Properties

Label 32.8.a.c.1.2
Level $32$
Weight $8$
Character 32.1
Self dual yes
Analytic conductor $9.996$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(\Q(\sqrt{15}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 15 \) 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 1.2
Root \(3.87298\) 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)
 
\(f(q)\) \(=\) \(q+61.9677 q^{3} +70.0000 q^{5} +1115.42 q^{7} +1653.00 q^{9} +O(q^{10})\) \(q+61.9677 q^{3} +70.0000 q^{5} +1115.42 q^{7} +1653.00 q^{9} -7250.22 q^{11} +13758.0 q^{13} +4337.74 q^{15} +16994.0 q^{17} +34020.3 q^{19} +69120.0 q^{21} -32347.2 q^{23} -73225.0 q^{25} -33090.8 q^{27} +34190.0 q^{29} -120465. q^{31} -449280. q^{33} +78079.3 q^{35} +35206.0 q^{37} +852552. q^{39} -484550. q^{41} -672040. q^{43} +115710. q^{45} -1.20688e6 q^{47} +420617. q^{49} +1.05308e6 q^{51} +851702. q^{53} -507516. q^{55} +2.10816e6 q^{57} +695464. q^{59} +71630.0 q^{61} +1.84379e6 q^{63} +963060. q^{65} -307298. q^{67} -2.00448e6 q^{69} -757370. q^{71} +3.91204e6 q^{73} -4.53759e6 q^{75} -8.08704e6 q^{77} +314548. q^{79} -5.66567e6 q^{81} -1.53649e6 q^{83} +1.18958e6 q^{85} +2.11868e6 q^{87} -2.51063e6 q^{89} +1.53459e7 q^{91} -7.46496e6 q^{93} +2.38142e6 q^{95} -50094.0 q^{97} -1.19846e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 140 q^{5} + 3306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 140 q^{5} + 3306 q^{9} + 27516 q^{13} + 33988 q^{17} + 138240 q^{21} - 146450 q^{25} + 68380 q^{29} - 898560 q^{33} + 70412 q^{37} - 969100 q^{41} + 231420 q^{45} + 841234 q^{49} + 1703404 q^{53} + 4216320 q^{57} + 143260 q^{61} + 1926120 q^{65} - 4008960 q^{69} + 7824084 q^{73} - 16174080 q^{77} - 11331342 q^{81} + 2379160 q^{85} - 5021260 q^{89} - 14929920 q^{93} - 100188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 61.9677 1.32508 0.662539 0.749028i \(-0.269479\pi\)
0.662539 + 0.749028i \(0.269479\pi\)
\(4\) 0 0
\(5\) 70.0000 0.250440 0.125220 0.992129i \(-0.460036\pi\)
0.125220 + 0.992129i \(0.460036\pi\)
\(6\) 0 0
\(7\) 1115.42 1.22912 0.614561 0.788869i \(-0.289333\pi\)
0.614561 + 0.788869i \(0.289333\pi\)
\(8\) 0 0
\(9\) 1653.00 0.755830
\(10\) 0 0
\(11\) −7250.22 −1.64239 −0.821196 0.570646i \(-0.806693\pi\)
−0.821196 + 0.570646i \(0.806693\pi\)
\(12\) 0 0
\(13\) 13758.0 1.73682 0.868408 0.495851i \(-0.165144\pi\)
0.868408 + 0.495851i \(0.165144\pi\)
\(14\) 0 0
\(15\) 4337.74 0.331852
\(16\) 0 0
\(17\) 16994.0 0.838927 0.419464 0.907772i \(-0.362218\pi\)
0.419464 + 0.907772i \(0.362218\pi\)
\(18\) 0 0
\(19\) 34020.3 1.13789 0.568945 0.822376i \(-0.307352\pi\)
0.568945 + 0.822376i \(0.307352\pi\)
\(20\) 0 0
\(21\) 69120.0 1.62868
\(22\) 0 0
\(23\) −32347.2 −0.554356 −0.277178 0.960819i \(-0.589399\pi\)
−0.277178 + 0.960819i \(0.589399\pi\)
\(24\) 0 0
\(25\) −73225.0 −0.937280
\(26\) 0 0
\(27\) −33090.8 −0.323544
\(28\) 0 0
\(29\) 34190.0 0.260319 0.130160 0.991493i \(-0.458451\pi\)
0.130160 + 0.991493i \(0.458451\pi\)
\(30\) 0 0
\(31\) −120465. −0.726266 −0.363133 0.931737i \(-0.618293\pi\)
−0.363133 + 0.931737i \(0.618293\pi\)
\(32\) 0 0
\(33\) −449280. −2.17630
\(34\) 0 0
\(35\) 78079.3 0.307821
\(36\) 0 0
\(37\) 35206.0 0.114264 0.0571322 0.998367i \(-0.481804\pi\)
0.0571322 + 0.998367i \(0.481804\pi\)
\(38\) 0 0
\(39\) 852552. 2.30141
\(40\) 0 0
\(41\) −484550. −1.09798 −0.548991 0.835828i \(-0.684988\pi\)
−0.548991 + 0.835828i \(0.684988\pi\)
\(42\) 0 0
\(43\) −672040. −1.28901 −0.644504 0.764601i \(-0.722936\pi\)
−0.644504 + 0.764601i \(0.722936\pi\)
\(44\) 0 0
\(45\) 115710. 0.189290
\(46\) 0 0
\(47\) −1.20688e6 −1.69560 −0.847799 0.530318i \(-0.822073\pi\)
−0.847799 + 0.530318i \(0.822073\pi\)
\(48\) 0 0
\(49\) 420617. 0.510741
\(50\) 0 0
\(51\) 1.05308e6 1.11164
\(52\) 0 0
\(53\) 851702. 0.785818 0.392909 0.919577i \(-0.371469\pi\)
0.392909 + 0.919577i \(0.371469\pi\)
\(54\) 0 0
\(55\) −507516. −0.411320
\(56\) 0 0
\(57\) 2.10816e6 1.50779
\(58\) 0 0
\(59\) 695464. 0.440852 0.220426 0.975404i \(-0.429255\pi\)
0.220426 + 0.975404i \(0.429255\pi\)
\(60\) 0 0
\(61\) 71630.0 0.0404055 0.0202028 0.999796i \(-0.493569\pi\)
0.0202028 + 0.999796i \(0.493569\pi\)
\(62\) 0 0
\(63\) 1.84379e6 0.929007
\(64\) 0 0
\(65\) 963060. 0.434967
\(66\) 0 0
\(67\) −307298. −0.124824 −0.0624120 0.998050i \(-0.519879\pi\)
−0.0624120 + 0.998050i \(0.519879\pi\)
\(68\) 0 0
\(69\) −2.00448e6 −0.734564
