Properties

Label 3381.2.a.bg.1.9
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
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] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 100x^{3} - 17x^{2} - 28x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-1.54861\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.54861 q^{2} -1.00000 q^{3} +0.398191 q^{4} -1.02582 q^{5} -1.54861 q^{6} -2.48058 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.54861 q^{2} -1.00000 q^{3} +0.398191 q^{4} -1.02582 q^{5} -1.54861 q^{6} -2.48058 q^{8} +1.00000 q^{9} -1.58859 q^{10} -6.58109 q^{11} -0.398191 q^{12} -0.0967108 q^{13} +1.02582 q^{15} -4.63783 q^{16} -2.79145 q^{17} +1.54861 q^{18} +5.99168 q^{19} -0.408472 q^{20} -10.1915 q^{22} +1.00000 q^{23} +2.48058 q^{24} -3.94770 q^{25} -0.149767 q^{26} -1.00000 q^{27} +2.41057 q^{29} +1.58859 q^{30} +10.1033 q^{31} -2.22103 q^{32} +6.58109 q^{33} -4.32286 q^{34} +0.398191 q^{36} +0.972553 q^{37} +9.27877 q^{38} +0.0967108 q^{39} +2.54462 q^{40} +7.47823 q^{41} +6.72262 q^{43} -2.62053 q^{44} -1.02582 q^{45} +1.54861 q^{46} -5.72919 q^{47} +4.63783 q^{48} -6.11344 q^{50} +2.79145 q^{51} -0.0385094 q^{52} -4.91819 q^{53} -1.54861 q^{54} +6.75100 q^{55} -5.99168 q^{57} +3.73303 q^{58} +4.38060 q^{59} +0.408472 q^{60} +14.4067 q^{61} +15.6460 q^{62} +5.83615 q^{64} +0.0992076 q^{65} +10.1915 q^{66} -4.09150 q^{67} -1.11153 q^{68} -1.00000 q^{69} +1.63827 q^{71} -2.48058 q^{72} -7.70434 q^{73} +1.50610 q^{74} +3.94770 q^{75} +2.38583 q^{76} +0.149767 q^{78} +7.79626 q^{79} +4.75756 q^{80} +1.00000 q^{81} +11.5809 q^{82} -12.4798 q^{83} +2.86351 q^{85} +10.4107 q^{86} -2.41057 q^{87} +16.3249 q^{88} -2.59708 q^{89} -1.58859 q^{90} +0.398191 q^{92} -10.1033 q^{93} -8.87228 q^{94} -6.14637 q^{95} +2.22103 q^{96} -0.921222 q^{97} -6.58109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} - 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 12 q^{8} + 10 q^{9} + 8 q^{10} - 2 q^{11} - 8 q^{12} + 16 q^{13} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 4 q^{18} + 26 q^{19} - 8 q^{22} + 10 q^{23} + 12 q^{24} + 14 q^{25} - 12 q^{26} - 10 q^{27} - 16 q^{29} - 8 q^{30} + 20 q^{31} - 8 q^{32} + 2 q^{33} - 4 q^{34} + 8 q^{36} + 8 q^{37} - 8 q^{38} - 16 q^{39} - 12 q^{40} + 22 q^{41} - 4 q^{43} - 24 q^{44} + 4 q^{45} - 4 q^{46} + 6 q^{47} - 4 q^{48} - 48 q^{50} - 12 q^{51} + 24 q^{52} - 30 q^{53} + 4 q^{54} + 48 q^{55} - 26 q^{57} + 24 q^{58} + 42 q^{59} + 14 q^{61} + 40 q^{62} + 8 q^{64} - 44 q^{65} + 8 q^{66} + 8 q^{68} - 10 q^{69} + 8 q^{71} - 12 q^{72} + 24 q^{73} + 8 q^{74} - 14 q^{75} + 32 q^{76} + 12 q^{78} + 32 q^{79} + 28 q^{80} + 10 q^{81} - 64 q^{82} + 28 q^{83} - 4 q^{85} - 4 q^{86} + 16 q^{87} + 20 q^{88} + 8 q^{90} + 8 q^{92} - 20 q^{93} + 8 q^{94} - 16 q^{95} + 8 q^{96} - 12 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.54861 1.09503 0.547516 0.836795i \(-0.315573\pi\)
0.547516 + 0.836795i \(0.315573\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.398191 0.199096
\(5\) −1.02582 −0.458760 −0.229380 0.973337i \(-0.573670\pi\)
−0.229380 + 0.973337i \(0.573670\pi\)
\(6\) −1.54861 −0.632217
\(7\) 0 0
\(8\) −2.48058 −0.877016
\(9\) 1.00000 0.333333
\(10\) −1.58859 −0.502356
\(11\) −6.58109 −1.98427 −0.992137 0.125157i \(-0.960057\pi\)
−0.992137 + 0.125157i \(0.960057\pi\)
\(12\) −0.398191 −0.114948
\(13\) −0.0967108 −0.0268228 −0.0134114 0.999910i \(-0.504269\pi\)
−0.0134114 + 0.999910i \(0.504269\pi\)
\(14\) 0 0
\(15\) 1.02582 0.264865
\(16\) −4.63783 −1.15946
\(17\) −2.79145 −0.677025 −0.338513 0.940962i \(-0.609924\pi\)
−0.338513 + 0.940962i \(0.609924\pi\)
\(18\) 1.54861 0.365011
\(19\) 5.99168 1.37459 0.687293 0.726381i \(-0.258799\pi\)
0.687293 + 0.726381i \(0.258799\pi\)
\(20\) −0.408472 −0.0913370
\(21\) 0 0
\(22\) −10.1915 −2.17284
\(23\) 1.00000 0.208514
\(24\) 2.48058 0.506345
\(25\) −3.94770 −0.789540
\(26\) −0.149767 −0.0293718
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.41057 0.447632 0.223816 0.974631i \(-0.428149\pi\)
0.223816 + 0.974631i \(0.428149\pi\)
\(30\) 1.58859 0.290036
\(31\) 10.1033 1.81460 0.907301 0.420481i \(-0.138139\pi\)
0.907301 + 0.420481i \(0.138139\pi\)
\(32\) −2.22103 −0.392626
\(33\) 6.58109 1.14562
\(34\) −4.32286 −0.741364
\(35\) 0 0
\(36\) 0.398191 0.0663652
\(37\) 0.972553 0.159887 0.0799434 0.996799i \(-0.474526\pi\)
0.0799434 + 0.996799i \(0.474526\pi\)
\(38\) 9.27877 1.50522
\(39\) 0.0967108 0.0154861
\(40\) 2.54462 0.402339
\(41\) 7.47823 1.16790 0.583952 0.811788i \(-0.301506\pi\)
0.583952 + 0.811788i \(0.301506\pi\)
\(42\) 0 0
\(43\) 6.72262 1.02519 0.512595 0.858630i \(-0.328684\pi\)
0.512595 + 0.858630i \(0.328684\pi\)
\(44\) −2.62053 −0.395060
\(45\) −1.02582 −0.152920
\(46\) 1.54861 0.228330
\(47\) −5.72919 −0.835689 −0.417844 0.908519i \(-0.637214\pi\)
−0.417844 + 0.908519i \(0.637214\pi\)
\(48\) 4.63783 0.669413
\(49\) 0 0
\(50\) −6.11344 −0.864571
\(51\) 2.79145 0.390881
