Properties

Label 32.8.a.b.1.1
Level $32$
Weight $8$
Character 32.1
Self dual yes
Analytic conductor $9.996$
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] = [32,8,Mod(1,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("32.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 32.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: \(9.99632081549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.16228\) of defining polynomial
Character \(\chi\) \(=\) 32.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-58.5964 q^{3} -494.772 q^{5} +1332.35 q^{7} +1246.54 q^{9} +O(q^{10})\) \(q-58.5964 q^{3} -494.772 q^{5} +1332.35 q^{7} +1246.54 q^{9} +725.267 q^{11} +4821.57 q^{13} +28991.9 q^{15} -3533.15 q^{17} +44922.4 q^{19} -78071.0 q^{21} -56324.6 q^{23} +166674. q^{25} +55107.4 q^{27} -11201.8 q^{29} -14447.8 q^{31} -42498.1 q^{33} -659209. q^{35} -380730. q^{37} -282527. q^{39} +449262. q^{41} +331342. q^{43} -616754. q^{45} +562571. q^{47} +951614. q^{49} +207030. q^{51} +1.58053e6 q^{53} -358841. q^{55} -2.63230e6 q^{57} +2.72292e6 q^{59} -556745. q^{61} +1.66083e6 q^{63} -2.38558e6 q^{65} +1.11092e6 q^{67} +3.30042e6 q^{69} -299313. q^{71} +3.93379e6 q^{73} -9.76650e6 q^{75} +966309. q^{77} -1.15331e6 q^{79} -5.95529e6 q^{81} -1.33540e6 q^{83} +1.74810e6 q^{85} +656387. q^{87} -1.58276e6 q^{89} +6.42402e6 q^{91} +846591. q^{93} -2.22263e7 q^{95} -1.11303e7 q^{97} +904076. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{3} - 180 q^{5} + 1248 q^{7} + 874 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{3} - 180 q^{5} + 1248 q^{7} + 874 q^{9} + 9040 q^{11} - 2500 q^{13} + 42400 q^{15} + 17220 q^{17} + 74160 q^{19} - 81664 q^{21} + 19104 q^{23} + 187630 q^{25} - 53920 q^{27} - 245028 q^{29} - 251520 q^{31} + 311680 q^{33} - 685760 q^{35} - 530740 q^{37} - 594400 q^{39} + 726900 q^{41} + 44496 q^{43} - 734020 q^{45} + 494912 q^{47} + 135186 q^{49} + 1091040 q^{51} + 1319340 q^{53} + 2258400 q^{55} - 1386880 q^{57} + 2618000 q^{59} + 836700 q^{61} + 1692256 q^{63} - 4690200 q^{65} + 2374128 q^{67} + 6513408 q^{69} - 2836000 q^{71} - 171180 q^{73} - 8873840 q^{75} + 264960 q^{77} - 2498880 q^{79} - 9784718 q^{81} - 9531984 q^{83} + 8280600 q^{85} - 9303776 q^{87} + 7318068 q^{89} + 7041600 q^{91} - 9251840 q^{93} - 13023200 q^{95} - 9316060 q^{97} - 2193520 q^{99}+O(q^{100}) \) 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
\(3\) −58.5964 −1.25299 −0.626494 0.779426i \(-0.715511\pi\)
−0.626494 + 0.779426i \(0.715511\pi\)
\(4\) 0 0
\(5\) −494.772 −1.77015 −0.885074 0.465450i \(-0.845893\pi\)
−0.885074 + 0.465450i \(0.845893\pi\)
\(6\) 0 0
\(7\) 1332.35 1.46817 0.734083 0.679060i \(-0.237612\pi\)
0.734083 + 0.679060i \(0.237612\pi\)
\(8\) 0 0
\(9\) 1246.54 0.569979
\(10\) 0 0
\(11\) 725.267 0.164295 0.0821473 0.996620i \(-0.473822\pi\)
0.0821473 + 0.996620i \(0.473822\pi\)
\(12\) 0 0
\(13\) 4821.57 0.608677 0.304339 0.952564i \(-0.401565\pi\)
0.304339 + 0.952564i \(0.401565\pi\)
\(14\) 0 0
\(15\) 28991.9 2.21797
\(16\) 0 0
\(17\) −3533.15 −0.174418 −0.0872088 0.996190i \(-0.527795\pi\)
−0.0872088 + 0.996190i \(0.527795\pi\)
\(18\) 0 0
\(19\) 44922.4 1.50254 0.751270 0.659995i \(-0.229442\pi\)
0.751270 + 0.659995i \(0.229442\pi\)
\(20\) 0 0
\(21\) −78071.0 −1.83959
\(22\) 0 0
\(23\) −56324.6 −0.965274 −0.482637 0.875821i \(-0.660321\pi\)
−0.482637 + 0.875821i \(0.660321\pi\)
\(24\) 0 0
\(25\) 166674. 2.13343
\(26\) 0 0
\(27\) 55107.4 0.538812
\(28\) 0 0
\(29\) −11201.8 −0.0852896 −0.0426448 0.999090i \(-0.513578\pi\)
−0.0426448 + 0.999090i \(0.513578\pi\)
\(30\) 0 0
\(31\) −14447.8 −0.0871037 −0.0435518 0.999051i \(-0.513867\pi\)
−0.0435518 + 0.999051i \(0.513867\pi\)
\(32\) 0 0
\(33\) −42498.1 −0.205859
\(34\) 0 0
\(35\) −659209. −2.59887
\(36\) 0 0
\(37\) −380730. −1.23569 −0.617847 0.786298i \(-0.711995\pi\)
−0.617847 + 0.786298i \(0.711995\pi\)
\(38\) 0 0
\(39\) −282527. −0.762665
\(40\) 0 0
\(41\) 449262. 1.01802 0.509009 0.860761i \(-0.330012\pi\)
0.509009 + 0.860761i \(0.330012\pi\)
\(42\) 0 0
\(43\) 331342. 0.635531 0.317765 0.948169i \(-0.397068\pi\)
0.317765 + 0.948169i \(0.397068\pi\)
\(44\) 0 0
\(45\) −616754. −1.00895
\(46\) 0 0
\(47\) 562571. 0.790377 0.395189 0.918600i \(-0.370679\pi\)
0.395189 + 0.918600i \(0.370679\pi\)
\(48\) 0 0
\(49\) 951614. 1.15551
\(50\) 0 0
\(51\) 207030. 0.218543
\(52\) 0 0
\(53\) 1.58053e6 1.45826 0.729132 0.684374i \(-0.239924\pi\)
0.729132 + 0.684374i \(0.239924\pi\)