\(70\) 0 0
\(71\) −757370. −0.251133 −0.125566 0.992085i \(-0.540075\pi\)
−0.125566 + 0.992085i \(0.540075\pi\)
\(72\) 0 0
\(73\) 3.91204e6 1.17699 0.588496 0.808500i \(-0.299720\pi\)
0.588496 + 0.808500i \(0.299720\pi\)
\(74\) 0 0
\(75\) −4.53759e6 −1.24197
\(76\) 0 0
\(77\) −8.08704e6 −2.01870
\(78\) 0 0
\(79\) 314548. 0.0717782 0.0358891 0.999356i \(-0.488574\pi\)
0.0358891 + 0.999356i \(0.488574\pi\)
\(80\) 0 0
\(81\) −5.66567e6 −1.18455
\(82\) 0 0
\(83\) −1.53649e6 −0.294955 −0.147478 0.989065i \(-0.547115\pi\)
−0.147478 + 0.989065i \(0.547115\pi\)
\(84\) 0 0
\(85\) 1.18958e6 0.210101
\(86\) 0 0
\(87\) 2.11868e6 0.344943
\(88\) 0 0
\(89\) −2.51063e6 −0.377501 −0.188750 0.982025i \(-0.560444\pi\)
−0.188750 + 0.982025i \(0.560444\pi\)
\(90\) 0 0
\(91\) 1.53459e7 2.13476
\(92\) 0 0
\(93\) −7.46496e6 −0.962359
\(94\) 0 0
\(95\) 2.38142e6 0.284973
\(96\) 0 0
\(97\) −50094.0 −0.00557294 −0.00278647 0.999996i \(-0.500887\pi\)
−0.00278647 + 0.999996i \(0.500887\pi\)
\(98\) 0 0
\(99\) −1.19846e7 −1.24137
\(100\) 0 0
\(101\) 1.50354e7 1.45208 0.726039 0.687653i \(-0.241359\pi\)
0.726039 + 0.687653i \(0.241359\pi\)
\(102\) 0 0
\(103\) −1.99917e7 −1.80268 −0.901340 0.433113i \(-0.857415\pi\)
−0.901340 + 0.433113i \(0.857415\pi\)
\(104\) 0 0
\(105\) 4.83840e6 0.407886
\(106\) 0 0
\(107\) 8.75102e6 0.690582 0.345291 0.938496i \(-0.387780\pi\)
0.345291 + 0.938496i \(0.387780\pi\)
\(108\) 0 0
\(109\) 1.78070e7 1.31704 0.658518 0.752565i \(-0.271184\pi\)
0.658518 + 0.752565i \(0.271184\pi\)
\(110\) 0 0
\(111\) 2.18164e6 0.151409
\(112\) 0 0
\(113\) −1.88499e7 −1.22895 −0.614475 0.788936i \(-0.710632\pi\)
−0.614475 + 0.788936i \(0.710632\pi\)
\(114\) 0 0
\(115\) −2.26430e6 −0.138833
\(116\) 0 0
\(117\) 2.27420e7 1.31274
\(118\) 0 0
\(119\) 1.89554e7 1.03114
\(120\) 0 0
\(121\) 3.30786e7 1.69745
\(122\) 0 0
\(123\) −3.00265e7 −1.45491
\(124\) 0 0
\(125\) −1.05945e7 −0.485172
\(126\) 0 0
\(127\) 3.31235e7 1.43490 0.717452 0.696608i \(-0.245308\pi\)
0.717452 + 0.696608i \(0.245308\pi\)
\(128\) 0 0
\(129\) −4.16448e7 −1.70804
\(130\) 0 0
\(131\) 2.63468e7 1.02395 0.511974 0.859001i \(-0.328915\pi\)
0.511974 + 0.859001i \(0.328915\pi\)
\(132\) 0 0
\(133\) 3.79469e7 1.39861
\(134\) 0 0
\(135\) −2.31635e6 −0.0810283
\(136\) 0 0
\(137\) −1.88745e7 −0.627125 −0.313563 0.949568i \(-0.601523\pi\)
−0.313563 + 0.949568i \(0.601523\pi\)
\(138\) 0 0
\(139\) −1.08848e7 −0.343771 −0.171886 0.985117i \(-0.554986\pi\)
−0.171886 + 0.985117i \(0.554986\pi\)
\(140\) 0 0
\(141\) −7.47878e7 −2.24680
\(142\) 0 0
\(143\) −9.97486e7 −2.85253
\(144\) 0 0
\(145\) 2.39330e6 0.0651942
\(146\) 0 0
\(147\) 2.60647e7 0.676771
\(148\) 0 0
\(149\) −3.44462e7 −0.853078 −0.426539 0.904469i \(-0.640267\pi\)
−0.426539 + 0.904469i \(0.640267\pi\)
\(150\) 0 0
\(151\) 1.30493e7 0.308438 0.154219 0.988037i \(-0.450714\pi\)
0.154219 + 0.988037i \(0.450714\pi\)
\(152\) 0 0
\(153\) 2.80911e7 0.634086
\(154\) 0 0
\(155\) −8.43257e6 −0.181886
\(156\) 0 0
\(157\) −2.04587e7 −0.421919 −0.210960 0.977495i \(-0.567659\pi\)
−0.210960 + 0.977495i \(0.567659\pi\)
\(158\) 0 0
\(159\) 5.27780e7 1.04127
\(160\) 0 0
\(161\) −3.60806e7 −0.681371
\(162\) 0 0
\(163\) 1.07978e7 0.195290 0.0976448 0.995221i \(-0.468869\pi\)
0.0976448 + 0.995221i \(0.468869\pi\)
\(164\) 0 0
\(165\) −3.14496e7 −0.545031
\(166\) 0 0
\(167\) 5.58122e7 0.927303 0.463652 0.886018i \(-0.346539\pi\)
0.463652 + 0.886018i \(0.346539\pi\)
\(168\) 0 0
\(169\) 1.26534e8 2.01653
\(170\) 0 0
\(171\) 5.62355e7 0.860051
\(172\) 0 0
\(173\) −8.94250e6 −0.131310 −0.0656550 0.997842i \(-0.520914\pi\)
−0.0656550 + 0.997842i \(0.520914\pi\)
\(174\) 0 0
\(175\) −8.16766e7 −1.15203
\(176\) 0 0
\(177\) 4.30963e7 0.584163
\(178\) 0 0
\(179\) −9.39334e7 −1.22415 −0.612074 0.790800i \(-0.709665\pi\)
−0.612074 + 0.790800i \(0.709665\pi\)
\(180\) 0 0
\(181\) 9.62269e7 1.20621 0.603103 0.797663i \(-0.293931\pi\)
0.603103 + 0.797663i \(0.293931\pi\)
\(182\) 0 0
\(183\) 4.43875e6 0.0535404
\(184\) 0 0
\(185\) 2.46442e6 0.0286163
\(186\) 0 0
\(187\) −1.23210e8 −1.37785
\(188\) 0 0
\(189\) −3.69101e7 −0.397675
\(190\) 0 0
\(191\) 1.06875e8 1.10984 0.554919 0.831905i \(-0.312749\pi\)
0.554919 + 0.831905i \(0.312749\pi\)
\(192\) 0 0
\(193\) 1.34693e8 1.34863 0.674317 0.738442i \(-0.264438\pi\)
0.674317 + 0.738442i \(0.264438\pi\)
\(194\) 0 0
\(195\) 5.96786e7 0.576365
\(196\) 0 0