\(52\) −0.0385094 −0.00534029
\(53\) −4.91819 −0.675566 −0.337783 0.941224i \(-0.609677\pi\)
−0.337783 + 0.941224i \(0.609677\pi\)
\(54\) −1.54861 −0.210739
\(55\) 6.75100 0.910305
\(56\) 0 0
\(57\) −5.99168 −0.793617
\(58\) 3.73303 0.490171
\(59\) 4.38060 0.570305 0.285153 0.958482i \(-0.407956\pi\)
0.285153 + 0.958482i \(0.407956\pi\)
\(60\) 0.408472 0.0527335
\(61\) 14.4067 1.84459 0.922294 0.386490i \(-0.126313\pi\)
0.922294 + 0.386490i \(0.126313\pi\)
\(62\) 15.6460 1.98705
\(63\) 0 0
\(64\) 5.83615 0.729518
\(65\) 0.0992076 0.0123052
\(66\) 10.1915 1.25449
\(67\) −4.09150 −0.499856 −0.249928 0.968264i \(-0.580407\pi\)
−0.249928 + 0.968264i \(0.580407\pi\)
\(68\) −1.11153 −0.134793
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 1.63827 0.194427 0.0972134 0.995264i \(-0.469007\pi\)
0.0972134 + 0.995264i \(0.469007\pi\)
\(72\) −2.48058 −0.292339
\(73\) −7.70434 −0.901725 −0.450862 0.892594i \(-0.648883\pi\)
−0.450862 + 0.892594i \(0.648883\pi\)
\(74\) 1.50610 0.175081
\(75\) 3.94770 0.455841
\(76\) 2.38583 0.273674
\(77\) 0 0
\(78\) 0.149767 0.0169578
\(79\) 7.79626 0.877148 0.438574 0.898695i \(-0.355484\pi\)
0.438574 + 0.898695i \(0.355484\pi\)
\(80\) 4.75756 0.531912
\(81\) 1.00000 0.111111
\(82\) 11.5809 1.27889
\(83\) −12.4798 −1.36983 −0.684916 0.728622i \(-0.740161\pi\)
−0.684916 + 0.728622i \(0.740161\pi\)
\(84\) 0 0
\(85\) 2.86351 0.310592
\(86\) 10.4107 1.12262
\(87\) −2.41057 −0.258440
\(88\) 16.3249 1.74024
\(89\) −2.59708 −0.275290 −0.137645 0.990482i \(-0.543953\pi\)
−0.137645 + 0.990482i \(0.543953\pi\)
\(90\) −1.58859 −0.167452
\(91\) 0 0
\(92\) 0.398191 0.0415143
\(93\) −10.1033 −1.04766
\(94\) −8.87228 −0.915106
\(95\) −6.14637 −0.630604
\(96\) 2.22103 0.226683
\(97\) −0.921222 −0.0935359 −0.0467679 0.998906i \(-0.514892\pi\)
−0.0467679 + 0.998906i \(0.514892\pi\)
\(98\) 0 0
\(99\) −6.58109 −0.661425
\(100\) −1.57194 −0.157194
\(101\) 16.4828 1.64010 0.820052 0.572289i \(-0.193944\pi\)
0.820052 + 0.572289i \(0.193944\pi\)
\(102\) 4.32286 0.428027
\(103\) 2.02573 0.199602 0.0998008 0.995007i \(-0.468179\pi\)
0.0998008 + 0.995007i \(0.468179\pi\)
\(104\) 0.239899 0.0235240
\(105\) 0 0
\(106\) −7.61636 −0.739766
\(107\) −16.9125 −1.63500 −0.817499 0.575930i \(-0.804640\pi\)
−0.817499 + 0.575930i \(0.804640\pi\)
\(108\) −0.398191 −0.0383160
\(109\) 11.3564 1.08775 0.543875 0.839167i \(-0.316957\pi\)
0.543875 + 0.839167i \(0.316957\pi\)
\(110\) 10.4547 0.996813
\(111\) −0.972553 −0.0923106
\(112\) 0 0
\(113\) −13.0458 −1.22724 −0.613620 0.789601i \(-0.710288\pi\)
−0.613620 + 0.789601i \(0.710288\pi\)
\(114\) −9.27877 −0.869036
\(115\) −1.02582 −0.0956580
\(116\) 0.959869 0.0891216
\(117\) −0.0967108 −0.00894092
\(118\) 6.78384 0.624503
\(119\) 0 0
\(120\) −2.54462 −0.232291
\(121\) 32.3108 2.93734
\(122\) 22.3103 2.01988
\(123\) −7.47823 −0.674290
\(124\) 4.02304 0.361280
\(125\) 9.17871 0.820968
\(126\) 0 0
\(127\) −14.0382 −1.24569 −0.622846 0.782344i \(-0.714024\pi\)
−0.622846 + 0.782344i \(0.714024\pi\)
\(128\) 13.4800 1.19147
\(129\) −6.72262 −0.591894
\(130\) 0.153634 0.0134746
\(131\) 2.72724 0.238280 0.119140 0.992877i \(-0.461986\pi\)
0.119140 + 0.992877i \(0.461986\pi\)
\(132\) 2.62053 0.228088
\(133\) 0 0
\(134\) −6.33614 −0.547359
\(135\) 1.02582 0.0882883
\(136\) 6.92439 0.593762
\(137\) 14.5708 1.24487 0.622435 0.782672i \(-0.286144\pi\)
0.622435 + 0.782672i \(0.286144\pi\)
\(138\) −1.54861 −0.131826
\(139\) 3.91027 0.331665 0.165832 0.986154i \(-0.446969\pi\)
0.165832 + 0.986154i \(0.446969\pi\)
\(140\) 0 0
\(141\) 5.72919 0.482485
\(142\) 2.53704 0.212904
\(143\) 0.636463 0.0532237
\(144\) −4.63783 −0.386486
\(145\) −2.47281 −0.205355
\(146\) −11.9310 −0.987417
\(147\) 0 0
\(148\) 0.387262 0.0318327
\(149\) 20.7291 1.69819 0.849097 0.528237i \(-0.177147\pi\)
0.849097 + 0.528237i \(0.177147\pi\)
\(150\) 6.11344 0.499161
\(151\) 4.02391 0.327461 0.163730 0.986505i \(-0.447647\pi\)
0.163730 + 0.986505i \(0.447647\pi\)
\(152\) −14.8628 −1.20553
\(153\) −2.79145 −0.225675
\(154\) 0 0
\(155\) −10.3641 −0.832466
\(156\) 0.0385094 0.00308322
\(157\) 5.63781 0.449947 0.224973 0.974365i \(-0.427771\pi\)
0.224973 + 0.974365i \(0.427771\pi\)
\(158\) 12.0734 0.960505
\(159\) 4.91819 0.390038
\(160\) 2.27837 0.180121
\(161\) 0 0
\(162\) 1.54861 0.121670
\(163\) 18.0803 1.41616 0.708079 0.706133i \(-0.249562\pi\)
0.708079 + 0.706133i \(0.249562\pi\)
\(164\) 2.97777 0.232525
\(165\) −6.75100 −0.525565
\(166\) −19.3263 −1.50001
\(167\) 16.9232 1.30955 0.654776 0.755823i \(-0.272763\pi\)
0.654776 + 0.755823i \(0.272763\pi\)
\(168\) 0 0
\(169\) −12.9906 −0.999281
\(170\) 4.43447 0.340108
\(171\) 5.99168 0.458195
\(172\) 2.67689 0.204111
\(173\) −5.89586 −0.448254 −0.224127 0.974560i \(-0.571953\pi\)
−0.224127 + 0.974560i \(0.571953\pi\)
\(174\) −3.73303 −0.283001
\(175\) 0 0
\(176\) 30.5220 2.30068
\(177\) −4.38060 −0.329266
\(178\) −4.02186 −0.301451
\(179\) −18.2272 −1.36237 −0.681183 0.732113i \(-0.738534\pi\)