\(54\) 0 0
\(55\) −358841. −0.290826
\(56\) 0 0
\(57\) −2.63230e6 −1.88266
\(58\) 0 0
\(59\) 2.72292e6 1.72605 0.863023 0.505164i \(-0.168568\pi\)
0.863023 + 0.505164i \(0.168568\pi\)
\(60\) 0 0
\(61\) −556745. −0.314052 −0.157026 0.987594i \(-0.550191\pi\)
−0.157026 + 0.987594i \(0.550191\pi\)
\(62\) 0 0
\(63\) 1.66083e6 0.836823
\(64\) 0 0
\(65\) −2.38558e6 −1.07745
\(66\) 0 0
\(67\) 1.11092e6 0.451252 0.225626 0.974214i \(-0.427557\pi\)
0.225626 + 0.974214i \(0.427557\pi\)
\(68\) 0 0
\(69\) 3.30042e6 1.20948
\(70\) 0 0
\(71\) −299313. −0.0992478 −0.0496239 0.998768i \(-0.515802\pi\)
−0.0496239 + 0.998768i \(0.515802\pi\)
\(72\) 0 0
\(73\) 3.93379e6 1.18354 0.591768 0.806108i \(-0.298430\pi\)
0.591768 + 0.806108i \(0.298430\pi\)
\(74\) 0 0
\(75\) −9.76650e6 −2.67316
\(76\) 0 0
\(77\) 966309. 0.241212
\(78\) 0 0
\(79\) −1.15331e6 −0.263178 −0.131589 0.991304i \(-0.542008\pi\)
−0.131589 + 0.991304i \(0.542008\pi\)
\(80\) 0 0
\(81\) −5.95529e6 −1.24510
\(82\) 0 0
\(83\) −1.33540e6 −0.256353 −0.128177 0.991751i \(-0.540912\pi\)
−0.128177 + 0.991751i \(0.540912\pi\)
\(84\) 0 0
\(85\) 1.74810e6 0.308745
\(86\) 0 0
\(87\) 656387. 0.106867
\(88\) 0 0
\(89\) −1.58276e6 −0.237985 −0.118992 0.992895i \(-0.537966\pi\)
−0.118992 + 0.992895i \(0.537966\pi\)
\(90\) 0 0
\(91\) 6.42402e6 0.893639
\(92\) 0 0
\(93\) 846591. 0.109140
\(94\) 0 0
\(95\) −2.22263e7 −2.65972
\(96\) 0 0
\(97\) −1.11303e7 −1.23825 −0.619123 0.785294i \(-0.712512\pi\)
−0.619123 + 0.785294i \(0.712512\pi\)
\(98\) 0 0
\(99\) 904076. 0.0936444
\(100\) 0 0
\(101\) −5.23576e6 −0.505656 −0.252828 0.967511i \(-0.581361\pi\)
−0.252828 + 0.967511i \(0.581361\pi\)
\(102\) 0 0
\(103\) 1.86835e7 1.68472 0.842362 0.538912i \(-0.181164\pi\)
0.842362 + 0.538912i \(0.181164\pi\)
\(104\) 0 0
\(105\) 3.86273e7 3.25636
\(106\) 0 0
\(107\) 2.09418e6 0.165261 0.0826306 0.996580i \(-0.473668\pi\)
0.0826306 + 0.996580i \(0.473668\pi\)
\(108\) 0 0
\(109\) −1.79197e7 −1.32537 −0.662686 0.748897i \(-0.730584\pi\)
−0.662686 + 0.748897i \(0.730584\pi\)
\(110\) 0 0
\(111\) 2.23094e7 1.54831
\(112\) 0 0
\(113\) 1.22948e7 0.801580 0.400790 0.916170i \(-0.368736\pi\)
0.400790 + 0.916170i \(0.368736\pi\)
\(114\) 0 0
\(115\) 2.78678e7 1.70868
\(116\) 0 0
\(117\) 6.01030e6 0.346933
\(118\) 0 0
\(119\) −4.70739e6 −0.256074
\(120\) 0 0
\(121\) −1.89612e7 −0.973007
\(122\) 0 0
\(123\) −2.63251e7 −1.27557
\(124\) 0 0
\(125\) −4.38115e7 −2.00633
\(126\) 0 0
\(127\) 1.55866e7 0.675208 0.337604 0.941288i \(-0.390384\pi\)
0.337604 + 0.941288i \(0.390384\pi\)
\(128\) 0 0
\(129\) −1.94154e7 −0.796312
\(130\) 0 0
\(131\) −2.26851e7 −0.881638 −0.440819 0.897596i \(-0.645312\pi\)
−0.440819 + 0.897596i \(0.645312\pi\)
\(132\) 0 0
\(133\) 5.98524e7 2.20598
\(134\) 0 0
\(135\) −2.72656e7 −0.953777
\(136\) 0 0
\(137\) 3.82819e7 1.27195 0.635977 0.771708i \(-0.280597\pi\)
0.635977 + 0.771708i \(0.280597\pi\)
\(138\) 0 0
\(139\) 3.36838e7 1.06382 0.531911 0.846800i \(-0.321474\pi\)
0.531911 + 0.846800i \(0.321474\pi\)
\(140\) 0 0
\(141\) −3.29646e7 −0.990333
\(142\) 0 0
\(143\) 3.49693e6 0.100002
\(144\) 0 0
\(145\) 5.54234e6 0.150975
\(146\) 0 0
\(147\) −5.57612e7 −1.44784
\(148\) 0 0
\(149\) 1.81432e7 0.449327 0.224663 0.974436i \(-0.427872\pi\)
0.224663 + 0.974436i \(0.427872\pi\)
\(150\) 0 0
\(151\) −5.63250e7 −1.33132 −0.665659 0.746256i \(-0.731849\pi\)
−0.665659 + 0.746256i \(0.731849\pi\)
\(152\) 0 0
\(153\) −4.40422e6 −0.0994143
\(154\) 0 0
\(155\) 7.14837e6 0.154186
\(156\) 0 0
\(157\) 1.45391e7 0.299839 0.149920 0.988698i \(-0.452099\pi\)
0.149920 + 0.988698i \(0.452099\pi\)
\(158\) 0 0
\(159\) −9.26132e7 −1.82719
\(160\) 0 0
\(161\) −7.50440e7 −1.41718
\(162\) 0 0
\(163\) 3.20141e7 0.579008 0.289504 0.957177i \(-0.406510\pi\)
0.289504 + 0.957177i \(0.406510\pi\)
\(164\) 0 0
\(165\) 2.10268e7 0.364401
\(166\) 0 0
\(167\) 4.97668e7 0.826860 0.413430 0.910536i \(-0.364331\pi\)
0.413430 + 0.910536i \(0.364331\pi\)
\(168\) 0 0
\(169\) −3.95009e7 −0.629512
\(170\) 0 0
\(171\) 5.59978e7 0.856415
\(172\) 0 0
\(173\) 6.05260e7 0.888752 0.444376 0.895840i \(-0.353425\pi\)
0.444376 + 0.895840i \(0.353425\pi\)
\(174\) 0 0
\(175\) 2.22068e8 3.13222
\(176\) 0 0
\(177\) −1.59553e8 −2.16272
\(178\) 0 0
\(179\) −9.70858e7 −1.26523 −0.632616 0.774466i \(-0.718019\pi\)
−0.632616 + 0.774466i \(0.718019\pi\)
\(180\) 0 0
\(181\) 4.98470e7 0.624833 0.312417 0.949945i \(-0.398862\pi\)
0.312417 + 0.949945i \(0.398862\pi\)
\(182\) 0 0