\(197\) 9.36611e7 0.872826 0.436413 0.899747i \(-0.356249\pi\)
0.436413 + 0.899747i \(0.356249\pi\)
\(198\) 0 0
\(199\) −5.37621e7 −0.483605 −0.241802 0.970326i \(-0.577739\pi\)
−0.241802 + 0.970326i \(0.577739\pi\)
\(200\) 0 0
\(201\) −1.90426e7 −0.165401
\(202\) 0 0
\(203\) 3.81362e7 0.319964
\(204\) 0 0
\(205\) −3.39185e7 −0.274978
\(206\) 0 0
\(207\) −5.34699e7 −0.418999
\(208\) 0 0
\(209\) −2.46655e8 −1.86886
\(210\) 0 0
\(211\) 1.26710e8 0.928586 0.464293 0.885682i \(-0.346308\pi\)
0.464293 + 0.885682i \(0.346308\pi\)
\(212\) 0 0
\(213\) −4.69325e7 −0.332771
\(214\) 0 0
\(215\) −4.70428e7 −0.322819
\(216\) 0 0
\(217\) −1.34369e8 −0.892670
\(218\) 0 0
\(219\) 2.42420e8 1.55961
\(220\) 0 0
\(221\) 2.33803e8 1.45706
\(222\) 0 0
\(223\) 2.00325e8 1.20967 0.604836 0.796350i \(-0.293239\pi\)
0.604836 + 0.796350i \(0.293239\pi\)
\(224\) 0 0
\(225\) −1.21041e8 −0.708424
\(226\) 0 0
\(227\) −2.46632e8 −1.39946 −0.699728 0.714410i \(-0.746695\pi\)
−0.699728 + 0.714410i \(0.746695\pi\)
\(228\) 0 0
\(229\) −2.24690e8 −1.23640 −0.618201 0.786020i \(-0.712138\pi\)
−0.618201 + 0.786020i \(0.712138\pi\)
\(230\) 0 0
\(231\) −5.01136e8 −2.67494
\(232\) 0 0
\(233\) −2.49045e8 −1.28983 −0.644914 0.764256i \(-0.723107\pi\)
−0.644914 + 0.764256i \(0.723107\pi\)
\(234\) 0 0
\(235\) −8.44819e7 −0.424645
\(236\) 0 0
\(237\) 1.94918e7 0.0951116
\(238\) 0 0
\(239\) 1.04740e8 0.496273 0.248136 0.968725i \(-0.420182\pi\)
0.248136 + 0.968725i \(0.420182\pi\)
\(240\) 0 0
\(241\) 8.04571e7 0.370258 0.185129 0.982714i \(-0.440730\pi\)
0.185129 + 0.982714i \(0.440730\pi\)
\(242\) 0 0
\(243\) −2.78719e8 −1.24608
\(244\) 0 0
\(245\) 2.94432e7 0.127910
\(246\) 0 0
\(247\) 4.68051e8 1.97630
\(248\) 0 0
\(249\) −9.52128e7 −0.390839
\(250\) 0 0
\(251\) 1.60093e8 0.639021 0.319511 0.947583i \(-0.396482\pi\)
0.319511 + 0.947583i \(0.396482\pi\)
\(252\) 0 0
\(253\) 2.34524e8 0.910470
\(254\) 0 0
\(255\) 7.37156e7 0.278400
\(256\) 0 0
\(257\) 1.76666e8 0.649213 0.324607 0.945849i \(-0.394768\pi\)
0.324607 + 0.945849i \(0.394768\pi\)
\(258\) 0 0
\(259\) 3.92694e7 0.140445
\(260\) 0 0
\(261\) 5.65161e7 0.196757
\(262\) 0 0
\(263\) 3.49243e8 1.18381 0.591906 0.806007i \(-0.298376\pi\)
0.591906 + 0.806007i \(0.298376\pi\)
\(264\) 0 0
\(265\) 5.96191e7 0.196800
\(266\) 0 0
\(267\) −1.55578e8 −0.500218
\(268\) 0 0
\(269\) 1.91355e8 0.599387 0.299693 0.954036i \(-0.403116\pi\)
0.299693 + 0.954036i \(0.403116\pi\)
\(270\) 0 0
\(271\) −5.62669e8 −1.71736 −0.858678 0.512516i \(-0.828714\pi\)
−0.858678 + 0.512516i \(0.828714\pi\)
\(272\) 0 0
\(273\) 9.50953e8 2.82872
\(274\) 0 0
\(275\) 5.30898e8 1.53938
\(276\) 0 0
\(277\) −5.25734e8 −1.48623 −0.743116 0.669162i \(-0.766653\pi\)
−0.743116 + 0.669162i \(0.766653\pi\)
\(278\) 0 0
\(279\) −1.99129e8 −0.548934
\(280\) 0 0
\(281\) −4.24101e7 −0.114024 −0.0570122 0.998373i \(-0.518157\pi\)
−0.0570122 + 0.998373i \(0.518157\pi\)
\(282\) 0 0
\(283\) −9.44052e7 −0.247596 −0.123798 0.992307i \(-0.539507\pi\)
−0.123798 + 0.992307i \(0.539507\pi\)
\(284\) 0 0
\(285\) 1.47571e8 0.377611
\(286\) 0 0
\(287\) −5.40476e8 −1.34955
\(288\) 0 0
\(289\) −1.21543e8 −0.296201
\(290\) 0 0
\(291\) −3.10421e6 −0.00738458
\(292\) 0 0
\(293\) 4.01296e8 0.932027 0.466014 0.884778i \(-0.345690\pi\)
0.466014 + 0.884778i \(0.345690\pi\)
\(294\) 0 0
\(295\) 4.86825e7 0.110407
\(296\) 0 0
\(297\) 2.39916e8 0.531387
\(298\) 0 0
\(299\) −4.45032e8 −0.962814
\(300\) 0 0
\(301\) −7.49606e8 −1.58435
\(302\) 0 0
\(303\) 9.31710e8 1.92412
\(304\) 0 0
\(305\) 5.01410e6 0.0101191
\(306\) 0 0
\(307\) 2.71076e8 0.534697 0.267348 0.963600i \(-0.413853\pi\)
0.267348 + 0.963600i \(0.413853\pi\)
\(308\) 0 0
\(309\) −1.23884e9 −2.38869
\(310\) 0 0
\(311\) −4.12911e8 −0.778387 −0.389193 0.921156i \(-0.627246\pi\)
−0.389193 + 0.921156i \(0.627246\pi\)
\(312\) 0 0
\(313\) −1.81101e8 −0.333823 −0.166911 0.985972i \(-0.553379\pi\)
−0.166911 + 0.985972i \(0.553379\pi\)
\(314\) 0 0
\(315\) 1.29065e8 0.232660
\(316\) 0 0
\(317\) 7.48850e8 1.32034 0.660172 0.751114i \(-0.270483\pi\)
0.660172 + 0.751114i \(0.270483\pi\)
\(318\) 0 0
\(319\) −2.47885e8 −0.427546
\(320\) 0 0
\(321\) 5.42281e8 0.915075
\(322\) 0 0
\(323\) 5.78141e8 0.954607
\(324\) 0 0
\(325\) −1.00743e9 −1.62788
\(326\) 0 0
\(327\) 1.10346e9 1.74517
\(328\) 0 0
\(329\) −1.34618e9 −2.08410
\(330\) 0 0
\(331\) −1.59530e8 −0.241793 −0.120897 0.992665i \(-0.538577\pi\)
