9 0.319700
\(842\) 0 0
\(843\) −8.18209e9 −0.470401
\(844\) 0 0
\(845\) −1.33775e10 −0.762741
\(846\) 0 0
\(847\) 7.90331e9 0.446907
\(848\) 0 0
\(849\) 1.43544e10 0.805025
\(850\) 0 0
\(851\) 3.85045e10 2.14170
\(852\) 0 0
\(853\) 2.00633e10 1.10683 0.553414 0.832906i \(-0.313325\pi\)
0.553414 + 0.832906i \(0.313325\pi\)
\(854\) 0 0
\(855\) 8.11198e9 0.443860
\(856\) 0 0
\(857\) −2.22093e10 −1.20532 −0.602661 0.797998i \(-0.705893\pi\)
−0.602661 + 0.797998i \(0.705893\pi\)
\(858\) 0 0
\(859\) −2.71682e10 −1.46246 −0.731232 0.682128i \(-0.761054\pi\)
−0.731232 + 0.682128i \(0.761054\pi\)
\(860\) 0 0
\(861\) 1.15979e10 0.619255
\(862\) 0 0
\(863\) −3.23767e10 −1.71473 −0.857364 0.514710i \(-0.827900\pi\)
−0.857364 + 0.514710i \(0.827900\pi\)
\(864\) 0 0
\(865\) 8.14060e9 0.427661
\(866\) 0 0
\(867\) 5.60301e9 0.291981
\(868\) 0 0
\(869\) 1.23473e10 0.638267
\(870\) 0 0
\(871\) −3.16675e10 −1.62386
\(872\) 0 0
\(873\) −7.37934e9 −0.375377
\(874\) 0 0
\(875\) −1.04416e10 −0.526911
\(876\) 0 0
\(877\) 2.43044e10 1.21671 0.608354 0.793666i \(-0.291830\pi\)
0.608354 + 0.793666i \(0.291830\pi\)
\(878\) 0 0
\(879\) 4.56986e9 0.226956
\(880\) 0 0
\(881\) 2.63625e10 1.29889 0.649444 0.760409i \(-0.275002\pi\)
0.649444 + 0.760409i \(0.275002\pi\)
\(882\) 0 0
\(883\) 1.59986e10 0.782021 0.391011 0.920386i \(-0.372126\pi\)
0.391011 + 0.920386i \(0.372126\pi\)
\(884\) 0 0
\(885\) −1.64568e10 −0.798076
\(886\) 0 0
\(887\) −3.86500e10 −1.85959 −0.929794 0.368081i \(-0.880015\pi\)
−0.929794 + 0.368081i \(0.880015\pi\)
\(888\) 0 0
\(889\) 1.90155e10 0.907718
\(890\) 0 0
\(891\) −1.53674e9 −0.0727829
\(892\) 0 0
\(893\) −5.28782e9 −0.248483
\(894\) 0 0
\(895\) 3.45739e10 1.61201
\(896\) 0 0
\(897\) −2.06184e10 −0.953854
\(898\) 0 0
\(899\) −8.31655e9 −0.381755
\(900\) 0 0
\(901\) 4.54307e9 0.206925
\(902\) 0 0
\(903\) 9.20293e9 0.415929
\(904\) 0 0
\(905\) −2.26259e9 −0.101470
\(906\) 0 0
\(907\) −5.99871e9 −0.266951 −0.133476 0.991052i \(-0.542614\pi\)
−0.133476 + 0.991052i \(0.542614\pi\)
\(908\) 0 0
\(909\) 2.91861e9 0.128885
\(910\) 0 0
\(911\) 1.01805e10 0.446121 0.223061 0.974805i \(-0.428395\pi\)
0.223061 + 0.974805i \(0.428395\pi\)
\(912\) 0 0
\(913\) −2.25062e10 −0.978712
\(914\) 0 0
\(915\) −2.72379e10 −1.17544
\(916\) 0 0
\(917\) 2.74846e10 1.17705
\(918\) 0 0
\(919\) −3.51377e10 −1.49338 −0.746688 0.665175i \(-0.768357\pi\)
−0.746688 + 0.665175i \(0.768357\pi\)
\(920\) 0 0
\(921\) 2.13184e10 0.899180
\(922\) 0 0
\(923\) −4.20614e10 −1.76067
\(924\) 0 0
\(925\) −1.75144e10 −0.727610
\(926\) 0 0
\(927\) 9.18100e9 0.378539
\(928\) 0 0
\(929\) 1.77161e10 0.724958 0.362479 0.931992i \(-0.381930\pi\)
0.362479 + 0.931992i \(0.381930\pi\)
\(930\) 0 0