\(183\) 3.26233e7 0.393503
\(184\) 0 0
\(185\) 1.88374e8 2.18736
\(186\) 0 0
\(187\) −2.56247e6 −0.0286559
\(188\) 0 0
\(189\) 7.34224e7 0.791065
\(190\) 0 0
\(191\) −1.03610e8 −1.07594 −0.537968 0.842965i \(-0.680808\pi\)
−0.537968 + 0.842965i \(0.680808\pi\)
\(192\) 0 0
\(193\) 7.55867e7 0.756824 0.378412 0.925637i \(-0.376470\pi\)
0.378412 + 0.925637i \(0.376470\pi\)
\(194\) 0 0
\(195\) 1.39786e8 1.35003
\(196\) 0 0
\(197\) −1.25368e8 −1.16830 −0.584150 0.811646i \(-0.698572\pi\)
−0.584150 + 0.811646i \(0.698572\pi\)
\(198\) 0 0
\(199\) 9.05037e7 0.814105 0.407052 0.913405i \(-0.366557\pi\)
0.407052 + 0.913405i \(0.366557\pi\)
\(200\) 0 0
\(201\) −6.50957e7 −0.565414
\(202\) 0 0
\(203\) −1.49248e7 −0.125219
\(204\) 0 0
\(205\) −2.22282e8 −1.80204
\(206\) 0 0
\(207\) −7.02110e7 −0.550185
\(208\) 0 0
\(209\) 3.25808e7 0.246859
\(210\) 0 0
\(211\) 1.60759e8 1.17811 0.589056 0.808092i \(-0.299500\pi\)
0.589056 + 0.808092i \(0.299500\pi\)
\(212\) 0 0
\(213\) 1.75387e7 0.124356
\(214\) 0 0
\(215\) −1.63938e8 −1.12498
\(216\) 0 0
\(217\) −1.92496e7 −0.127883
\(218\) 0 0
\(219\) −2.30506e8 −1.48296
\(220\) 0 0
\(221\) −1.70353e7 −0.106164
\(222\) 0 0
\(223\) −1.59627e8 −0.963915 −0.481958 0.876195i \(-0.660074\pi\)
−0.481958 + 0.876195i \(0.660074\pi\)
\(224\) 0 0
\(225\) 2.07766e8 1.21601
\(226\) 0 0
\(227\) −1.42550e8 −0.808866 −0.404433 0.914568i \(-0.632531\pi\)
−0.404433 + 0.914568i \(0.632531\pi\)
\(228\) 0 0
\(229\) 1.20136e8 0.661070 0.330535 0.943794i \(-0.392771\pi\)
0.330535 + 0.943794i \(0.392771\pi\)
\(230\) 0 0
\(231\) −5.66223e7 −0.302236
\(232\) 0 0
\(233\) −1.21419e8 −0.628841 −0.314420 0.949284i \(-0.601810\pi\)
−0.314420 + 0.949284i \(0.601810\pi\)
\(234\) 0 0
\(235\) −2.78344e8 −1.39909
\(236\) 0 0
\(237\) 6.75797e7 0.329759
\(238\) 0 0
\(239\) 7.73882e7 0.366676 0.183338 0.983050i \(-0.441310\pi\)
0.183338 + 0.983050i \(0.441310\pi\)
\(240\) 0 0
\(241\) 2.93813e8 1.35211 0.676054 0.736852i \(-0.263689\pi\)
0.676054 + 0.736852i \(0.263689\pi\)
\(242\) 0 0
\(243\) 2.28439e8 1.02129
\(244\) 0 0
\(245\) −4.70832e8 −2.04543
\(246\) 0 0
\(247\) 2.16597e8 0.914561
\(248\) 0 0
\(249\) 7.82498e7 0.321207
\(250\) 0 0
\(251\) 3.76211e8 1.50167 0.750834 0.660491i \(-0.229652\pi\)
0.750834 + 0.660491i \(0.229652\pi\)
\(252\) 0 0
\(253\) −4.08503e7 −0.158589
\(254\) 0 0
\(255\) −1.02432e8 −0.386854
\(256\) 0 0
\(257\) 3.31425e8 1.21792 0.608961 0.793200i \(-0.291586\pi\)
0.608961 + 0.793200i \(0.291586\pi\)
\(258\) 0 0
\(259\) −5.07266e8 −1.81420
\(260\) 0 0
\(261\) −1.39636e7 −0.0486132
\(262\) 0 0
\(263\) 5.14878e8 1.74526 0.872628 0.488386i \(-0.162414\pi\)
0.872628 + 0.488386i \(0.162414\pi\)
\(264\) 0 0
\(265\) −7.81999e8 −2.58134
\(266\) 0 0
\(267\) 9.27440e7 0.298192
\(268\) 0 0
\(269\) −1.86030e7 −0.0582708 −0.0291354 0.999575i \(-0.509275\pi\)
−0.0291354 + 0.999575i \(0.509275\pi\)
\(270\) 0 0
\(271\) −8.04683e7 −0.245602 −0.122801 0.992431i \(-0.539188\pi\)
−0.122801 + 0.992431i \(0.539188\pi\)
\(272\) 0 0
\(273\) −3.76425e8 −1.11972
\(274\) 0 0
\(275\) 1.20883e8 0.350510
\(276\) 0 0
\(277\) 3.67049e8 1.03764 0.518818 0.854885i \(-0.326372\pi\)
0.518818 + 0.854885i \(0.326372\pi\)
\(278\) 0 0
\(279\) −1.80098e7 −0.0496472
\(280\) 0 0
\(281\) 3.45888e8 0.929958 0.464979 0.885322i \(-0.346062\pi\)
0.464979 + 0.885322i \(0.346062\pi\)
\(282\) 0 0
\(283\) 6.91779e8 1.81432 0.907162 0.420781i \(-0.138244\pi\)
0.907162 + 0.420781i \(0.138244\pi\)
\(284\) 0 0
\(285\) 1.30238e9 3.33259
\(286\) 0 0
\(287\) 5.98574e8 1.49462
\(288\) 0 0
\(289\) −3.97856e8 −0.969578
\(290\) 0 0
\(291\) 6.52198e8 1.55151
\(292\) 0 0
\(293\) −5.13809e8 −1.19334 −0.596671 0.802486i \(-0.703510\pi\)
−0.596671 + 0.802486i \(0.703510\pi\)
\(294\) 0 0
\(295\) −1.34722e9 −3.05536
\(296\) 0 0
\(297\) 3.99676e7 0.0885239
\(298\) 0 0
\(299\) −2.71573e8 −0.587540
\(300\) 0 0
\(301\) 4.41463e8 0.933065
\(302\) 0 0
\(303\) 3.06797e8 0.633580
\(304\) 0 0
\(305\) 2.75461e8 0.555919
\(306\) 0 0
\(307\) −8.42633e8 −1.66209 −0.831044 0.556207i \(-0.812256\pi\)
−0.831044 + 0.556207i \(0.812256\pi\)
\(308\) 0 0
\(309\) −1.09479e9 −2.11094
\(310\) 0 0
\(311\) −5.43692e8 −1.02492 −0.512462 0.858710i \(-0.671266\pi\)
−0.512462 + 0.858710i \(0.671266\pi\)
\(312\) 0 0
\(313\) −9.28626e8 −1.71173 −0.855866 0.517198i \(-0.826975\pi\)
−0.855866 + 0.517198i \(0.826975\pi\)
\(314\) 0 0
\(315\) −8.21732e8 −1.48130
\(316\) 0 0
\(317\) −6.28050e7 −0.110735 −0.0553677 0.998466i \(-0.517633\pi\)