−0.120897 + 0.992665i \(0.538577\pi\)
\(332\) 0 0
\(333\) 5.81955e7 0.0863644
\(334\) 0 0
\(335\) −2.15109e7 −0.0312609
\(336\) 0 0
\(337\) −5.50578e8 −0.783636 −0.391818 0.920043i \(-0.628154\pi\)
−0.391818 + 0.920043i \(0.628154\pi\)
\(338\) 0 0
\(339\) −1.16809e9 −1.62846
\(340\) 0 0
\(341\) 8.73400e8 1.19281
\(342\) 0 0
\(343\) −4.49431e8 −0.601359
\(344\) 0 0
\(345\) −1.40314e8 −0.183964
\(346\) 0 0
\(347\) −1.20856e9 −1.55279 −0.776397 0.630244i \(-0.782955\pi\)
−0.776397 + 0.630244i \(0.782955\pi\)
\(348\) 0 0
\(349\) 1.10448e8 0.139081 0.0695404 0.997579i \(-0.477847\pi\)
0.0695404 + 0.997579i \(0.477847\pi\)
\(350\) 0 0
\(351\) −4.55263e8 −0.561937
\(352\) 0 0
\(353\) 6.42950e8 0.777975 0.388988 0.921243i \(-0.372825\pi\)
0.388988 + 0.921243i \(0.372825\pi\)
\(354\) 0 0
\(355\) −5.30159e7 −0.0628936
\(356\) 0 0
\(357\) 1.17463e9 1.36635
\(358\) 0 0
\(359\) 1.97988e8 0.225844 0.112922 0.993604i \(-0.463979\pi\)
0.112922 + 0.993604i \(0.463979\pi\)
\(360\) 0 0
\(361\) 2.63508e8 0.294794
\(362\) 0 0
\(363\) 2.04981e9 2.24926
\(364\) 0 0
\(365\) 2.73843e8 0.294765
\(366\) 0 0
\(367\) 2.68287e8 0.283315 0.141657 0.989916i \(-0.454757\pi\)
0.141657 + 0.989916i \(0.454757\pi\)
\(368\) 0 0
\(369\) −8.00961e8 −0.829887
\(370\) 0 0
\(371\) 9.50005e8 0.965866
\(372\) 0 0
\(373\) 4.72832e8 0.471765 0.235883 0.971782i \(-0.424202\pi\)
0.235883 + 0.971782i \(0.424202\pi\)
\(374\) 0 0
\(375\) −6.56517e8 −0.642890
\(376\) 0 0
\(377\) 4.70386e8 0.452126
\(378\) 0 0
\(379\) 1.27673e9 1.20465 0.602324 0.798251i \(-0.294241\pi\)
0.602324 + 0.798251i \(0.294241\pi\)
\(380\) 0 0
\(381\) 2.05259e9 1.90136
\(382\) 0 0
\(383\) −7.29377e8 −0.663371 −0.331685 0.943390i \(-0.607617\pi\)
−0.331685 + 0.943390i \(0.607617\pi\)
\(384\) 0 0
\(385\) −5.66093e8 −0.505563
\(386\) 0 0
\(387\) −1.11088e9 −0.974271
\(388\) 0 0
\(389\) 5.29163e8 0.455791 0.227896 0.973686i \(-0.426815\pi\)
0.227896 + 0.973686i \(0.426815\pi\)
\(390\) 0 0
\(391\) −5.49708e8 −0.465064
\(392\) 0 0
\(393\) 1.63265e9 1.35681
\(394\) 0 0
\(395\) 2.20184e7 0.0179761
\(396\) 0 0
\(397\) −1.59191e9 −1.27688 −0.638442 0.769670i \(-0.720421\pi\)
−0.638442 + 0.769670i \(0.720421\pi\)
\(398\) 0 0
\(399\) 2.35148e9 1.85326
\(400\) 0 0
\(401\) −1.03748e9 −0.803480 −0.401740 0.915754i \(-0.631594\pi\)
−0.401740 + 0.915754i \(0.631594\pi\)
\(402\) 0 0
\(403\) −1.65736e9 −1.26139
\(404\) 0 0
\(405\) −3.96597e8 −0.296659
\(406\) 0 0
\(407\) −2.55251e8 −0.187667
\(408\) 0 0
\(409\) −5.08439e8 −0.367458 −0.183729 0.982977i \(-0.558817\pi\)
−0.183729 + 0.982977i \(0.558817\pi\)
\(410\) 0 0
\(411\) −1.16961e9 −0.830989
\(412\) 0 0
\(413\) 7.75734e8 0.541861
\(414\) 0 0
\(415\) −1.07554e8 −0.0738685
\(416\) 0 0
\(417\) −6.74508e8 −0.455523
\(418\) 0 0
\(419\) −2.96853e8 −0.197148 −0.0985739 0.995130i \(-0.531428\pi\)
−0.0985739 + 0.995130i \(0.531428\pi\)
\(420\) 0 0
\(421\) 8.73091e8 0.570259 0.285130 0.958489i \(-0.407963\pi\)
0.285130 + 0.958489i \(0.407963\pi\)
\(422\) 0 0
\(423\) −1.99498e9 −1.28158
\(424\) 0 0
\(425\) −1.24439e9 −0.786310
\(426\) 0 0
\(427\) 7.98975e7 0.0496633
\(428\) 0 0
\(429\) −6.18119e9 −3.77983
\(430\) 0 0
\(431\) 2.90031e9 1.74491 0.872457 0.488691i \(-0.162525\pi\)
0.872457 + 0.488691i \(0.162525\pi\)
\(432\) 0 0
\(433\) 2.46197e9 1.45739 0.728695 0.684838i \(-0.240127\pi\)
0.728695 + 0.684838i \(0.240127\pi\)
\(434\) 0 0
\(435\) 1.48307e8 0.0863874
\(436\) 0 0
\(437\) −1.10046e9 −0.630796
\(438\) 0 0
\(439\) 1.88969e9 1.06602 0.533009 0.846110i \(-0.321061\pi\)
0.533009 + 0.846110i \(0.321061\pi\)
\(440\) 0 0
\(441\) 6.95280e8 0.386033
\(442\) 0 0
\(443\) 2.14187e8 0.117052 0.0585261 0.998286i \(-0.481360\pi\)
0.0585261 + 0.998286i \(0.481360\pi\)
\(444\) 0 0
\(445\) −1.75744e8 −0.0945411
\(446\) 0 0
\(447\) −2.13455e9 −1.13039
\(448\) 0 0
\(449\) −2.48824e9 −1.29727 −0.648635 0.761100i \(-0.724660\pi\)
−0.648635 + 0.761100i \(0.724660\pi\)
\(450\) 0 0
\(451\) 3.51310e9 1.80332
\(452\) 0 0
\(453\) 8.08635e8 0.408704
\(454\) 0 0
\(455\) 1.07422e9 0.534628
\(456\) 0 0
\(457\) 2.18777e9 1.07225 0.536123 0.844140i \(-0.319888\pi\)
0.536123 + 0.844140i \(0.319888\pi\)
\(458\) 0 0
\(459\) −5.62345e8 −0.271430
\(460\) 0 0
\(461\) −1.48625e9 −0.706541 −0.353271 0.935521i \(-0.614930\pi\)
−0.353271 + 0.935521i \(0.614930\pi\)
\(462\) 0 0
\(463\) −2.59340e9 −1.21433 −0.607164 0.794576i \(-0.707693\pi\)