−0.681183 + 0.732113i \(0.738534\pi\)
\(180\) −0.408472 −0.0304457
\(181\) −24.5989 −1.82842 −0.914210 0.405242i \(-0.867187\pi\)
−0.914210 + 0.405242i \(0.867187\pi\)
\(182\) 0 0
\(183\) −14.4067 −1.06497
\(184\) −2.48058 −0.182870
\(185\) −0.997662 −0.0733495
\(186\) −15.6460 −1.14722
\(187\) 18.3708 1.34340
\(188\) −2.28132 −0.166382
\(189\) 0 0
\(190\) −9.51832 −0.690532
\(191\) 19.6107 1.41898 0.709490 0.704716i \(-0.248925\pi\)
0.709490 + 0.704716i \(0.248925\pi\)
\(192\) −5.83615 −0.421188
\(193\) −10.5101 −0.756533 −0.378267 0.925697i \(-0.623480\pi\)
−0.378267 + 0.925697i \(0.623480\pi\)
\(194\) −1.42661 −0.102425
\(195\) −0.0992076 −0.00710441
\(196\) 0 0
\(197\) 7.33222 0.522399 0.261199 0.965285i \(-0.415882\pi\)
0.261199 + 0.965285i \(0.415882\pi\)
\(198\) −10.1915 −0.724281
\(199\) 7.36088 0.521799 0.260899 0.965366i \(-0.415981\pi\)
0.260899 + 0.965366i \(0.415981\pi\)
\(200\) 9.79257 0.692439
\(201\) 4.09150 0.288592
\(202\) 25.5255 1.79597
\(203\) 0 0
\(204\) 1.11153 0.0778226
\(205\) −7.67130 −0.535787
\(206\) 3.13707 0.218570
\(207\) 1.00000 0.0695048
\(208\) 0.448528 0.0310998
\(209\) −39.4318 −2.72755
\(210\) 0 0
\(211\) −0.515133 −0.0354632 −0.0177316 0.999843i \(-0.505644\pi\)
−0.0177316 + 0.999843i \(0.505644\pi\)
\(212\) −1.95838 −0.134502
\(213\) −1.63827 −0.112252
\(214\) −26.1909 −1.79038
\(215\) −6.89619 −0.470316
\(216\) 2.48058 0.168782
\(217\) 0 0
\(218\) 17.5867 1.19112
\(219\) 7.70434 0.520611
\(220\) 2.68819 0.181238
\(221\) 0.269963 0.0181597
\(222\) −1.50610 −0.101083
\(223\) 16.5692 1.10956 0.554778 0.831998i \(-0.312803\pi\)
0.554778 + 0.831998i \(0.312803\pi\)
\(224\) 0 0
\(225\) −3.94770 −0.263180
\(226\) −20.2028 −1.34387
\(227\) 2.13077 0.141424 0.0707120 0.997497i \(-0.477473\pi\)
0.0707120 + 0.997497i \(0.477473\pi\)
\(228\) −2.38583 −0.158006
\(229\) −14.4259 −0.953290 −0.476645 0.879096i \(-0.658147\pi\)
−0.476645 + 0.879096i \(0.658147\pi\)
\(230\) −1.58859 −0.104749
\(231\) 0 0
\(232\) −5.97961 −0.392580
\(233\) −2.22006 −0.145441 −0.0727206 0.997352i \(-0.523168\pi\)
−0.0727206 + 0.997352i \(0.523168\pi\)
\(234\) −0.149767 −0.00979059
\(235\) 5.87711 0.383380
\(236\) 1.74432 0.113545
\(237\) −7.79626 −0.506422
\(238\) 0 0
\(239\) 14.4611 0.935412 0.467706 0.883884i \(-0.345081\pi\)
0.467706 + 0.883884i \(0.345081\pi\)
\(240\) −4.75756 −0.307099
\(241\) −17.8066 −1.14703 −0.573513 0.819196i \(-0.694420\pi\)
−0.573513 + 0.819196i \(0.694420\pi\)
\(242\) 50.0368 3.21649
\(243\) −1.00000 −0.0641500
\(244\) 5.73662 0.367249
\(245\) 0 0
\(246\) −11.5809 −0.738369
\(247\) −0.579460 −0.0368702
\(248\) −25.0620 −1.59144
\(249\) 12.4798 0.790873
\(250\) 14.2142 0.898987
\(251\) 27.5761 1.74059 0.870293 0.492535i \(-0.163930\pi\)
0.870293 + 0.492535i \(0.163930\pi\)
\(252\) 0 0
\(253\) −6.58109 −0.413750
\(254\) −21.7397 −1.36407
\(255\) −2.86351 −0.179320
\(256\) 9.20292 0.575182
\(257\) −17.0021 −1.06056 −0.530280 0.847822i \(-0.677913\pi\)
−0.530280 + 0.847822i \(0.677913\pi\)
\(258\) −10.4107 −0.648143
\(259\) 0 0
\(260\) 0.0395036 0.00244991
\(261\) 2.41057 0.149211
\(262\) 4.22344 0.260925
\(263\) −20.0349 −1.23540 −0.617702 0.786412i \(-0.711936\pi\)
−0.617702 + 0.786412i \(0.711936\pi\)
\(264\) −16.3249 −1.00473
\(265\) 5.04517 0.309922
\(266\) 0 0
\(267\) 2.59708 0.158939
\(268\) −1.62920 −0.0995192
\(269\) 18.1520 1.10675 0.553373 0.832934i \(-0.313340\pi\)
0.553373 + 0.832934i \(0.313340\pi\)
\(270\) 1.58859 0.0966785
\(271\) 7.58594 0.460813 0.230406 0.973094i \(-0.425994\pi\)
0.230406 + 0.973094i \(0.425994\pi\)
\(272\) 12.9462 0.784981
\(273\) 0 0
\(274\) 22.5645 1.36317
\(275\) 25.9802 1.56666
\(276\) −0.398191 −0.0239683
\(277\) 19.5130 1.17242 0.586212 0.810158i \(-0.300619\pi\)
0.586212 + 0.810158i \(0.300619\pi\)
\(278\) 6.05548 0.363184
\(279\) 10.1033 0.604868
\(280\) 0 0
\(281\) 1.47664 0.0880892 0.0440446 0.999030i \(-0.485976\pi\)
0.0440446 + 0.999030i \(0.485976\pi\)
\(282\) 8.87228 0.528337
\(283\) −1.23515 −0.0734218 −0.0367109 0.999326i \(-0.511688\pi\)
−0.0367109 + 0.999326i \(0.511688\pi\)
\(284\) 0.652345 0.0387095
\(285\) 6.14637 0.364079
\(286\) 0.985632 0.0582817
\(287\) 0 0
\(288\) −2.22103 −0.130875
\(289\) −9.20783 −0.541637
\(290\) −3.82941 −0.224871
\(291\) 0.921222 0.0540030
\(292\) −3.06780 −0.179529
\(293\) 9.74857 0.569517 0.284759 0.958599i \(-0.408087\pi\)
0.284759 + 0.958599i \(0.408087\pi\)
\(294\) 0 0
\(295\) −4.49369 −0.261633
\(296\) −2.41249 −0.140223
\(297\) 6.58109 0.381874
\(298\) 32.1013 1.85958
\(299\) −0.0967108 −0.00559293
\(300\) 1.57194 0.0907560
\(301\) 0 0
\(302\) 6.23146 0.358580
\(303\) −16.4828 −0.946915
\(304\) −27.7884 −1.59377
\(305\) −14.7786 −0.846222
\(306\) −4.32286 −0.247121
\(307\) 27.7926 1.58621 0.793105 0.609085i \(-0.208463\pi\)
0.793105 + 0.609085i \(0.208463\pi\)
\(308\) 0 0
\(309\) −2.02573 −0.115240
\(310\) −16.0500 −0.911577
\(311\) −3.96040 −0.224574 −0.112287 0.993676i \(-0.535818\pi\)