−0.0553677 + 0.998466i \(0.517633\pi\)
\(318\) 0 0
\(319\) −8.12431e6 −0.0140126
\(320\) 0 0
\(321\) −1.22712e8 −0.207070
\(322\) 0 0
\(323\) −1.58718e8 −0.262069
\(324\) 0 0
\(325\) 8.03630e8 1.29857
\(326\) 0 0
\(327\) 1.05003e9 1.66067
\(328\) 0 0
\(329\) 7.49541e8 1.16041
\(330\) 0 0
\(331\) 6.87877e7 0.104259 0.0521294 0.998640i \(-0.483399\pi\)
0.0521294 + 0.998640i \(0.483399\pi\)
\(332\) 0 0
\(333\) −4.74596e8 −0.704319
\(334\) 0 0
\(335\) −5.49650e8 −0.798784
\(336\) 0 0
\(337\) 8.32867e8 1.18542 0.592708 0.805417i \(-0.298059\pi\)
0.592708 + 0.805417i \(0.298059\pi\)
\(338\) 0 0
\(339\) −7.20431e8 −1.00437
\(340\) 0 0
\(341\) −1.04785e7 −0.0143107
\(342\) 0 0
\(343\) 1.70635e8 0.228318
\(344\) 0 0
\(345\) −1.63295e9 −2.14095
\(346\) 0 0
\(347\) 4.00233e8 0.514232 0.257116 0.966380i \(-0.417228\pi\)
0.257116 + 0.966380i \(0.417228\pi\)
\(348\) 0 0
\(349\) 7.68178e8 0.967327 0.483663 0.875254i \(-0.339306\pi\)
0.483663 + 0.875254i \(0.339306\pi\)
\(350\) 0 0
\(351\) 2.65705e8 0.327962
\(352\) 0 0
\(353\) 9.54035e8 1.15439 0.577195 0.816606i \(-0.304147\pi\)
0.577195 + 0.816606i \(0.304147\pi\)
\(354\) 0 0
\(355\) 1.48091e8 0.175683
\(356\) 0 0
\(357\) 2.75836e8 0.320858
\(358\) 0 0
\(359\) −1.08981e9 −1.24314 −0.621570 0.783359i \(-0.713505\pi\)
−0.621570 + 0.783359i \(0.713505\pi\)
\(360\) 0 0
\(361\) 1.12415e9 1.25762
\(362\) 0 0
\(363\) 1.11106e9 1.21917
\(364\) 0 0
\(365\) −1.94633e9 −2.09503
\(366\) 0 0
\(367\) −1.06975e9 −1.12967 −0.564833 0.825205i \(-0.691060\pi\)
−0.564833 + 0.825205i \(0.691060\pi\)
\(368\) 0 0
\(369\) 5.60024e8 0.580249
\(370\) 0 0
\(371\) 2.10581e9 2.14097
\(372\) 0 0
\(373\) −1.08817e9 −1.08572 −0.542859 0.839824i \(-0.682658\pi\)
−0.542859 + 0.839824i \(0.682658\pi\)
\(374\) 0 0
\(375\) 2.56720e9 2.51391
\(376\) 0 0
\(377\) −5.40104e7 −0.0519138
\(378\) 0 0
\(379\) −1.02117e9 −0.963523 −0.481761 0.876302i \(-0.660003\pi\)
−0.481761 + 0.876302i \(0.660003\pi\)
\(380\) 0 0
\(381\) −9.13318e8 −0.846027
\(382\) 0 0
\(383\) 1.94316e9 1.76731 0.883657 0.468135i \(-0.155074\pi\)
0.883657 + 0.468135i \(0.155074\pi\)
\(384\) 0 0
\(385\) −4.78102e8 −0.426981
\(386\) 0 0
\(387\) 4.13032e8 0.362239
\(388\) 0 0
\(389\) −6.68612e7 −0.0575904 −0.0287952 0.999585i \(-0.509167\pi\)
−0.0287952 + 0.999585i \(0.509167\pi\)
\(390\) 0 0
\(391\) 1.99003e8 0.168361
\(392\) 0 0
\(393\) 1.32926e9 1.10468
\(394\) 0 0
\(395\) 5.70623e8 0.465865
\(396\) 0 0
\(397\) −1.78931e9 −1.43522 −0.717612 0.696443i \(-0.754765\pi\)
−0.717612 + 0.696443i \(0.754765\pi\)
\(398\) 0 0
\(399\) −3.50714e9 −2.76406
\(400\) 0 0
\(401\) −5.14443e7 −0.0398411 −0.0199206 0.999802i \(-0.506341\pi\)
−0.0199206 + 0.999802i \(0.506341\pi\)
\(402\) 0 0
\(403\) −6.96613e7 −0.0530180
\(404\) 0 0
\(405\) 2.94651e9 2.20402
\(406\) 0 0
\(407\) −2.76131e8 −0.203018
\(408\) 0 0
\(409\) 7.71845e8 0.557826 0.278913 0.960316i \(-0.410026\pi\)
0.278913 + 0.960316i \(0.410026\pi\)
\(410\) 0 0
\(411\) −2.24318e9 −1.59374
\(412\) 0 0
\(413\) 3.62788e9 2.53412
\(414\) 0 0
\(415\) 6.60719e8 0.453783
\(416\) 0 0
\(417\) −1.97375e9 −1.33296
\(418\) 0 0
\(419\) −9.35613e8 −0.621366 −0.310683 0.950514i \(-0.600558\pi\)
−0.310683 + 0.950514i \(0.600558\pi\)
\(420\) 0 0
\(421\) −5.83597e8 −0.381176 −0.190588 0.981670i \(-0.561039\pi\)
−0.190588 + 0.981670i \(0.561039\pi\)
\(422\) 0 0
\(423\) 7.01269e8 0.450498
\(424\) 0 0
\(425\) −5.88883e8 −0.372107
\(426\) 0 0
\(427\) −7.41779e8 −0.461081
\(428\) 0 0
\(429\) −2.04907e8 −0.125302
\(430\) 0 0
\(431\) 1.70017e9 1.02287 0.511436 0.859322i \(-0.329114\pi\)
0.511436 + 0.859322i \(0.329114\pi\)
\(432\) 0 0
\(433\) −2.79589e9 −1.65506 −0.827529 0.561423i \(-0.810254\pi\)
−0.827529 + 0.561423i \(0.810254\pi\)
\(434\) 0 0
\(435\) −3.24762e8 −0.189170
\(436\) 0 0
\(437\) −2.53024e9 −1.45036
\(438\) 0 0
\(439\) −1.80716e9 −1.01946 −0.509729 0.860335i \(-0.670254\pi\)
−0.509729 + 0.860335i \(0.670254\pi\)
\(440\) 0 0
\(441\) 1.18623e9 0.658617
\(442\) 0 0
\(443\) −1.29517e9 −0.707807 −0.353904 0.935282i \(-0.615146\pi\)
−0.353904 + 0.935282i \(0.615146\pi\)
\(444\) 0 0
\(445\) 7.83103e8 0.421269
\(446\) 0 0
\(447\) −1.06313e9 −0.563001
\(448\) 0 0
\(449\) −1.37411e9 −0.716404 −0.358202 0.933644i \(-0.616610\pi\)
−0.358202 + 0.933644i \(0.616610\pi\)
\(450\) 0 0
\(451\) 3.25835e8 0.167255
\(452\) 0 0
\(453\) 3.30044e9 1.66813
\(454\) 0 0
\(455\) −3.17842e9 −1.58187
\(456\) 0 0
\(457\) 1.53742e9 0.753504 0.376752 0.926314i \(-0.377041\pi\)