−0.607164 + 0.794576i \(0.707693\pi\)
\(464\) 0 0
\(465\) −5.22547e8 −0.241013
\(466\) 0 0
\(467\) 1.84718e9 0.839266 0.419633 0.907694i \(-0.362159\pi\)
0.419633 + 0.907694i \(0.362159\pi\)
\(468\) 0 0
\(469\) −3.42766e8 −0.153424
\(470\) 0 0
\(471\) −1.26778e9 −0.559076
\(472\) 0 0
\(473\) 4.87244e9 2.11706
\(474\) 0 0
\(475\) −2.49114e9 −1.06652
\(476\) 0 0
\(477\) 1.40786e9 0.593945
\(478\) 0 0
\(479\) −3.45543e9 −1.43657 −0.718287 0.695747i \(-0.755073\pi\)
−0.718287 + 0.695747i \(0.755073\pi\)
\(480\) 0 0
\(481\) 4.84364e8 0.198456
\(482\) 0 0
\(483\) −2.23584e9 −0.902869
\(484\) 0 0
\(485\) −3.50658e6 −0.00139569
\(486\) 0 0
\(487\) 3.63187e9 1.42488 0.712442 0.701731i \(-0.247589\pi\)
0.712442 + 0.701731i \(0.247589\pi\)
\(488\) 0 0
\(489\) 6.69116e8 0.258774
\(490\) 0 0
\(491\) −1.32674e9 −0.505827 −0.252914 0.967489i \(-0.581389\pi\)
−0.252914 + 0.967489i \(0.581389\pi\)
\(492\) 0 0
\(493\) 5.81025e8 0.218389
\(494\) 0 0
\(495\) −8.38924e8 −0.310888
\(496\) 0 0
\(497\) −8.44785e8 −0.308673
\(498\) 0 0
\(499\) 2.32010e8 0.0835901 0.0417950 0.999126i \(-0.486692\pi\)
0.0417950 + 0.999126i \(0.486692\pi\)
\(500\) 0 0
\(501\) 3.45856e9 1.22875
\(502\) 0 0
\(503\) 1.22500e9 0.429187 0.214594 0.976703i \(-0.431157\pi\)
0.214594 + 0.976703i \(0.431157\pi\)
\(504\) 0 0
\(505\) 1.05248e9 0.363658
\(506\) 0 0
\(507\) 7.84103e9 2.67205
\(508\) 0 0
\(509\) 6.54747e8 0.220070 0.110035 0.993928i \(-0.464904\pi\)
0.110035 + 0.993928i \(0.464904\pi\)
\(510\) 0 0
\(511\) 4.36357e9 1.44667
\(512\) 0 0
\(513\) −1.12576e9 −0.368158
\(514\) 0 0
\(515\) −1.39942e9 −0.451462
\(516\) 0 0
\(517\) 8.75018e9 2.78484
\(518\) 0 0
\(519\) −5.54146e8 −0.173996
\(520\) 0 0
\(521\) −2.39015e8 −0.0740445 −0.0370222 0.999314i \(-0.511787\pi\)
−0.0370222 + 0.999314i \(0.511787\pi\)
\(522\) 0 0
\(523\) −5.76480e9 −1.76209 −0.881045 0.473032i \(-0.843159\pi\)
−0.881045 + 0.473032i \(0.843159\pi\)
\(524\) 0 0
\(525\) −5.06131e9 −1.52653
\(526\) 0 0
\(527\) −2.04719e9 −0.609285
\(528\) 0 0
\(529\) −2.35849e9 −0.692690
\(530\) 0 0
\(531\) 1.14960e9 0.333209
\(532\) 0 0
\(533\) −6.66644e9 −1.90699
\(534\) 0 0
\(535\) 6.12571e8 0.172949
\(536\) 0 0
\(537\) −5.82084e9 −1.62209
\(538\) 0 0
\(539\) −3.04957e9 −0.838837
\(540\) 0 0
\(541\) 5.89419e9 1.60042 0.800210 0.599720i \(-0.204722\pi\)
0.800210 + 0.599720i \(0.204722\pi\)
\(542\) 0 0
\(543\) 5.96296e9 1.59832
\(544\) 0 0
\(545\) 1.24649e9 0.329838
\(546\) 0 0
\(547\) −1.11664e9 −0.291714 −0.145857 0.989306i \(-0.546594\pi\)
−0.145857 + 0.989306i \(0.546594\pi\)
\(548\) 0 0
\(549\) 1.18404e8 0.0305397
\(550\) 0 0
\(551\) 1.16315e9 0.296215
\(552\) 0 0
\(553\) 3.50853e8 0.0882241
\(554\) 0 0
\(555\) 1.52715e8 0.0379188
\(556\) 0 0
\(557\) −2.94322e9 −0.721656 −0.360828 0.932632i \(-0.617506\pi\)
−0.360828 + 0.932632i \(0.617506\pi\)
\(558\) 0 0
\(559\) −9.24593e9 −2.23877
\(560\) 0 0
\(561\) −7.63506e9 −1.82576
\(562\) 0 0
\(563\) 5.84572e8 0.138057 0.0690285 0.997615i \(-0.478010\pi\)
0.0690285 + 0.997615i \(0.478010\pi\)
\(564\) 0 0
\(565\) −1.31949e9 −0.307778
\(566\) 0 0
\(567\) −6.31960e9 −1.45596
\(568\) 0 0
\(569\) −7.40531e9 −1.68519 −0.842597 0.538544i \(-0.818975\pi\)
−0.842597 + 0.538544i \(0.818975\pi\)
\(570\) 0 0
\(571\) 3.06829e8 0.0689715 0.0344858 0.999405i \(-0.489021\pi\)
0.0344858 + 0.999405i \(0.489021\pi\)
\(572\) 0 0
\(573\) 6.62280e9 1.47062
\(574\) 0 0
\(575\) 2.36862e9 0.519587
\(576\) 0 0
\(577\) −3.60748e8 −0.0781786 −0.0390893 0.999236i \(-0.512446\pi\)
−0.0390893 + 0.999236i \(0.512446\pi\)
\(578\) 0 0
\(579\) 8.34662e9 1.78705
\(580\) 0 0
\(581\) −1.71383e9 −0.362536
\(582\) 0 0
\(583\) −6.17503e9 −1.29062
\(584\) 0 0
\(585\) 1.59194e9 0.328761
\(586\) 0 0
\(587\) 6.35297e9 1.29641 0.648206 0.761465i \(-0.275520\pi\)
0.648206 + 0.761465i \(0.275520\pi\)
\(588\) 0 0
\(589\) −4.09826e9 −0.826411
\(590\) 0 0
\(591\) 5.80397e9 1.15656
\(592\) 0 0
\(593\) 8.89548e9 1.75177 0.875886 0.482517i \(-0.160277\pi\)
0.875886 + 0.482517i \(0.160277\pi\)
\(594\) 0 0
\(595\) 1.32688e9 0.258239
\(596\) 0 0
\(597\) −3.33151e9 −0.640813
\(598\) 0 0
\(599\) 5.27230e9 1.00232 0.501160 0.865355i \(-0.332907\pi\)
0.501160 + 0.865355i \(0.332907\pi\)
\(600\) 0 0
\(601\) 1.96322e9 0.368899 0.184449 0.982842i \(-0.440950\pi\)
0.184449 + 0.982842i \(0.440950\pi\)
\(602\) 0 0
\(603\) −5.07964e8 −0.0943457
\(604\) 0 0
\(605\) 2.31550e9 0.425110