−0.112287 + 0.993676i \(0.535818\pi\)
\(312\) −0.239899 −0.0135816
\(313\) −4.81964 −0.272422 −0.136211 0.990680i \(-0.543492\pi\)
−0.136211 + 0.990680i \(0.543492\pi\)
\(314\) 8.73077 0.492706
\(315\) 0 0
\(316\) 3.10440 0.174636
\(317\) −16.8489 −0.946326 −0.473163 0.880975i \(-0.656888\pi\)
−0.473163 + 0.880975i \(0.656888\pi\)
\(318\) 7.61636 0.427104
\(319\) −15.8642 −0.888225
\(320\) −5.98682 −0.334673
\(321\) 16.9125 0.943966
\(322\) 0 0
\(323\) −16.7254 −0.930629
\(324\) 0.398191 0.0221217
\(325\) 0.381785 0.0211776
\(326\) 27.9993 1.55074
\(327\) −11.3564 −0.628012
\(328\) −18.5503 −1.02427
\(329\) 0 0
\(330\) −10.4547 −0.575510
\(331\) 10.0553 0.552689 0.276344 0.961059i \(-0.410877\pi\)
0.276344 + 0.961059i \(0.410877\pi\)
\(332\) −4.96933 −0.272728
\(333\) 0.972553 0.0532956
\(334\) 26.2074 1.43400
\(335\) 4.19713 0.229314
\(336\) 0 0
\(337\) −28.6099 −1.55848 −0.779240 0.626725i \(-0.784395\pi\)
−0.779240 + 0.626725i \(0.784395\pi\)
\(338\) −20.1174 −1.09424
\(339\) 13.0458 0.708548
\(340\) 1.14023 0.0618375
\(341\) −66.4906 −3.60067
\(342\) 9.27877 0.501738
\(343\) 0 0
\(344\) −16.6760 −0.899109
\(345\) 1.02582 0.0552282
\(346\) −9.13039 −0.490853
\(347\) 19.4460 1.04392 0.521958 0.852971i \(-0.325202\pi\)
0.521958 + 0.852971i \(0.325202\pi\)
\(348\) −0.959869 −0.0514544
\(349\) −11.4642 −0.613663 −0.306832 0.951764i \(-0.599269\pi\)
−0.306832 + 0.951764i \(0.599269\pi\)
\(350\) 0 0
\(351\) 0.0967108 0.00516204
\(352\) 14.6168 0.779078
\(353\) 31.1454 1.65770 0.828850 0.559471i \(-0.188996\pi\)
0.828850 + 0.559471i \(0.188996\pi\)
\(354\) −6.78384 −0.360557
\(355\) −1.68057 −0.0891952
\(356\) −1.03413 −0.0548090
\(357\) 0 0
\(358\) −28.2268 −1.49183
\(359\) −15.8708 −0.837628 −0.418814 0.908072i \(-0.637554\pi\)
−0.418814 + 0.908072i \(0.637554\pi\)
\(360\) 2.54462 0.134113
\(361\) 16.9002 0.889485
\(362\) −38.0940 −2.00218
\(363\) −32.3108 −1.69588
\(364\) 0 0
\(365\) 7.90324 0.413675
\(366\) −22.3103 −1.16618
\(367\) 0.586635 0.0306221 0.0153111 0.999883i \(-0.495126\pi\)
0.0153111 + 0.999883i \(0.495126\pi\)
\(368\) −4.63783 −0.241763
\(369\) 7.47823 0.389301
\(370\) −1.54499 −0.0803201
\(371\) 0 0
\(372\) −4.02304 −0.208585
\(373\) −10.9030 −0.564537 −0.282268 0.959335i \(-0.591087\pi\)
−0.282268 + 0.959335i \(0.591087\pi\)
\(374\) 28.4491 1.47107
\(375\) −9.17871 −0.473986
\(376\) 14.2117 0.732913
\(377\) −0.233128 −0.0120067
\(378\) 0 0
\(379\) 22.2093 1.14081 0.570407 0.821362i \(-0.306786\pi\)
0.570407 + 0.821362i \(0.306786\pi\)
\(380\) −2.44743 −0.125551
\(381\) 14.0382 0.719201
\(382\) 30.3693 1.55383
\(383\) 16.1733 0.826417 0.413209 0.910636i \(-0.364408\pi\)
0.413209 + 0.910636i \(0.364408\pi\)
\(384\) −13.4800 −0.687897
\(385\) 0 0
\(386\) −16.2760 −0.828429
\(387\) 6.72262 0.341730
\(388\) −0.366822 −0.0186226
\(389\) −8.60710 −0.436397 −0.218199 0.975904i \(-0.570018\pi\)
−0.218199 + 0.975904i \(0.570018\pi\)
\(390\) −0.153634 −0.00777955
\(391\) −2.79145 −0.141170
\(392\) 0 0
\(393\) −2.72724 −0.137571
\(394\) 11.3547 0.572044
\(395\) −7.99754 −0.402400
\(396\) −2.62053 −0.131687
\(397\) 5.16626 0.259287 0.129644 0.991561i \(-0.458617\pi\)
0.129644 + 0.991561i \(0.458617\pi\)
\(398\) 11.3991 0.571386
\(399\) 0 0
\(400\) 18.3087 0.915437
\(401\) 25.6945 1.28312 0.641561 0.767072i \(-0.278287\pi\)
0.641561 + 0.767072i \(0.278287\pi\)
\(402\) 6.33614 0.316018
\(403\) −0.977096 −0.0486726
\(404\) 6.56333 0.326538
\(405\) −1.02582 −0.0509733
\(406\) 0 0
\(407\) −6.40046 −0.317259
\(408\) −6.92439 −0.342809
\(409\) −14.7987 −0.731749 −0.365874 0.930664i \(-0.619230\pi\)
−0.365874 + 0.930664i \(0.619230\pi\)
\(410\) −11.8799 −0.586704
\(411\) −14.5708 −0.718726
\(412\) 0.806630 0.0397398
\(413\) 0 0
\(414\) 1.54861 0.0761100
\(415\) 12.8020 0.628424
\(416\) 0.214798 0.0105313
\(417\) −3.91027 −0.191487
\(418\) −61.0644 −2.98676
\(419\) −6.43007 −0.314130 −0.157065 0.987588i \(-0.550203\pi\)
−0.157065 + 0.987588i \(0.550203\pi\)
\(420\) 0 0
\(421\) −18.1314 −0.883670 −0.441835 0.897096i \(-0.645672\pi\)
−0.441835 + 0.897096i \(0.645672\pi\)
\(422\) −0.797740 −0.0388334
\(423\) −5.72919 −0.278563
\(424\) 12.2000 0.592482
\(425\) 11.0198 0.534538
\(426\) −2.53704 −0.122920
\(427\) 0 0
\(428\) −6.73443 −0.325521
\(429\) −0.636463 −0.0307287
\(430\) −10.6795 −0.515011
\(431\) −15.6712 −0.754856 −0.377428 0.926039i \(-0.623191\pi\)
−0.377428 + 0.926039i \(0.623191\pi\)
\(432\) 4.63783 0.223138
\(433\) −23.4459 −1.12674 −0.563368 0.826206i \(-0.690495\pi\)
−0.563368 + 0.826206i \(0.690495\pi\)
\(434\) 0 0
\(435\) 2.47281 0.118562
\(436\) 4.52203 0.216566
\(437\) 5.99168 0.286621
\(438\) 11.9310 0.570086
\(439\) 6.77828 0.323510 0.161755 0.986831i \(-0.448285\pi\)
0.161755 + 0.986831i \(0.448285\pi\)
\(440\) −16.7464 −0.798352
\(441\) 0 0
\(442\) 0.418067 0.0198854
\(443\) −29.1655 −1.38569 −0.692847 0.721085i \(-0.743644\pi\)
−0.692847 + 0.721085i \(0.743644\pi\)
\(444\) −0.387262 −0.0183786
\(445\) 2.66413 0.126292