0.376752 + 0.926314i \(0.377041\pi\)
\(458\) 0 0
\(459\) −1.94703e8 −0.0939783
\(460\) 0 0
\(461\) −3.69895e9 −1.75843 −0.879214 0.476427i \(-0.841932\pi\)
−0.879214 + 0.476427i \(0.841932\pi\)
\(462\) 0 0
\(463\) 1.76669e9 0.827232 0.413616 0.910451i \(-0.364266\pi\)
0.413616 + 0.910451i \(0.364266\pi\)
\(464\) 0 0
\(465\) −4.18869e8 −0.193194
\(466\) 0 0
\(467\) −1.53429e9 −0.697103 −0.348552 0.937290i \(-0.613326\pi\)
−0.348552 + 0.937290i \(0.613326\pi\)
\(468\) 0 0
\(469\) 1.48013e9 0.662514
\(470\) 0 0
\(471\) −8.51939e8 −0.375695
\(472\) 0 0
\(473\) 2.40311e8 0.104414
\(474\) 0 0
\(475\) 7.48740e9 3.20556
\(476\) 0 0
\(477\) 1.97019e9 0.831179
\(478\) 0 0
\(479\) −8.94277e8 −0.371790 −0.185895 0.982570i \(-0.559518\pi\)
−0.185895 + 0.982570i \(0.559518\pi\)
\(480\) 0 0
\(481\) −1.83572e9 −0.752139
\(482\) 0 0
\(483\) 4.39731e9 1.77571
\(484\) 0 0
\(485\) 5.50697e9 2.19188
\(486\) 0 0
\(487\) 4.20006e9 1.64780 0.823899 0.566736i \(-0.191794\pi\)
0.823899 + 0.566736i \(0.191794\pi\)
\(488\) 0 0
\(489\) −1.87591e9 −0.725490
\(490\) 0 0
\(491\) −6.21508e8 −0.236953 −0.118476 0.992957i \(-0.537801\pi\)
−0.118476 + 0.992957i \(0.537801\pi\)
\(492\) 0 0
\(493\) 3.95777e7 0.0148760
\(494\) 0 0
\(495\) −4.47311e8 −0.165765
\(496\) 0 0
\(497\) −3.98789e8 −0.145712
\(498\) 0 0
\(499\) −3.24307e9 −1.16844 −0.584218 0.811597i \(-0.698599\pi\)
−0.584218 + 0.811597i \(0.698599\pi\)
\(500\) 0 0
\(501\) −2.91616e9 −1.03605
\(502\) 0 0
\(503\) 3.42333e9 1.19939 0.599696 0.800228i \(-0.295288\pi\)
0.599696 + 0.800228i \(0.295288\pi\)
\(504\) 0 0
\(505\) 2.59050e9 0.895085
\(506\) 0 0
\(507\) 2.31462e9 0.788771
\(508\) 0 0
\(509\) −2.37250e9 −0.797432 −0.398716 0.917074i \(-0.630544\pi\)
−0.398716 + 0.917074i \(0.630544\pi\)
\(510\) 0 0
\(511\) 5.24119e9 1.73763
\(512\) 0 0
\(513\) 2.47556e9 0.809586
\(514\) 0 0
\(515\) −9.24409e9 −2.98221
\(516\) 0 0
\(517\) 4.08014e8 0.129855
\(518\) 0 0
\(519\) −3.54661e9 −1.11360
\(520\) 0 0
\(521\) −2.79017e9 −0.864367 −0.432184 0.901786i \(-0.642257\pi\)
−0.432184 + 0.901786i \(0.642257\pi\)
\(522\) 0 0
\(523\) 5.40390e8 0.165178 0.0825889 0.996584i \(-0.473681\pi\)
0.0825889 + 0.996584i \(0.473681\pi\)
\(524\) 0 0
\(525\) −1.30124e10 −3.92464
\(526\) 0 0
\(527\) 5.10463e7 0.0151924
\(528\) 0 0
\(529\) −2.32368e8 −0.0682468
\(530\) 0 0
\(531\) 3.39423e9 0.983809
\(532\) 0 0
\(533\) 2.16615e9 0.619645
\(534\) 0 0
\(535\) −1.03614e9 −0.292537
\(536\) 0 0
\(537\) 5.68888e9 1.58532
\(538\) 0 0
\(539\) 6.90174e8 0.189844
\(540\) 0 0
\(541\) 4.31485e9 1.17159 0.585794 0.810460i \(-0.300783\pi\)
0.585794 + 0.810460i \(0.300783\pi\)
\(542\) 0 0
\(543\) −2.92086e9 −0.782909
\(544\) 0 0
\(545\) 8.86615e9 2.34610
\(546\) 0 0
\(547\) 4.39068e9 1.14703 0.573516 0.819194i \(-0.305579\pi\)
0.573516 + 0.819194i \(0.305579\pi\)
\(548\) 0 0
\(549\) −6.94006e8 −0.179003
\(550\) 0 0
\(551\) −5.03213e8 −0.128151
\(552\) 0 0
\(553\) −1.53661e9 −0.386389
\(554\) 0 0
\(555\) −1.10381e10 −2.74074
\(556\) 0 0
\(557\) 1.66990e9 0.409446 0.204723 0.978820i \(-0.434371\pi\)
0.204723 + 0.978820i \(0.434371\pi\)
\(558\) 0 0
\(559\) 1.59759e9 0.386833
\(560\) 0 0
\(561\) 1.50152e8 0.0359055
\(562\) 0 0
\(563\) 5.74928e9 1.35780 0.678898 0.734233i \(-0.262458\pi\)
0.678898 + 0.734233i \(0.262458\pi\)
\(564\) 0 0
\(565\) −6.08312e9 −1.41892
\(566\) 0 0
\(567\) −7.93453e9 −1.82802
\(568\) 0 0
\(569\) −1.80817e8 −0.0411478 −0.0205739 0.999788i \(-0.506549\pi\)
−0.0205739 + 0.999788i \(0.506549\pi\)
\(570\) 0 0
\(571\) −7.36079e9 −1.65462 −0.827310 0.561746i \(-0.810130\pi\)
−0.827310 + 0.561746i \(0.810130\pi\)
\(572\) 0 0
\(573\) 6.07120e9 1.34814
\(574\) 0 0
\(575\) −9.38783e9 −2.05934
\(576\) 0 0
\(577\) −4.18503e9 −0.906949 −0.453474 0.891269i \(-0.649816\pi\)
−0.453474 + 0.891269i \(0.649816\pi\)
\(578\) 0 0
\(579\) −4.42911e9 −0.948291
\(580\) 0 0
\(581\) −1.77922e9 −0.376369
\(582\) 0 0
\(583\) 1.14630e9 0.239585
\(584\) 0 0
\(585\) −2.97372e9 −0.614123
\(586\) 0 0
\(587\) 9.28639e9 1.89502 0.947510 0.319727i \(-0.103591\pi\)
0.947510 + 0.319727i \(0.103591\pi\)
\(588\) 0 0
\(589\) −6.49032e8 −0.130877
\(590\) 0 0
\(591\) 7.34610e9 1.46386
\(592\) 0 0
\(593\) −9.05149e9 −1.78250 −0.891248 0.453516i \(-0.850169\pi\)
−0.891248 + 0.453516i \(0.850169\pi\)
\(594\) 0 0
\(595\) 2.32908e9 0.453289
\(596\) 0 0
\(597\) −5.30319e9 −1.02006
\(598\) 0 0
\(599\) 5.58913e9 1.06255 0.531276 0.847199i \(-0.321713\pi\)