\(606\) 0 0
\(607\) 6.98364e9 1.26742 0.633712 0.773569i \(-0.281531\pi\)
0.633712 + 0.773569i \(0.281531\pi\)
\(608\) 0 0
\(609\) 2.36321e9 0.423977
\(610\) 0 0
\(611\) −1.66043e10 −2.94494
\(612\) 0 0
\(613\) 1.02824e9 0.180295 0.0901476 0.995928i \(-0.471266\pi\)
0.0901476 + 0.995928i \(0.471266\pi\)
\(614\) 0 0
\(615\) −2.10185e9 −0.364367
\(616\) 0 0
\(617\) 1.41933e8 0.0243267 0.0121634 0.999926i \(-0.496128\pi\)
0.0121634 + 0.999926i \(0.496128\pi\)
\(618\) 0 0
\(619\) 7.00946e9 1.18787 0.593933 0.804515i \(-0.297574\pi\)
0.593933 + 0.804515i \(0.297574\pi\)
\(620\) 0 0
\(621\) 1.07039e9 0.179359
\(622\) 0 0
\(623\) −2.80040e9 −0.463994
\(624\) 0 0
\(625\) 4.97909e9 0.815774
\(626\) 0 0
\(627\) −1.52846e10 −2.47639
\(628\) 0 0
\(629\) 5.98291e8 0.0958595
\(630\) 0 0
\(631\) 6.69697e8 0.106115 0.0530573 0.998591i \(-0.483103\pi\)
0.0530573 + 0.998591i \(0.483103\pi\)
\(632\) 0 0
\(633\) 7.85193e9 1.23045
\(634\) 0 0
\(635\) 2.31864e9 0.359357
\(636\) 0 0
\(637\) 5.78685e9 0.887062
\(638\) 0 0
\(639\) −1.25193e9 −0.189814
\(640\) 0 0
\(641\) 1.05455e10 1.58149 0.790743 0.612148i \(-0.209694\pi\)
0.790743 + 0.612148i \(0.209694\pi\)
\(642\) 0 0
\(643\) 7.39491e9 1.09697 0.548485 0.836160i \(-0.315205\pi\)
0.548485 + 0.836160i \(0.315205\pi\)
\(644\) 0 0
\(645\) −2.91514e9 −0.427760
\(646\) 0 0
\(647\) −7.79266e9 −1.13115 −0.565575 0.824697i \(-0.691346\pi\)
−0.565575 + 0.824697i \(0.691346\pi\)
\(648\) 0 0
\(649\) −5.04227e9 −0.724052
\(650\) 0 0
\(651\) −8.32656e9 −1.18286
\(652\) 0 0
\(653\) −1.23260e10 −1.73232 −0.866159 0.499768i \(-0.833419\pi\)
−0.866159 + 0.499768i \(0.833419\pi\)
\(654\) 0 0
\(655\) 1.84427e9 0.256437
\(656\) 0 0
\(657\) 6.46661e9 0.889606
\(658\) 0 0
\(659\) 1.03521e10 1.40906 0.704532 0.709672i \(-0.251157\pi\)
0.704532 + 0.709672i \(0.251157\pi\)
\(660\) 0 0
\(661\) −2.58143e9 −0.347661 −0.173830 0.984776i \(-0.555614\pi\)
−0.173830 + 0.984776i \(0.555614\pi\)
\(662\) 0 0
\(663\) 1.44883e10 1.93072
\(664\) 0 0
\(665\) 2.65628e9 0.350266
\(666\) 0 0
\(667\) −1.10595e9 −0.144309
\(668\) 0 0
\(669\) 1.24137e10 1.60291
\(670\) 0 0
\(671\) −5.19334e8 −0.0663617
\(672\) 0 0
\(673\) −6.53171e9 −0.825989 −0.412995 0.910734i \(-0.635517\pi\)
−0.412995 + 0.910734i \(0.635517\pi\)
\(674\) 0 0
\(675\) 2.42307e9 0.303252
\(676\) 0 0
\(677\) −1.13228e10 −1.40247 −0.701233 0.712932i \(-0.747367\pi\)
−0.701233 + 0.712932i \(0.747367\pi\)
\(678\) 0 0
\(679\) −5.58758e7 −0.00684983
\(680\) 0 0
\(681\) −1.52832e10 −1.85439
\(682\) 0 0
\(683\) 1.30342e10 1.56535 0.782676 0.622429i \(-0.213854\pi\)
0.782676 + 0.622429i \(0.213854\pi\)
\(684\) 0 0
\(685\) −1.32122e9 −0.157057
\(686\) 0 0
\(687\) −1.39235e10 −1.63833
\(688\) 0 0
\(689\) 1.17177e10 1.36482
\(690\) 0 0
\(691\) −1.79137e9 −0.206544 −0.103272 0.994653i \(-0.532931\pi\)
−0.103272 + 0.994653i \(0.532931\pi\)
\(692\) 0 0
\(693\) −1.33679e10 −1.52579
\(694\) 0 0
\(695\) −7.61937e8 −0.0860939
\(696\) 0 0
\(697\) −8.23444e9 −0.921127
\(698\) 0 0
\(699\) −1.54327e10 −1.70912
\(700\) 0 0
\(701\) 1.08521e10 1.18987 0.594937 0.803772i \(-0.297177\pi\)
0.594937 + 0.803772i \(0.297177\pi\)
\(702\) 0 0
\(703\) 1.19772e9 0.130020
\(704\) 0 0
\(705\) −5.23515e9 −0.562687
\(706\) 0 0
\(707\) 1.67708e10 1.78478
\(708\) 0 0
\(709\) −7.12248e9 −0.750532 −0.375266 0.926917i \(-0.622449\pi\)
−0.375266 + 0.926917i \(0.622449\pi\)
\(710\) 0 0
\(711\) 5.19948e8 0.0542521
\(712\) 0 0
\(713\) 3.89671e9 0.402610
\(714\) 0 0
\(715\) −6.98240e9 −0.714387
\(716\) 0 0
\(717\) 6.49051e9 0.657600
\(718\) 0 0
\(719\) −5.34850e9 −0.536638 −0.268319 0.963330i \(-0.586468\pi\)
−0.268319 + 0.963330i \(0.586468\pi\)
\(720\) 0 0
\(721\) −2.22991e10 −2.21571
\(722\) 0 0
\(723\) 4.98574e9 0.490621
\(724\) 0 0
\(725\) −2.50356e9 −0.243992
\(726\) 0 0
\(727\) −7.41219e9 −0.715445 −0.357722 0.933828i \(-0.616447\pi\)
−0.357722 + 0.933828i \(0.616447\pi\)
\(728\) 0 0
\(729\) −4.88078e9 −0.466598
\(730\) 0 0
\(731\) −1.14206e10 −1.08138
\(732\) 0 0
\(733\) −1.23085e10 −1.15436 −0.577182 0.816615i \(-0.695848\pi\)
−0.577182 + 0.816615i \(0.695848\pi\)
\(734\) 0 0
\(735\) 1.82453e9 0.169490
\(736\) 0 0
\(737\) 2.22798e9 0.205010
\(738\) 0 0
\(739\) −1.11431e10 −1.01567 −0.507833 0.861455i \(-0.669553\pi\)
−0.507833 + 0.861455i \(0.669553\pi\)
\(740\) 0 0
\(741\) 2.90041e10 2.61876
\(742\) 0 0