\(446\) 25.6592 1.21500
\(447\) −20.7291 −0.980453
\(448\) 0 0
\(449\) 36.9760 1.74501 0.872503 0.488609i \(-0.162496\pi\)
0.872503 + 0.488609i \(0.162496\pi\)
\(450\) −6.11344 −0.288190
\(451\) −49.2150 −2.31744
\(452\) −5.19470 −0.244338
\(453\) −4.02391 −0.189060
\(454\) 3.29973 0.154864
\(455\) 0 0
\(456\) 14.8628 0.696015
\(457\) 27.7671 1.29889 0.649445 0.760409i \(-0.275001\pi\)
0.649445 + 0.760409i \(0.275001\pi\)
\(458\) −22.3401 −1.04388
\(459\) 2.79145 0.130294
\(460\) −0.408472 −0.0190451
\(461\) 26.9409 1.25476 0.627380 0.778713i \(-0.284127\pi\)
0.627380 + 0.778713i \(0.284127\pi\)
\(462\) 0 0
\(463\) 26.8087 1.24591 0.622954 0.782258i \(-0.285932\pi\)
0.622954 + 0.782258i \(0.285932\pi\)
\(464\) −11.1798 −0.519010
\(465\) 10.3641 0.480625
\(466\) −3.43801 −0.159263
\(467\) 16.0527 0.742833 0.371416 0.928466i \(-0.378872\pi\)
0.371416 + 0.928466i \(0.378872\pi\)
\(468\) −0.0385094 −0.00178010
\(469\) 0 0
\(470\) 9.10134 0.419814
\(471\) −5.63781 −0.259777
\(472\) −10.8664 −0.500167
\(473\) −44.2422 −2.03426
\(474\) −12.0734 −0.554548
\(475\) −23.6533 −1.08529
\(476\) 0 0
\(477\) −4.91819 −0.225189
\(478\) 22.3946 1.02431
\(479\) −8.08384 −0.369360 −0.184680 0.982799i \(-0.559125\pi\)
−0.184680 + 0.982799i \(0.559125\pi\)
\(480\) −2.27837 −0.103993
\(481\) −0.0940564 −0.00428860
\(482\) −27.5755 −1.25603
\(483\) 0 0
\(484\) 12.8659 0.584812
\(485\) 0.945005 0.0429105
\(486\) −1.54861 −0.0702464
\(487\) −22.2419 −1.00787 −0.503937 0.863740i \(-0.668116\pi\)
−0.503937 + 0.863740i \(0.668116\pi\)
\(488\) −35.7369 −1.61773
\(489\) −18.0803 −0.817619
\(490\) 0 0
\(491\) −25.0527 −1.13061 −0.565307 0.824881i \(-0.691242\pi\)
−0.565307 + 0.824881i \(0.691242\pi\)
\(492\) −2.97777 −0.134248
\(493\) −6.72898 −0.303058
\(494\) −0.897357 −0.0403740
\(495\) 6.75100 0.303435
\(496\) −46.8573 −2.10395
\(497\) 0 0
\(498\) 19.3263 0.866031
\(499\) 22.4671 1.00576 0.502882 0.864355i \(-0.332273\pi\)
0.502882 + 0.864355i \(0.332273\pi\)
\(500\) 3.65488 0.163451
\(501\) −16.9232 −0.756071
\(502\) 42.7045 1.90600
\(503\) 14.3313 0.638999 0.319500 0.947586i \(-0.396485\pi\)
0.319500 + 0.947586i \(0.396485\pi\)
\(504\) 0 0
\(505\) −16.9084 −0.752413
\(506\) −10.1915 −0.453069
\(507\) 12.9906 0.576935
\(508\) −5.58990 −0.248012
\(509\) −20.5708 −0.911786 −0.455893 0.890035i \(-0.650680\pi\)
−0.455893 + 0.890035i \(0.650680\pi\)
\(510\) −4.43447 −0.196361
\(511\) 0 0
\(512\) −12.7082 −0.561629
\(513\) −5.99168 −0.264539
\(514\) −26.3296 −1.16135
\(515\) −2.07803 −0.0915691
\(516\) −2.67689 −0.117844
\(517\) 37.7044 1.65824
\(518\) 0 0
\(519\) 5.89586 0.258800
\(520\) −0.246092 −0.0107919
\(521\) −30.8216 −1.35032 −0.675160 0.737672i \(-0.735925\pi\)
−0.675160 + 0.737672i \(0.735925\pi\)
\(522\) 3.73303 0.163390
\(523\) 3.49880 0.152992 0.0764958 0.997070i \(-0.475627\pi\)
0.0764958 + 0.997070i \(0.475627\pi\)
\(524\) 1.08597 0.0474406
\(525\) 0 0
\(526\) −31.0262 −1.35281
\(527\) −28.2028 −1.22853
\(528\) −30.5220 −1.32830
\(529\) 1.00000 0.0434783
\(530\) 7.81299 0.339375
\(531\) 4.38060 0.190102
\(532\) 0 0
\(533\) −0.723226 −0.0313264
\(534\) 4.02186 0.174043
\(535\) 17.3492 0.750071
\(536\) 10.1493 0.438382
\(537\) 18.2272 0.786563
\(538\) 28.1103 1.21192
\(539\) 0 0
\(540\) 0.408472 0.0175778
\(541\) −35.7811 −1.53835 −0.769175 0.639038i \(-0.779333\pi\)
−0.769175 + 0.639038i \(0.779333\pi\)
\(542\) 11.7477 0.504605
\(543\) 24.5989 1.05564
\(544\) 6.19988 0.265818
\(545\) −11.6496 −0.499015
\(546\) 0 0
\(547\) 28.2676 1.20864 0.604318 0.796743i \(-0.293446\pi\)
0.604318 + 0.796743i \(0.293446\pi\)
\(548\) 5.80198 0.247848
\(549\) 14.4067 0.614862
\(550\) 40.2331 1.71555
\(551\) 14.4434 0.615308
\(552\) 2.48058 0.105580
\(553\) 0 0
\(554\) 30.2180 1.28384
\(555\) 0.997662 0.0423484
\(556\) 1.55704 0.0660330
\(557\) 13.3099 0.563960 0.281980 0.959420i \(-0.409009\pi\)
0.281980 + 0.959420i \(0.409009\pi\)
\(558\) 15.6460 0.662350
\(559\) −0.650151 −0.0274984
\(560\) 0 0
\(561\) −18.3708 −0.775614
\(562\) 2.28674 0.0964605
\(563\) −20.4179 −0.860513 −0.430257 0.902707i \(-0.641577\pi\)
−0.430257 + 0.902707i \(0.641577\pi\)
\(564\) 2.28132 0.0960607
\(565\) 13.3826 0.563008
\(566\) −1.91276 −0.0803992
\(567\) 0 0
\(568\) −4.06385 −0.170515
\(569\) −7.25109 −0.303982 −0.151991 0.988382i \(-0.548568\pi\)
−0.151991 + 0.988382i \(0.548568\pi\)
\(570\) 9.51832 0.398679
\(571\) 20.9061 0.874893 0.437446 0.899244i \(-0.355883\pi\)
0.437446 + 0.899244i \(0.355883\pi\)
\(572\) 0.253434 0.0105966
\(573\) −19.6107 −0.819248
\(574\) 0 0
\(575\) −3.94770 −0.164630
\(576\) 5.83615 0.243173
\(577\) −28.3400 −1.17981 −0.589905 0.807473i \(-0.700835\pi\)
−0.589905 + 0.807473i \(0.700835\pi\)
\(578\) −14.2593 −0.593110
\(579\) 10.5101 0.436785
\(580\) −0.984650 −0.0408854
\(581\) 0 0
\(582\) 1.42661 0.0591350
\(583\) 32.3671 1.34051
\(584\) 19.1112 0.790827
\(585\) 0.0992076 0.00410173
\(586\) 15.0967 0.623640
\(587\) 23.3149 0.962310 0.481155 0.876635i \(-0.340217\pi\)