0.531276 + 0.847199i \(0.321713\pi\)
\(600\) 0 0
\(601\) 9.35393e9 1.75766 0.878828 0.477140i \(-0.158326\pi\)
0.878828 + 0.477140i \(0.158326\pi\)
\(602\) 0 0
\(603\) 1.38481e9 0.257204
\(604\) 0 0
\(605\) 9.38144e9 1.72237
\(606\) 0 0
\(607\) −6.26278e9 −1.13660 −0.568299 0.822822i \(-0.692398\pi\)
−0.568299 + 0.822822i \(0.692398\pi\)
\(608\) 0 0
\(609\) 8.74538e8 0.156898
\(610\) 0 0
\(611\) 2.71248e9 0.481085
\(612\) 0 0
\(613\) −2.08628e9 −0.365815 −0.182907 0.983130i \(-0.558551\pi\)
−0.182907 + 0.983130i \(0.558551\pi\)
\(614\) 0 0
\(615\) 1.30249e10 2.25794
\(616\) 0 0
\(617\) −4.15947e9 −0.712919 −0.356459 0.934311i \(-0.616016\pi\)
−0.356459 + 0.934311i \(0.616016\pi\)
\(618\) 0 0
\(619\) 3.59117e9 0.608581 0.304291 0.952579i \(-0.401581\pi\)
0.304291 + 0.952579i \(0.401581\pi\)
\(620\) 0 0
\(621\) −3.10390e9 −0.520101
\(622\) 0 0
\(623\) −2.10879e9 −0.349401
\(624\) 0 0
\(625\) 8.65527e9 1.41808
\(626\) 0 0
\(627\) −1.90912e9 −0.309312
\(628\) 0 0
\(629\) 1.34517e9 0.215527
\(630\) 0 0
\(631\) 1.31712e9 0.208700 0.104350 0.994541i \(-0.466724\pi\)
0.104350 + 0.994541i \(0.466724\pi\)
\(632\) 0 0
\(633\) −9.41991e9 −1.47616
\(634\) 0 0
\(635\) −7.71179e9 −1.19522
\(636\) 0 0
\(637\) 4.58828e9 0.703334
\(638\) 0 0
\(639\) −3.73106e8 −0.0565691
\(640\) 0 0
\(641\) −3.66349e8 −0.0549404 −0.0274702 0.999623i \(-0.508745\pi\)
−0.0274702 + 0.999623i \(0.508745\pi\)
\(642\) 0 0
\(643\) 4.40404e9 0.653301 0.326650 0.945145i \(-0.394080\pi\)
0.326650 + 0.945145i \(0.394080\pi\)
\(644\) 0 0
\(645\) 9.60621e9 1.40959
\(646\) 0 0
\(647\) 4.85799e9 0.705166 0.352583 0.935781i \(-0.385303\pi\)
0.352583 + 0.935781i \(0.385303\pi\)
\(648\) 0 0
\(649\) 1.97484e9 0.283580
\(650\) 0 0
\(651\) 1.12796e9 0.160235
\(652\) 0 0
\(653\) 9.96158e9 1.40001 0.700007 0.714136i \(-0.253180\pi\)
0.700007 + 0.714136i \(0.253180\pi\)
\(654\) 0 0
\(655\) 1.12239e10 1.56063
\(656\) 0 0
\(657\) 4.90364e9 0.674590
\(658\) 0 0
\(659\) −5.58699e9 −0.760464 −0.380232 0.924891i \(-0.624156\pi\)
−0.380232 + 0.924891i \(0.624156\pi\)
\(660\) 0 0
\(661\) −5.09996e9 −0.686849 −0.343425 0.939180i \(-0.611587\pi\)
−0.343425 + 0.939180i \(0.611587\pi\)
\(662\) 0 0
\(663\) 9.98209e8 0.133022
\(664\) 0 0
\(665\) −2.96133e10 −3.90491
\(666\) 0 0
\(667\) 6.30938e8 0.0823278
\(668\) 0 0
\(669\) 9.35356e9 1.20777
\(670\) 0 0
\(671\) −4.03788e8 −0.0515971
\(672\) 0 0
\(673\) −1.00616e10 −1.27238 −0.636188 0.771534i \(-0.719490\pi\)
−0.636188 + 0.771534i \(0.719490\pi\)
\(674\) 0 0
\(675\) 9.18497e9 1.14951
\(676\) 0 0
\(677\) −3.91470e9 −0.484885 −0.242442 0.970166i \(-0.577949\pi\)
−0.242442 + 0.970166i \(0.577949\pi\)
\(678\) 0 0
\(679\) −1.48295e10 −1.81795
\(680\) 0 0
\(681\) 8.35292e9 1.01350
\(682\) 0 0
\(683\) 2.62271e9 0.314977 0.157488 0.987521i \(-0.449660\pi\)
0.157488 + 0.987521i \(0.449660\pi\)
\(684\) 0 0
\(685\) −1.89408e10 −2.25155
\(686\) 0 0
\(687\) −7.03952e9 −0.828313
\(688\) 0 0
\(689\) 7.62062e9 0.887611
\(690\) 0 0
\(691\) −9.80668e9 −1.13070 −0.565352 0.824850i \(-0.691260\pi\)
−0.565352 + 0.824850i \(0.691260\pi\)
\(692\) 0 0
\(693\) 1.20455e9 0.137486
\(694\) 0 0
\(695\) −1.66658e10 −1.88312
\(696\) 0 0
\(697\) −1.58731e9 −0.177560
\(698\) 0 0
\(699\) 7.11472e9 0.787930
\(700\) 0 0
\(701\) 9.29229e9 1.01885 0.509424 0.860515i \(-0.329858\pi\)
0.509424 + 0.860515i \(0.329858\pi\)
\(702\) 0 0
\(703\) −1.71033e10 −1.85668
\(704\) 0 0
\(705\) 1.63100e10 1.75304
\(706\) 0 0
\(707\) −6.97586e9 −0.742386
\(708\) 0 0
\(709\) −3.37224e9 −0.355350 −0.177675 0.984089i \(-0.556858\pi\)
−0.177675 + 0.984089i \(0.556858\pi\)
\(710\) 0 0
\(711\) −1.43765e9 −0.150006
\(712\) 0 0
\(713\) 8.13768e8 0.0840789
\(714\) 0 0
\(715\) −1.73018e9 −0.177019
\(716\) 0 0
\(717\) −4.53467e9 −0.459440
\(718\) 0 0
\(719\) 1.58951e10 1.59482 0.797410 0.603438i \(-0.206203\pi\)
0.797410 + 0.603438i \(0.206203\pi\)
\(720\) 0 0
\(721\) 2.48930e10 2.47346
\(722\) 0 0
\(723\) −1.72164e10 −1.69418
\(724\) 0 0
\(725\) −1.86705e9 −0.181959
\(726\) 0 0
\(727\) −2.57648e8 −0.0248689 −0.0124344 0.999923i \(-0.503958\pi\)
−0.0124344 + 0.999923i \(0.503958\pi\)
\(728\) 0 0
\(729\) −3.61484e8 −0.0345575
\(730\) 0 0
\(731\) −1.17068e9 −0.110848
\(732\) 0 0
\(733\) −5.80778e9 −0.544686 −0.272343 0.962200i \(-0.587799\pi\)
−0.272343 + 0.962200i \(0.587799\pi\)
\(734\) 0 0
\(735\) 2.75891e10 2.56290
\(736\) 0 0
\(737\) 8.05711e8 0.0741384
\(738\) 0 0
\(739\) 6.85330e9 0.624661 0.312330 0.949974i \(-0.398890\pi\)