\(743\) 5.48011e9 0.490150 0.245075 0.969504i \(-0.421187\pi\)
0.245075 + 0.969504i \(0.421187\pi\)
\(744\) 0 0
\(745\) −2.41123e9 −0.213645
\(746\) 0 0
\(747\) −2.53982e9 −0.222936
\(748\) 0 0
\(749\) 9.76106e9 0.848810
\(750\) 0 0
\(751\) 7.76351e9 0.668834 0.334417 0.942425i \(-0.391461\pi\)
0.334417 + 0.942425i \(0.391461\pi\)
\(752\) 0 0
\(753\) 9.92062e9 0.846752
\(754\) 0 0
\(755\) 9.13450e8 0.0772450
\(756\) 0 0
\(757\) 5.58618e9 0.468036 0.234018 0.972232i \(-0.424813\pi\)
0.234018 + 0.972232i \(0.424813\pi\)
\(758\) 0 0
\(759\) 1.45329e10 1.20644
\(760\) 0 0
\(761\) −1.26117e10 −1.03736 −0.518679 0.854969i \(-0.673576\pi\)
−0.518679 + 0.854969i \(0.673576\pi\)
\(762\) 0 0
\(763\) 1.98622e10 1.61880
\(764\) 0 0
\(765\) 1.96638e9 0.158800
\(766\) 0 0
\(767\) 9.56819e9 0.765678
\(768\) 0 0
\(769\) 1.33475e10 1.05842 0.529208 0.848492i \(-0.322489\pi\)
0.529208 + 0.848492i \(0.322489\pi\)
\(770\) 0 0
\(771\) 1.09476e10 0.860258
\(772\) 0 0
\(773\) −1.75882e8 −0.0136959 −0.00684797 0.999977i \(-0.502180\pi\)
−0.00684797 + 0.999977i \(0.502180\pi\)
\(774\) 0 0
\(775\) 8.82107e9 0.680715
\(776\) 0 0
\(777\) 2.43344e9 0.186100
\(778\) 0 0
\(779\) −1.64845e10 −1.24938
\(780\) 0 0
\(781\) 5.49110e9 0.412459
\(782\) 0 0
\(783\) −1.13137e9 −0.0842248
\(784\) 0 0
\(785\) −1.43211e9 −0.105665
\(786\) 0 0
\(787\) −1.27829e10 −0.934802 −0.467401 0.884046i \(-0.654809\pi\)
−0.467401 + 0.884046i \(0.654809\pi\)
\(788\) 0 0
\(789\) 2.16418e10 1.56864
\(790\) 0 0
\(791\) −2.10255e10 −1.51053
\(792\) 0 0
\(793\) 9.85486e8 0.0701769
\(794\) 0 0
\(795\) 3.69446e9 0.260775
\(796\) 0 0
\(797\) 2.22622e10 1.55763 0.778814 0.627255i \(-0.215822\pi\)
0.778814 + 0.627255i \(0.215822\pi\)
\(798\) 0 0
\(799\) −2.05098e10 −1.42248
\(800\) 0 0
\(801\) −4.15007e9 −0.285326
\(802\) 0 0
\(803\) −2.83632e10 −1.93308
\(804\) 0 0
\(805\) −2.52564e9 −0.170642
\(806\) 0 0
\(807\) 1.18578e10 0.794234
\(808\) 0 0
\(809\) 2.37057e10 1.57410 0.787050 0.616889i \(-0.211607\pi\)
0.787050 + 0.616889i \(0.211607\pi\)
\(810\) 0 0
\(811\) 2.02345e10 1.33205 0.666025 0.745930i \(-0.267995\pi\)
0.666025 + 0.745930i \(0.267995\pi\)
\(812\) 0 0
\(813\) −3.48673e10 −2.27563
\(814\) 0 0
\(815\) 7.55847e8 0.0489083
\(816\) 0 0
\(817\) −2.28630e10 −1.46675
\(818\) 0 0
\(819\) 2.53668e10 1.61351
\(820\) 0 0
\(821\) −2.64348e10 −1.66715 −0.833575 0.552407i \(-0.813710\pi\)
−0.833575 + 0.552407i \(0.813710\pi\)
\(822\) 0 0
\(823\) −2.73175e10 −1.70821 −0.854104 0.520102i \(-0.825894\pi\)
−0.854104 + 0.520102i \(0.825894\pi\)
\(824\) 0 0
\(825\) 3.28985e10 2.03980
\(826\) 0 0
\(827\) 1.63558e10 1.00555 0.502773 0.864418i \(-0.332313\pi\)
0.502773 + 0.864418i \(0.332313\pi\)
\(828\) 0 0
\(829\) −6.74626e9 −0.411266 −0.205633 0.978629i \(-0.565925\pi\)
−0.205633 + 0.978629i \(0.565925\pi\)
\(830\) 0 0
\(831\) −3.25785e10 −1.96937
\(832\) 0 0
\(833\) 7.14797e9 0.428474
\(834\) 0 0
\(835\) 3.90686e9 0.232233
\(836\) 0 0
\(837\) 3.98629e9 0.234979
\(838\) 0 0
\(839\) 1.37908e10 0.806161 0.403080 0.915165i \(-0.367940\pi\)
0.403080 + 0.915165i \(0.367940\pi\)
\(840\) 0 0
\(841\) −1.60809e10 −0.932234
\(842\) 0 0
\(843\) −2.62806e9 −0.151091
\(844\) 0 0
\(845\) 8.85738e9 0.505018
\(846\) 0 0
\(847\) 3.68965e10 2.08638
\(848\) 0 0
\(849\) −5.85007e9 −0.328084
\(850\) 0 0
\(851\) −1.13881e9 −0.0633431
\(852\) 0 0
\(853\) −7.09615e9 −0.391472 −0.195736 0.980657i \(-0.562710\pi\)
−0.195736 + 0.980657i \(0.562710\pi\)
\(854\) 0 0
\(855\) 3.93649e9 0.215391
\(856\) 0 0
\(857\) −2.09501e10 −1.13698 −0.568490 0.822690i \(-0.692472\pi\)
−0.568490 + 0.822690i \(0.692472\pi\)
\(858\) 0 0
\(859\) −1.36920e10 −0.737040 −0.368520 0.929620i \(-0.620135\pi\)
−0.368520 + 0.929620i \(0.620135\pi\)
\(860\) 0 0
\(861\) −3.34921e10 −1.78826
\(862\) 0 0
\(863\) 2.83553e10 1.50174 0.750872 0.660447i \(-0.229633\pi\)
0.750872 + 0.660447i \(0.229633\pi\)
\(864\) 0 0
\(865\) −6.25975e8 −0.0328852
\(866\) 0 0
\(867\) −7.53172e9 −0.392489
\(868\) 0 0
\(869\) −2.28055e9 −0.117888
\(870\) 0 0
\(871\) −4.22781e9 −0.216796
\(872\) 0 0
\(873\) −8.28054e7 −0.00421220
\(874\) 0 0
\(875\) −1.18173e10 −0.596335
\(876\) 0 0
\(877\) −2.71632e10 −1.35982 −0.679912 0.733294i \(-0.737982\pi\)
−0.679912 + 0.733294i \(0.737982\pi\)
\(878\) 0 0
\(879\) 2.48674e10 1.23501
\(880\) 0 0
\(881\) 1.39431e10 0.686978 0.343489 0.939157i \(-0.388391\pi\)
0.343489 + 0.939157i \(0.388391\pi\)
\(882\) 0 0