0.481155 + 0.876635i \(0.340217\pi\)
\(588\) 0 0
\(589\) 60.5356 2.49433
\(590\) −6.95898 −0.286497
\(591\) −7.33222 −0.301607
\(592\) −4.51053 −0.185382
\(593\) −8.07712 −0.331687 −0.165844 0.986152i \(-0.553035\pi\)
−0.165844 + 0.986152i \(0.553035\pi\)
\(594\) 10.1915 0.418164
\(595\) 0 0
\(596\) 8.25415 0.338103
\(597\) −7.36088 −0.301261
\(598\) −0.149767 −0.00612444
\(599\) 22.2393 0.908673 0.454337 0.890830i \(-0.349876\pi\)
0.454337 + 0.890830i \(0.349876\pi\)
\(600\) −9.79257 −0.399780
\(601\) −34.9803 −1.42688 −0.713438 0.700719i \(-0.752863\pi\)
−0.713438 + 0.700719i \(0.752863\pi\)
\(602\) 0 0
\(603\) −4.09150 −0.166619
\(604\) 1.60228 0.0651960
\(605\) −33.1450 −1.34753
\(606\) −25.5255 −1.03690
\(607\) −14.3826 −0.583771 −0.291886 0.956453i \(-0.594283\pi\)
−0.291886 + 0.956453i \(0.594283\pi\)
\(608\) −13.3077 −0.539698
\(609\) 0 0
\(610\) −22.8863 −0.926640
\(611\) 0.554075 0.0224155
\(612\) −1.11153 −0.0449309
\(613\) −30.0109 −1.21213 −0.606065 0.795415i \(-0.707253\pi\)
−0.606065 + 0.795415i \(0.707253\pi\)
\(614\) 43.0400 1.73695
\(615\) 7.67130 0.309337
\(616\) 0 0
\(617\) −14.0763 −0.566690 −0.283345 0.959018i \(-0.591444\pi\)
−0.283345 + 0.959018i \(0.591444\pi\)
\(618\) −3.13707 −0.126192
\(619\) 27.4497 1.10330 0.551648 0.834077i \(-0.313999\pi\)
0.551648 + 0.834077i \(0.313999\pi\)
\(620\) −4.12690 −0.165740
\(621\) −1.00000 −0.0401286
\(622\) −6.13311 −0.245916
\(623\) 0 0
\(624\) −0.448528 −0.0179555
\(625\) 10.3228 0.412913
\(626\) −7.46373 −0.298311
\(627\) 39.4318 1.57475
\(628\) 2.24493 0.0895824
\(629\) −2.71483 −0.108247
\(630\) 0 0
\(631\) 22.9335 0.912970 0.456485 0.889731i \(-0.349108\pi\)
0.456485 + 0.889731i \(0.349108\pi\)
\(632\) −19.3392 −0.769273
\(633\) 0.515133 0.0204747
\(634\) −26.0923 −1.03626
\(635\) 14.4007 0.571473
\(636\) 1.95838 0.0776549
\(637\) 0 0
\(638\) −24.5674 −0.972635
\(639\) 1.63827 0.0648089
\(640\) −13.8280 −0.546599
\(641\) −4.93467 −0.194908 −0.0974539 0.995240i \(-0.531070\pi\)
−0.0974539 + 0.995240i \(0.531070\pi\)
\(642\) 26.1909 1.03367
\(643\) 2.11089 0.0832453 0.0416226 0.999133i \(-0.486747\pi\)
0.0416226 + 0.999133i \(0.486747\pi\)
\(644\) 0 0
\(645\) 6.89619 0.271537
\(646\) −25.9012 −1.01907
\(647\) 29.9015 1.17555 0.587774 0.809025i \(-0.300004\pi\)
0.587774 + 0.809025i \(0.300004\pi\)
\(648\) −2.48058 −0.0974462
\(649\) −28.8291 −1.13164
\(650\) 0.591236 0.0231902
\(651\) 0 0
\(652\) 7.19941 0.281951
\(653\) −31.7412 −1.24213 −0.621065 0.783759i \(-0.713300\pi\)
−0.621065 + 0.783759i \(0.713300\pi\)
\(654\) −17.5867 −0.687694
\(655\) −2.79766 −0.109313
\(656\) −34.6828 −1.35413
\(657\) −7.70434 −0.300575
\(658\) 0 0
\(659\) 2.60404 0.101439 0.0507195 0.998713i \(-0.483849\pi\)
0.0507195 + 0.998713i \(0.483849\pi\)
\(660\) −2.68819 −0.104638
\(661\) 38.2421 1.48744 0.743722 0.668489i \(-0.233059\pi\)
0.743722 + 0.668489i \(0.233059\pi\)
\(662\) 15.5717 0.605212
\(663\) −0.269963 −0.0104845
\(664\) 30.9570 1.20136
\(665\) 0 0
\(666\) 1.50610 0.0583604
\(667\) 2.41057 0.0933377
\(668\) 6.73865 0.260726
\(669\) −16.5692 −0.640603
\(670\) 6.49972 0.251106
\(671\) −94.8117 −3.66017
\(672\) 0 0
\(673\) 31.2699 1.20536 0.602682 0.797981i \(-0.294099\pi\)
0.602682 + 0.797981i \(0.294099\pi\)
\(674\) −44.3056 −1.70659
\(675\) 3.94770 0.151947
\(676\) −5.17276 −0.198952
\(677\) 10.1973 0.391915 0.195958 0.980612i \(-0.437219\pi\)
0.195958 + 0.980612i \(0.437219\pi\)
\(678\) 20.2028 0.775883
\(679\) 0 0
\(680\) −7.10317 −0.272394
\(681\) −2.13077 −0.0816512
\(682\) −102.968 −3.94285
\(683\) −12.6603 −0.484432 −0.242216 0.970222i \(-0.577874\pi\)
−0.242216 + 0.970222i \(0.577874\pi\)
\(684\) 2.38583 0.0912246
\(685\) −14.9470 −0.571096
\(686\) 0 0
\(687\) 14.4259 0.550382
\(688\) −31.1784 −1.18866
\(689\) 0.475642 0.0181205
\(690\) 1.58859 0.0604766
\(691\) −6.74340 −0.256531 −0.128266 0.991740i \(-0.540941\pi\)
−0.128266 + 0.991740i \(0.540941\pi\)
\(692\) −2.34768 −0.0892454
\(693\) 0 0
\(694\) 30.1143 1.14312
\(695\) −4.01122 −0.152154
\(696\) 5.97961 0.226656
\(697\) −20.8751 −0.790700
\(698\) −17.7535 −0.671981
\(699\) 2.22006 0.0839705
\(700\) 0 0
\(701\) 19.1107 0.721803 0.360901 0.932604i \(-0.382469\pi\)
0.360901 + 0.932604i \(0.382469\pi\)
\(702\) 0.149767 0.00565260
\(703\) 5.82722 0.219778
\(704\) −38.4082 −1.44756
\(705\) −5.87711 −0.221345
\(706\) 48.2320 1.81524
\(707\) 0 0
\(708\) −1.74432 −0.0655554
\(709\) 10.9432 0.410980 0.205490 0.978659i \(-0.434121\pi\)
0.205490 + 0.978659i \(0.434121\pi\)
\(710\) −2.60254 −0.0976716
\(711\) 7.79626 0.292383
\(712\) 6.44225 0.241434
\(713\) 10.1033 0.378371
\(714\) 0 0
\(715\) −0.652895 −0.0244169
\(716\) −7.25792 −0.271241
\(717\) −14.4611 −0.540060
\(718\) −24.5776 −0.917230
\(719\) 45.3780 1.69232 0.846158 0.532932i \(-0.178910\pi\)
0.846158 + 0.532932i \(0.178910\pi\)
\(720\) 4.75756 0.177304
\(721\) 0 0
\(722\) 26.1718 0.974014
\(723\) 17.8066 0.662236
\(724\) −9.79505 −0.364030