\(931\) 1.05843e10 0.429872
\(932\) 0 0
\(933\) −4.48327e9 −0.180721
\(934\) 0 0
\(935\) −1.38071e10 −0.552411
\(936\) 0 0
\(937\) −4.17094e10 −1.65633 −0.828163 0.560487i \(-0.810614\pi\)
−0.828163 + 0.560487i \(0.810614\pi\)
\(938\) 0 0
\(939\) −1.12919e10 −0.445079
\(940\) 0 0
\(941\) 2.80091e10 1.09581 0.547905 0.836541i \(-0.315425\pi\)
0.547905 + 0.836541i \(0.315425\pi\)
\(942\) 0 0
\(943\) −4.55767e10 −1.76991
\(944\) 0 0
\(945\) −4.68794e9 −0.180705
\(946\) 0 0
\(947\) −4.94004e10 −1.89019 −0.945094 0.326798i \(-0.894030\pi\)
−0.945094 + 0.326798i \(0.894030\pi\)
\(948\) 0 0
\(949\) −1.71683e10 −0.652074
\(950\) 0 0
\(951\) −1.55255e10 −0.585348
\(952\) 0 0
\(953\) −8.75488e9 −0.327661 −0.163831 0.986488i \(-0.552385\pi\)
−0.163831 + 0.986488i \(0.552385\pi\)
\(954\) 0 0
\(955\) −6.45555e9 −0.239840
\(956\) 0 0
\(957\) −1.17799e10 −0.434459
\(958\) 0 0
\(959\) 1.38922e10 0.508636
\(960\) 0 0
\(961\) −2.44743e10 −0.889568
\(962\) 0 0
\(963\) 1.05213e10 0.379646
\(964\) 0 0
\(965\) 5.65555e10 2.02595
\(966\) 0 0
\(967\) 2.37991e10 0.846385 0.423192 0.906040i \(-0.360909\pi\)
0.423192 + 0.906040i \(0.360909\pi\)
\(968\) 0 0
\(969\) −1.27619e10 −0.450592
\(970\) 0 0
\(971\) 6.16413e9 0.216075 0.108037 0.994147i \(-0.465543\pi\)
0.108037 + 0.994147i \(0.465543\pi\)
\(972\) 0 0
\(973\) −3.47204e10 −1.20834
\(974\) 0 0
\(975\) 9.37862e9 0.324058
\(976\) 0 0
\(977\) 1.77272e10 0.608147 0.304074 0.952649i \(-0.401653\pi\)
0.304074 + 0.952649i \(0.401653\pi\)
\(978\) 0 0
\(979\) 2.77091e10 0.943806
\(980\) 0 0
\(981\) 4.49860e9 0.152137
\(982\) 0 0
\(983\) 1.61779e10 0.543232 0.271616 0.962406i \(-0.412442\pi\)
0.271616 + 0.962406i \(0.412442\pi\)
\(984\) 0 0
\(985\) −5.27486e10 −1.75867
\(986\) 0 0
\(987\) 3.05585e9 0.101163
\(988\) 0 0
\(989\) −3.61650e10 −1.18878
\(990\) 0 0
\(991\) 2.06432e9 0.0673780 0.0336890 0.999432i \(-0.489274\pi\)
0.0336890 + 0.999432i \(0.489274\pi\)
\(992\) 0 0
\(993\) −3.07915e10 −0.997948
\(994\) 0 0
\(995\) 4.46993e10 1.43853
\(996\) 0 0
\(997\) 9.93049e9 0.317349 0.158675 0.987331i \(-0.449278\pi\)
0.158675 + 0.987331i \(0.449278\pi\)
\(998\) 0 0
\(999\) 1.00551e10 0.319086
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 192.8.a.r.1.1 2
3.2 odd 2 576.8.a.bm.1.2 2
4.3 odd 2 192.8.a.u.1.1 2
8.3 odd 2 96.8.a.d.1.2 2
8.5 even 2 96.8.a.g.1.2 yes 2
12.11 even 2 576.8.a.bl.1.2 2
24.5 odd 2 288.8.a.j.1.1 2
24.11 even 2 288.8.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.8.a.d.1.2 2 8.3 odd 2
96.8.a.g.1.2 yes 2 8.5 even 2
192.8.a.r.1.1 2 1.1 even 1 trivial
192.8.a.u.1.1 2 4.3 odd 2
288.8.a.i.1.1 2 24.11 even 2
288.8.a.j.1.1 2 24.5 odd 2
576.8.a.bl.1.2 2 12.11 even 2
576.8.a.bm.1.2 2 3.2 odd 2