0.312330 + 0.949974i \(0.398890\pi\)
\(740\) 0 0
\(741\) −1.26918e10 −1.14593
\(742\) 0 0
\(743\) 1.61728e9 0.144652 0.0723261 0.997381i \(-0.476958\pi\)
0.0723261 + 0.997381i \(0.476958\pi\)
\(744\) 0 0
\(745\) −8.97675e9 −0.795375
\(746\) 0 0
\(747\) −1.66464e9 −0.146116
\(748\) 0 0
\(749\) 2.79018e9 0.242631
\(750\) 0 0
\(751\) −1.22343e10 −1.05400 −0.526999 0.849866i \(-0.676683\pi\)
−0.526999 + 0.849866i \(0.676683\pi\)
\(752\) 0 0
\(753\) −2.20446e10 −1.88157
\(754\) 0 0
\(755\) 2.78680e10 2.35663
\(756\) 0 0
\(757\) 3.82416e9 0.320406 0.160203 0.987084i \(-0.448785\pi\)
0.160203 + 0.987084i \(0.448785\pi\)
\(758\) 0 0
\(759\) 2.39368e9 0.198710
\(760\) 0 0
\(761\) 9.74562e9 0.801610 0.400805 0.916163i \(-0.368731\pi\)
0.400805 + 0.916163i \(0.368731\pi\)
\(762\) 0 0
\(763\) −2.38753e10 −1.94587
\(764\) 0 0
\(765\) 2.17908e9 0.175978
\(766\) 0 0
\(767\) 1.31287e10 1.05061
\(768\) 0 0
\(769\) −6.69708e9 −0.531059 −0.265530 0.964103i \(-0.585547\pi\)
−0.265530 + 0.964103i \(0.585547\pi\)
\(770\) 0 0
\(771\) −1.94203e10 −1.52604
\(772\) 0 0
\(773\) −3.84770e9 −0.299621 −0.149811 0.988715i \(-0.547866\pi\)
−0.149811 + 0.988715i \(0.547866\pi\)
\(774\) 0 0
\(775\) −2.40808e9 −0.185829
\(776\) 0 0
\(777\) 2.97240e10 2.27318
\(778\) 0 0
\(779\) 2.01819e10 1.52961
\(780\) 0 0
\(781\) −2.17082e8 −0.0163059
\(782\) 0 0
\(783\) −6.17304e8 −0.0459550
\(784\) 0 0
\(785\) −7.19353e9 −0.530760
\(786\) 0 0
\(787\) 1.08120e9 0.0790666 0.0395333 0.999218i \(-0.487413\pi\)
0.0395333 + 0.999218i \(0.487413\pi\)
\(788\) 0 0
\(789\) −3.01700e10 −2.18678
\(790\) 0 0
\(791\) 1.63810e10 1.17685
\(792\) 0 0
\(793\) −2.68438e9 −0.191156
\(794\) 0 0
\(795\) 4.58224e10 3.23439
\(796\) 0 0
\(797\) −3.35998e9 −0.235089 −0.117545 0.993068i \(-0.537502\pi\)
−0.117545 + 0.993068i \(0.537502\pi\)
\(798\) 0 0
\(799\) −1.98764e9 −0.137856
\(800\) 0 0
\(801\) −1.97298e9 −0.135646
\(802\) 0 0
\(803\) 2.85305e9 0.194449
\(804\) 0 0
\(805\) 3.71297e10 2.50862
\(806\) 0 0
\(807\) 1.09007e9 0.0730125
\(808\) 0 0
\(809\) 1.79911e10 1.19465 0.597323 0.802001i \(-0.296231\pi\)
0.597323 + 0.802001i \(0.296231\pi\)
\(810\) 0 0
\(811\) −6.03707e9 −0.397423 −0.198711 0.980058i \(-0.563676\pi\)
−0.198711 + 0.980058i \(0.563676\pi\)
\(812\) 0 0
\(813\) 4.71515e9 0.307736
\(814\) 0 0
\(815\) −1.58397e10 −1.02493
\(816\) 0 0
\(817\) 1.48847e10 0.954910
\(818\) 0 0
\(819\) 8.00782e9 0.509355
\(820\) 0 0
\(821\) −1.62254e10 −1.02328 −0.511641 0.859200i \(-0.670962\pi\)
−0.511641 + 0.859200i \(0.670962\pi\)
\(822\) 0 0
\(823\) 1.87177e10 1.17045 0.585225 0.810871i \(-0.301006\pi\)
0.585225 + 0.810871i \(0.301006\pi\)
\(824\) 0 0
\(825\) −7.08332e9 −0.439185
\(826\) 0 0
\(827\) −2.09971e10 −1.29089 −0.645445 0.763806i \(-0.723328\pi\)
−0.645445 + 0.763806i \(0.723328\pi\)
\(828\) 0 0
\(829\) 6.77963e9 0.413300 0.206650 0.978415i \(-0.433744\pi\)
0.206650 + 0.978415i \(0.433744\pi\)
\(830\) 0 0
\(831\) −2.15078e10 −1.30015
\(832\) 0 0
\(833\) −3.36219e9 −0.201542
\(834\) 0 0
\(835\) −2.46232e10 −1.46367
\(836\) 0 0
\(837\) −7.96183e8 −0.0469325
\(838\) 0 0
\(839\) 8.76720e8 0.0512500 0.0256250 0.999672i \(-0.491842\pi\)
0.0256250 + 0.999672i \(0.491842\pi\)
\(840\) 0 0
\(841\) −1.71244e10 −0.992726
\(842\) 0 0
\(843\) −2.02678e10 −1.16523
\(844\) 0 0
\(845\) 1.95439e10 1.11433
\(846\) 0 0
\(847\) −2.52629e10 −1.42854
\(848\) 0 0
\(849\) −4.05358e10 −2.27333
\(850\) 0 0
\(851\) 2.14444e10 1.19278
\(852\) 0 0
\(853\) 3.48352e10 1.92175 0.960875 0.276984i \(-0.0893348\pi\)
0.960875 + 0.276984i \(0.0893348\pi\)
\(854\) 0 0
\(855\) −2.77061e10 −1.51598
\(856\) 0 0
\(857\) −4.85212e9 −0.263329 −0.131665 0.991294i \(-0.542032\pi\)
−0.131665 + 0.991294i \(0.542032\pi\)
\(858\) 0 0
\(859\) 1.09083e10 0.587191 0.293596 0.955930i \(-0.405148\pi\)
0.293596 + 0.955930i \(0.405148\pi\)
\(860\) 0 0
\(861\) −3.50743e10 −1.87274
\(862\) 0 0
\(863\) −8.92478e9 −0.472672 −0.236336 0.971671i \(-0.575947\pi\)
−0.236336 + 0.971671i \(0.575947\pi\)
\(864\) 0 0
\(865\) −2.99465e10 −1.57322
\(866\) 0 0
\(867\) 2.33129e10 1.21487
\(868\) 0 0
\(869\) −8.36455e8 −0.0432388
\(870\) 0 0
\(871\) 5.35636e9 0.274667
\(872\) 0 0
\(873\) −1.38744e10 −0.705774
\(874\) 0 0
\(875\) −5.83722e10 −2.94563
\(876\) 0 0
\(877\) −2.64337e10 −1.32331 −0.661653 0.749810i \(-0.730145\pi\)
−0.661653 + 0.749810i \(0.730145\pi\)
\(878\) 0 0
\(879\) 3.01074e10 1.49524
\(880\) 0 0
\(881\) 1.62164e10 0.798984 0.399492 0.916737i \(-0.369186\pi\)