\(883\) −3.30370e10 −1.61487 −0.807437 0.589954i \(-0.799146\pi\)
−0.807437 + 0.589954i \(0.799146\pi\)
\(884\) 0 0
\(885\) 3.01674e9 0.146297
\(886\) 0 0
\(887\) −1.66589e10 −0.801521 −0.400760 0.916183i \(-0.631254\pi\)
−0.400760 + 0.916183i \(0.631254\pi\)
\(888\) 0 0
\(889\) 3.69466e10 1.76367
\(890\) 0 0
\(891\) 4.10774e10 1.94550
\(892\) 0 0
\(893\) −4.10585e10 −1.92940
\(894\) 0 0
\(895\) −6.57533e9 −0.306575
\(896\) 0 0
\(897\) −2.75776e10 −1.27580
\(898\) 0 0
\(899\) −4.11871e9 −0.189061
\(900\) 0 0
\(901\) 1.44738e10 0.659244
\(902\) 0 0
\(903\) −4.64514e10 −2.09938
\(904\) 0 0
\(905\) 6.73588e9 0.302082
\(906\) 0 0
\(907\) −1.81507e10 −0.807731 −0.403865 0.914818i \(-0.632334\pi\)
−0.403865 + 0.914818i \(0.632334\pi\)
\(908\) 0 0
\(909\) 2.48535e10 1.09752
\(910\) 0 0
\(911\) 1.78139e10 0.780630 0.390315 0.920681i \(-0.372366\pi\)
0.390315 + 0.920681i \(0.372366\pi\)
\(912\) 0 0
\(913\) 1.11399e10 0.484433
\(914\) 0 0
\(915\) 3.10712e8 0.0134086
\(916\) 0 0
\(917\) 2.93877e10 1.25856
\(918\) 0 0
\(919\) 1.94899e10 0.828333 0.414166 0.910201i \(-0.364073\pi\)
0.414166 + 0.910201i \(0.364073\pi\)
\(920\) 0 0
\(921\) 1.67980e10 0.708514
\(922\) 0 0
\(923\) −1.04199e10 −0.436171
\(924\) 0 0
\(925\) −2.57796e9 −0.107098
\(926\) 0 0
\(927\) −3.30462e10 −1.36252
\(928\) 0 0
\(929\) −3.36640e10 −1.37756 −0.688780 0.724971i \(-0.741853\pi\)
−0.688780 + 0.724971i \(0.741853\pi\)
\(930\) 0 0
\(931\) 1.43095e10 0.581167
\(932\) 0 0
\(933\) −2.55872e10 −1.03142
\(934\) 0 0
\(935\) −8.62472e9 −0.345068
\(936\) 0 0
\(937\) 4.56214e10 1.81167 0.905837 0.423626i \(-0.139243\pi\)
0.905837 + 0.423626i \(0.139243\pi\)
\(938\) 0 0
\(939\) −1.12224e10 −0.442341
\(940\) 0 0
\(941\) 5.00373e10 1.95763 0.978814 0.204752i \(-0.0656388\pi\)
0.978814 + 0.204752i \(0.0656388\pi\)
\(942\) 0 0
\(943\) 1.56738e10 0.608673
\(944\) 0 0
\(945\) −2.58371e9 −0.0995937
\(946\) 0 0
\(947\) 3.19677e10 1.22317 0.611584 0.791180i \(-0.290533\pi\)
0.611584 + 0.791180i \(0.290533\pi\)
\(948\) 0 0
\(949\) 5.38219e10 2.04422
\(950\) 0 0
\(951\) 4.64045e10 1.74956
\(952\) 0 0
\(953\) −3.10999e10 −1.16395 −0.581974 0.813208i \(-0.697719\pi\)
−0.581974 + 0.813208i \(0.697719\pi\)
\(954\) 0 0
\(955\) 7.48125e9 0.277947
\(956\) 0 0
\(957\) −1.53609e10 −0.566532
\(958\) 0 0
\(959\) −2.10530e10 −0.770813
\(960\) 0 0
\(961\) −1.30007e10 −0.472537
\(962\) 0 0
\(963\) 1.44654e10 0.521963
\(964\) 0 0
\(965\) 9.42851e9 0.337752
\(966\) 0 0
\(967\) 5.30176e10 1.88550 0.942751 0.333497i \(-0.108229\pi\)
0.942751 + 0.333497i \(0.108229\pi\)
\(968\) 0 0
\(969\) 3.58261e10 1.26493
\(970\) 0 0
\(971\) 4.84951e10 1.69993 0.849964 0.526841i \(-0.176624\pi\)
0.849964 + 0.526841i \(0.176624\pi\)
\(972\) 0 0
\(973\) −1.21411e10 −0.422537
\(974\) 0 0
\(975\) −6.24281e10 −2.15707
\(976\) 0 0
\(977\) −2.30489e9 −0.0790714 −0.0395357 0.999218i \(-0.512588\pi\)
−0.0395357 + 0.999218i \(0.512588\pi\)
\(978\) 0 0
\(979\) 1.82026e10 0.620004
\(980\) 0 0
\(981\) 2.94349e10 0.995455
\(982\) 0 0
\(983\) −2.41653e10 −0.811438 −0.405719 0.913998i \(-0.632979\pi\)
−0.405719 + 0.913998i \(0.632979\pi\)
\(984\) 0 0
\(985\) 6.55628e9 0.218590
\(986\) 0 0
\(987\) −8.34198e10 −2.76159
\(988\) 0 0
\(989\) 2.17386e10 0.714569
\(990\) 0 0
\(991\) −5.22086e10 −1.70406 −0.852029 0.523495i \(-0.824628\pi\)
−0.852029 + 0.523495i \(0.824628\pi\)
\(992\) 0 0
\(993\) −9.88572e9 −0.320395
\(994\) 0 0
\(995\) −3.76335e9 −0.121114
\(996\) 0 0
\(997\) 7.83024e9 0.250232 0.125116 0.992142i \(-0.460070\pi\)
0.125116 + 0.992142i \(0.460070\pi\)
\(998\) 0 0
\(999\) −1.16499e9 −0.0369696
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.c.1.2 yes 2
3.2 odd 2 288.8.a.k.1.2 2
4.3 odd 2 inner 32.8.a.c.1.1 2
8.3 odd 2 64.8.a.i.1.2 2
8.5 even 2 64.8.a.i.1.1 2
12.11 even 2 288.8.a.k.1.1 2
16.3 odd 4 256.8.b.i.129.1 4
16.5 even 4 256.8.b.i.129.2 4
16.11 odd 4 256.8.b.i.129.4 4
16.13 even 4 256.8.b.i.129.3 4
24.5 odd 2 576.8.a.bk.1.2 2
24.11 even 2 576.8.a.bk.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.8.a.c.1.1 2 4.3 odd 2 inner
32.8.a.c.1.2 yes 2 1.1 even 1 trivial
64.8.a.i.1.1 2 8.5 even 2
64.8.a.i.1.2 2 8.3 odd 2
256.8.b.i.129.1 4 16.3 odd 4
256.8.b.i.129.2 4 16.5 even 4
256.8.b.i.129.3 4 16.13 even 4
256.8.b.i.129.4 4 16.11 odd 4
288.8.a.k.1.1 2 12.11 even 2
288.8.a.k.1.2 2 3.2 odd 2
576.8.a.bk.1.1 2 24.11 even 2
576.8.a.bk.1.2 2 24.5 odd 2