\(725\) −9.51621 −0.353423
\(726\) −50.0368 −1.85704
\(727\) 19.4657 0.721942 0.360971 0.932577i \(-0.382445\pi\)
0.360971 + 0.932577i \(0.382445\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.2390 0.452987
\(731\) −18.7658 −0.694080
\(732\) −5.73662 −0.212032
\(733\) 10.3542 0.382442 0.191221 0.981547i \(-0.438755\pi\)
0.191221 + 0.981547i \(0.438755\pi\)
\(734\) 0.908469 0.0335322
\(735\) 0 0
\(736\) −2.22103 −0.0818682
\(737\) 26.9265 0.991852
\(738\) 11.5809 0.426298
\(739\) −35.4220 −1.30302 −0.651510 0.758640i \(-0.725864\pi\)
−0.651510 + 0.758640i \(0.725864\pi\)
\(740\) −0.397260 −0.0146036
\(741\) 0.579460 0.0212870
\(742\) 0 0
\(743\) 42.5211 1.55995 0.779974 0.625812i \(-0.215232\pi\)
0.779974 + 0.625812i \(0.215232\pi\)
\(744\) 25.0620 0.918816
\(745\) −21.2643 −0.779063
\(746\) −16.8845 −0.618186
\(747\) −12.4798 −0.456611
\(748\) 7.31508 0.267466
\(749\) 0 0
\(750\) −14.2142 −0.519030
\(751\) −37.4114 −1.36516 −0.682581 0.730810i \(-0.739142\pi\)
−0.682581 + 0.730810i \(0.739142\pi\)
\(752\) 26.5710 0.968945
\(753\) −27.5761 −1.00493
\(754\) −0.361025 −0.0131477
\(755\) −4.12779 −0.150226
\(756\) 0 0
\(757\) 53.8670 1.95783 0.978914 0.204271i \(-0.0654822\pi\)
0.978914 + 0.204271i \(0.0654822\pi\)
\(758\) 34.3935 1.24923
\(759\) 6.58109 0.238879
\(760\) 15.2465 0.553050
\(761\) 19.9697 0.723900 0.361950 0.932197i \(-0.382111\pi\)
0.361950 + 0.932197i \(0.382111\pi\)
\(762\) 21.7397 0.787548
\(763\) 0 0
\(764\) 7.80881 0.282513
\(765\) 2.86351 0.103531
\(766\) 25.0461 0.904953
\(767\) −0.423651 −0.0152972
\(768\) −9.20292 −0.332082
\(769\) −46.6660 −1.68282 −0.841409 0.540399i \(-0.818273\pi\)
−0.841409 + 0.540399i \(0.818273\pi\)
\(770\) 0 0
\(771\) 17.0021 0.612315
\(772\) −4.18503 −0.150623
\(773\) 16.8883 0.607429 0.303714 0.952763i \(-0.401773\pi\)
0.303714 + 0.952763i \(0.401773\pi\)
\(774\) 10.4107 0.374206
\(775\) −39.8847 −1.43270
\(776\) 2.28516 0.0820325
\(777\) 0 0
\(778\) −13.3290 −0.477869
\(779\) 44.8072 1.60538
\(780\) −0.0395036 −0.00141446
\(781\) −10.7816 −0.385796
\(782\) −4.32286 −0.154585
\(783\) −2.41057 −0.0861468
\(784\) 0 0
\(785\) −5.78337 −0.206417
\(786\) −4.22344 −0.150645
\(787\) −32.9203 −1.17348 −0.586741 0.809774i \(-0.699590\pi\)
−0.586741 + 0.809774i \(0.699590\pi\)
\(788\) 2.91963 0.104007
\(789\) 20.0349 0.713261
\(790\) −12.3851 −0.440641
\(791\) 0 0
\(792\) 16.3249 0.580080
\(793\) −1.39328 −0.0494769
\(794\) 8.00052 0.283928
\(795\) −5.04517 −0.178934
\(796\) 2.93104 0.103888
\(797\) −21.6726 −0.767684 −0.383842 0.923399i \(-0.625399\pi\)
−0.383842 + 0.923399i \(0.625399\pi\)
\(798\) 0 0
\(799\) 15.9927 0.565782
\(800\) 8.76796 0.309994
\(801\) −2.59708 −0.0917632
\(802\) 39.7908 1.40506
\(803\) 50.7030 1.78927
\(804\) 1.62920 0.0574575
\(805\) 0 0
\(806\) −1.51314 −0.0532981
\(807\) −18.1520 −0.638980
\(808\) −40.8869 −1.43840
\(809\) −53.1040 −1.86704 −0.933519 0.358529i \(-0.883278\pi\)
−0.933519 + 0.358529i \(0.883278\pi\)
\(810\) −1.58859 −0.0558174
\(811\) 29.7443 1.04446 0.522232 0.852804i \(-0.325100\pi\)
0.522232 + 0.852804i \(0.325100\pi\)
\(812\) 0 0
\(813\) −7.58594 −0.266050
\(814\) −9.91181 −0.347409
\(815\) −18.5471 −0.649676
\(816\) −12.9462 −0.453209
\(817\) 40.2798 1.40921
\(818\) −22.9174 −0.801288
\(819\) 0 0
\(820\) −3.05465 −0.106673
\(821\) 26.3798 0.920663 0.460331 0.887747i \(-0.347731\pi\)
0.460331 + 0.887747i \(0.347731\pi\)
\(822\) −22.5645 −0.787028
\(823\) −5.25436 −0.183155 −0.0915777 0.995798i \(-0.529191\pi\)
−0.0915777 + 0.995798i \(0.529191\pi\)
\(824\) −5.02499 −0.175054
\(825\) −25.9802 −0.904513
\(826\) 0 0
\(827\) 1.44455 0.0502318 0.0251159 0.999685i \(-0.492005\pi\)
0.0251159 + 0.999685i \(0.492005\pi\)
\(828\) 0.398191 0.0138381
\(829\) 30.9757 1.07583 0.537915 0.842999i \(-0.319212\pi\)
0.537915 + 0.842999i \(0.319212\pi\)
\(830\) 19.8252 0.688144
\(831\) −19.5130 −0.676899
\(832\) −0.564418 −0.0195677
\(833\) 0 0
\(834\) −6.05548 −0.209684
\(835\) −17.3601 −0.600770
\(836\) −15.7014 −0.543044
\(837\) −10.1033 −0.349220
\(838\) −9.95767 −0.343982
\(839\) 27.1001 0.935598 0.467799 0.883835i \(-0.345047\pi\)
0.467799 + 0.883835i \(0.345047\pi\)
\(840\) 0 0
\(841\) −23.1891 −0.799626
\(842\) −28.0784 −0.967647
\(843\) −1.47664 −0.0508583
\(844\) −0.205121 −0.00706057
\(845\) 13.3260 0.458429
\(846\) −8.87228 −0.305035
\(847\) 0 0
\(848\) 22.8097 0.783289
\(849\) 1.23515 0.0423901
\(850\) 17.0653 0.585337
\(851\) 0.972553 0.0333387
\(852\) −0.652345 −0.0223490
\(853\) 44.5874 1.52664 0.763322 0.646019i \(-0.223567\pi\)
0.763322 + 0.646019i \(0.223567\pi\)
\(854\) 0 0
\(855\) −6.14637 −0.210201
\(856\) 41.9529 1.43392
\(857\) 10.7743 0.368045 0.184022 0.982922i \(-0.441088\pi\)
0.184022 + 0.982922i \(0.441088\pi\)
\(858\) −0.985632 −0.0336489
\(859\) 22.9846 0.784224 0.392112 0.919917i \(-0.371745\pi\)
0.392112 + 0.919917i \(0.371745\pi\)
\(860\) −2.74600 −0.0936379
\(861\) 0 0
\(862\) −24.2686 −0.826591