0.399492 + 0.916737i \(0.369186\pi\)
\(882\) 0 0
\(883\) −9.95689e9 −0.486700 −0.243350 0.969939i \(-0.578246\pi\)
−0.243350 + 0.969939i \(0.578246\pi\)
\(884\) 0 0
\(885\) 7.89424e10 3.82833
\(886\) 0 0
\(887\) 1.18915e10 0.572143 0.286072 0.958208i \(-0.407650\pi\)
0.286072 + 0.958208i \(0.407650\pi\)
\(888\) 0 0
\(889\) 2.07668e10 0.991317
\(890\) 0 0
\(891\) −4.31917e9 −0.204564
\(892\) 0 0
\(893\) 2.52721e10 1.18757
\(894\) 0 0
\(895\) 4.80353e10 2.23965
\(896\) 0 0
\(897\) 1.59132e10 0.736181
\(898\) 0 0
\(899\) 1.61842e8 0.00742904
\(900\) 0 0
\(901\) −5.58423e9 −0.254347
\(902\) 0 0
\(903\) −2.58682e10 −1.16912
\(904\) 0 0
\(905\) −2.46629e10 −1.10605
\(906\) 0 0
\(907\) −7.98425e9 −0.355311 −0.177655 0.984093i \(-0.556851\pi\)
−0.177655 + 0.984093i \(0.556851\pi\)
\(908\) 0 0
\(909\) −6.52660e9 −0.288213
\(910\) 0 0
\(911\) −4.41995e10 −1.93688 −0.968441 0.249242i \(-0.919818\pi\)
−0.968441 + 0.249242i \(0.919818\pi\)
\(912\) 0 0
\(913\) −9.68522e8 −0.0421174
\(914\) 0 0
\(915\) −1.61411e10 −0.696559
\(916\) 0 0
\(917\) −3.02244e10 −1.29439
\(918\) 0 0
\(919\) 2.34655e10 0.997299 0.498650 0.866804i \(-0.333829\pi\)
0.498650 + 0.866804i \(0.333829\pi\)
\(920\) 0 0
\(921\) 4.93753e10 2.08258
\(922\) 0 0
\(923\) −1.44316e9 −0.0604099
\(924\) 0 0
\(925\) −6.34577e10 −2.63626
\(926\) 0 0
\(927\) 2.32898e10 0.960257
\(928\) 0 0
\(929\) 2.61342e10 1.06944 0.534718 0.845031i \(-0.320418\pi\)
0.534718 + 0.845031i \(0.320418\pi\)
\(930\) 0 0
\(931\) 4.27488e10 1.73620
\(932\) 0 0
\(933\) 3.18584e10 1.28422
\(934\) 0 0
\(935\) 1.26784e9 0.0507252
\(936\) 0 0
\(937\) −2.03666e10 −0.808779 −0.404389 0.914587i \(-0.632516\pi\)
−0.404389 + 0.914587i \(0.632516\pi\)
\(938\) 0 0
\(939\) 5.44142e10 2.14478
\(940\) 0 0
\(941\) −1.75644e10 −0.687180 −0.343590 0.939120i \(-0.611643\pi\)
−0.343590 + 0.939120i \(0.611643\pi\)
\(942\) 0 0
\(943\) −2.53045e10 −0.982667
\(944\) 0 0
\(945\) −3.63273e10 −1.40030
\(946\) 0 0
\(947\) −1.43195e10 −0.547903 −0.273951 0.961744i \(-0.588331\pi\)
−0.273951 + 0.961744i \(0.588331\pi\)
\(948\) 0 0
\(949\) 1.89671e10 0.720391
\(950\) 0 0
\(951\) 3.68015e9 0.138750
\(952\) 0 0
\(953\) 1.93399e10 0.723817 0.361908 0.932214i \(-0.382125\pi\)
0.361908 + 0.932214i \(0.382125\pi\)
\(954\) 0 0
\(955\) 5.12635e10 1.90457
\(956\) 0 0
\(957\) 4.76056e8 0.0175576
\(958\) 0 0
\(959\) 5.10049e10 1.86744
\(960\) 0 0
\(961\) −2.73039e10 −0.992413
\(962\) 0 0
\(963\) 2.61049e9 0.0941954
\(964\) 0 0
\(965\) −3.73981e10 −1.33969
\(966\) 0 0
\(967\) 1.23800e10 0.440280 0.220140 0.975468i \(-0.429349\pi\)
0.220140 + 0.975468i \(0.429349\pi\)
\(968\) 0 0
\(969\) 9.30029e9 0.328370
\(970\) 0 0
\(971\) 4.53075e10 1.58819 0.794096 0.607792i \(-0.207945\pi\)
0.794096 + 0.607792i \(0.207945\pi\)
\(972\) 0 0
\(973\) 4.48786e10 1.56187
\(974\) 0 0
\(975\) −4.70899e10 −1.62709
\(976\) 0 0
\(977\) −2.32784e10 −0.798587 −0.399293 0.916823i \(-0.630744\pi\)
−0.399293 + 0.916823i \(0.630744\pi\)
\(978\) 0 0
\(979\) −1.14792e9 −0.0390996
\(980\) 0 0
\(981\) −2.23377e10 −0.755433
\(982\) 0 0
\(983\) −1.09683e10 −0.368300 −0.184150 0.982898i \(-0.558953\pi\)
−0.184150 + 0.982898i \(0.558953\pi\)
\(984\) 0 0
\(985\) 6.20284e10 2.06806
\(986\) 0 0
\(987\) −4.39204e10 −1.45397
\(988\) 0 0
\(989\) −1.86627e10 −0.613461
\(990\) 0 0
\(991\) −2.57317e10 −0.839868 −0.419934 0.907555i \(-0.637947\pi\)
−0.419934 + 0.907555i \(0.637947\pi\)
\(992\) 0 0
\(993\) −4.03071e9 −0.130635
\(994\) 0 0
\(995\) −4.47786e10 −1.44109
\(996\) 0 0
\(997\) −2.67530e10 −0.854948 −0.427474 0.904028i \(-0.640596\pi\)
−0.427474 + 0.904028i \(0.640596\pi\)
\(998\) 0 0
\(999\) −2.09810e10 −0.665806
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 32.8.a.b.1.1 2
3.2 odd 2 288.8.a.o.1.2 2
4.3 odd 2 32.8.a.d.1.2 yes 2
8.3 odd 2 64.8.a.h.1.1 2
8.5 even 2 64.8.a.j.1.2 2
12.11 even 2 288.8.a.n.1.2 2
16.3 odd 4 256.8.b.j.129.4 4
16.5 even 4 256.8.b.h.129.4 4
16.11 odd 4 256.8.b.j.129.1 4
16.13 even 4 256.8.b.h.129.1 4
24.5 odd 2 576.8.a.bf.1.1 2
24.11 even 2 576.8.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.8.a.b.1.1 2 1.1 even 1 trivial
32.8.a.d.1.2 yes 2 4.3 odd 2
64.8.a.h.1.1 2 8.3 odd 2
64.8.a.j.1.2 2 8.5 even 2
256.8.b.h.129.1 4 16.13 even 4
256.8.b.h.129.4 4 16.5 even 4
256.8.b.j.129.1 4 16.11 odd 4
256.8.b.j.129.4 4 16.3 odd 4
288.8.a.n.1.2 2 12.11 even 2
288.8.a.o.1.2 2 3.2 odd 2
576.8.a.be.1.1 2 24.11 even 2
576.8.a.bf.1.1 2 24.5 odd 2