\(863\) −23.9034 −0.813682 −0.406841 0.913499i \(-0.633370\pi\)
−0.406841 + 0.913499i \(0.633370\pi\)
\(864\) 2.22103 0.0755610
\(865\) 6.04808 0.205641
\(866\) −36.3085 −1.23381
\(867\) 9.20783 0.312714
\(868\) 0 0
\(869\) −51.3079 −1.74050
\(870\) 3.82941 0.129829
\(871\) 0.395692 0.0134075
\(872\) −28.1705 −0.953973
\(873\) −0.921222 −0.0311786
\(874\) 9.27877 0.313859
\(875\) 0 0
\(876\) 3.06780 0.103651
\(877\) 20.0265 0.676247 0.338123 0.941102i \(-0.390208\pi\)
0.338123 + 0.941102i \(0.390208\pi\)
\(878\) 10.4969 0.354253
\(879\) −9.74857 −0.328811
\(880\) −31.3100 −1.05546
\(881\) −8.33251 −0.280730 −0.140365 0.990100i \(-0.544828\pi\)
−0.140365 + 0.990100i \(0.544828\pi\)
\(882\) 0 0
\(883\) 49.0846 1.65183 0.825915 0.563795i \(-0.190659\pi\)
0.825915 + 0.563795i \(0.190659\pi\)
\(884\) 0.107497 0.00361551
\(885\) 4.49369 0.151054
\(886\) −45.1659 −1.51738
\(887\) 3.74487 0.125740 0.0628701 0.998022i \(-0.479975\pi\)
0.0628701 + 0.998022i \(0.479975\pi\)
\(888\) 2.41249 0.0809579
\(889\) 0 0
\(890\) 4.12569 0.138294
\(891\) −6.58109 −0.220475
\(892\) 6.59772 0.220908
\(893\) −34.3275 −1.14873
\(894\) −32.1013 −1.07363
\(895\) 18.6978 0.624998
\(896\) 0 0
\(897\) 0.0967108 0.00322908
\(898\) 57.2614 1.91084
\(899\) 24.3547 0.812274
\(900\) −1.57194 −0.0523980
\(901\) 13.7289 0.457375
\(902\) −76.2147 −2.53767
\(903\) 0 0
\(904\) 32.3610 1.07631
\(905\) 25.2339 0.838805
\(906\) −6.23146 −0.207026
\(907\) 10.0339 0.333170 0.166585 0.986027i \(-0.446726\pi\)
0.166585 + 0.986027i \(0.446726\pi\)
\(908\) 0.848453 0.0281569
\(909\) 16.4828 0.546701
\(910\) 0 0
\(911\) −6.63188 −0.219724 −0.109862 0.993947i \(-0.535041\pi\)
−0.109862 + 0.993947i \(0.535041\pi\)
\(912\) 27.7884 0.920165
\(913\) 82.1305 2.71812
\(914\) 43.0004 1.42233
\(915\) 14.7786 0.488567
\(916\) −5.74427 −0.189796
\(917\) 0 0
\(918\) 4.32286 0.142676
\(919\) 6.24684 0.206064 0.103032 0.994678i \(-0.467146\pi\)
0.103032 + 0.994678i \(0.467146\pi\)
\(920\) 2.54462 0.0838936
\(921\) −27.7926 −0.915799
\(922\) 41.7209 1.37400
\(923\) −0.158438 −0.00521506
\(924\) 0 0
\(925\) −3.83935 −0.126237
\(926\) 41.5163 1.36431
\(927\) 2.02573 0.0665339
\(928\) −5.35395 −0.175752
\(929\) 1.74181 0.0571471 0.0285735 0.999592i \(-0.490904\pi\)
0.0285735 + 0.999592i \(0.490904\pi\)
\(930\) 16.0500 0.526299
\(931\) 0 0
\(932\) −0.884010 −0.0289567
\(933\) 3.96040 0.129658
\(934\) 24.8594 0.813426
\(935\) −18.8451 −0.616299
\(936\) 0.239899 0.00784133
\(937\) 1.99662 0.0652267 0.0326134 0.999468i \(-0.489617\pi\)
0.0326134 + 0.999468i \(0.489617\pi\)
\(938\) 0 0
\(939\) 4.81964 0.157283
\(940\) 2.34021 0.0763293
\(941\) −33.7616 −1.10060 −0.550298 0.834968i \(-0.685486\pi\)
−0.550298 + 0.834968i \(0.685486\pi\)
\(942\) −8.73077 −0.284464
\(943\) 7.47823 0.243525
\(944\) −20.3165 −0.661244
\(945\) 0 0
\(946\) −68.5139 −2.22758
\(947\) −4.41249 −0.143387 −0.0716933 0.997427i \(-0.522840\pi\)
−0.0716933 + 0.997427i \(0.522840\pi\)
\(948\) −3.10440 −0.100826
\(949\) 0.745093 0.0241867
\(950\) −36.6298 −1.18843
\(951\) 16.8489 0.546361
\(952\) 0 0
\(953\) 26.9581 0.873258 0.436629 0.899642i \(-0.356172\pi\)
0.436629 + 0.899642i \(0.356172\pi\)
\(954\) −7.61636 −0.246589
\(955\) −20.1170 −0.650970
\(956\) 5.75829 0.186236
\(957\) 15.8642 0.512817
\(958\) −12.5187 −0.404461
\(959\) 0 0
\(960\) 5.98682 0.193224
\(961\) 71.0763 2.29278
\(962\) −0.145657 −0.00469616
\(963\) −16.9125 −0.544999
\(964\) −7.09045 −0.228368
\(965\) 10.7814 0.347067
\(966\) 0 0
\(967\) −25.8724 −0.832001 −0.416000 0.909364i \(-0.636568\pi\)
−0.416000 + 0.909364i \(0.636568\pi\)
\(968\) −80.1493 −2.57610
\(969\) 16.7254 0.537299
\(970\) 1.46344 0.0469884
\(971\) 16.1998 0.519877 0.259938 0.965625i \(-0.416298\pi\)
0.259938 + 0.965625i \(0.416298\pi\)
\(972\) −0.398191 −0.0127720
\(973\) 0 0
\(974\) −34.4439 −1.10366
\(975\) −0.381785 −0.0122269
\(976\) −66.8157 −2.13872
\(977\) −19.2228 −0.614992 −0.307496 0.951549i \(-0.599491\pi\)
−0.307496 + 0.951549i \(0.599491\pi\)
\(978\) −27.9993 −0.895319
\(979\) 17.0916 0.546250
\(980\) 0 0
\(981\) 11.3564 0.362583
\(982\) −38.7969 −1.23806
\(983\) −53.9905 −1.72203 −0.861014 0.508581i \(-0.830170\pi\)
−0.861014 + 0.508581i \(0.830170\pi\)
\(984\) 18.5503 0.591363
\(985\) −7.52152 −0.239655
\(986\) −10.4206 −0.331858
\(987\) 0 0
\(988\) −0.230736 −0.00734069
\(989\) 6.72262 0.213767
\(990\) 10.4547 0.332271
\(991\) 19.5181 0.620011 0.310006 0.950735i \(-0.399669\pi\)
0.310006 + 0.950735i \(0.399669\pi\)
\(992\) −22.4397 −0.712461
\(993\) −10.0553 −0.319095
\(994\) 0 0
\(995\) −7.55092 −0.239380
\(996\) 4.96933 0.157459
\(997\) 47.2150 1.49531 0.747657 0.664085i \(-0.231179\pi\)
0.747657 + 0.664085i \(0.231179\pi\)
\(998\) 34.7927 1.10134
\(999\) −0.972553 −0.0307702
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 3381.2.a.bg.1.9 10
7.6 odd 2 3381.2.a.bh.1.9 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.2.a.bg.1.9 10 1.1 even 1 trivial
3381.2.a.bh.1.9 yes 10 7.6 odd 2