Properties

Label 539.2.a.i.1.2
Level $539$
Weight $2$
Character 539.1
Self dual yes
Analytic conductor $4.304$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(1,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("539.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.91223\) of defining polynomial
Character \(\chi\) \(=\) 539.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.656620 q^{2} -1.91223 q^{3} -1.56885 q^{4} +3.56885 q^{5} -1.25561 q^{6} -2.34338 q^{8} +0.656620 q^{9} +O(q^{10})\) \(q+0.656620 q^{2} -1.91223 q^{3} -1.56885 q^{4} +3.56885 q^{5} -1.25561 q^{6} -2.34338 q^{8} +0.656620 q^{9} +2.34338 q^{10} -1.00000 q^{11} +3.00000 q^{12} +5.91223 q^{13} -6.82446 q^{15} +1.59899 q^{16} +1.65662 q^{17} +0.431150 q^{18} -1.48108 q^{19} -5.59899 q^{20} -0.656620 q^{22} +3.34338 q^{23} +4.48108 q^{24} +7.73669 q^{25} +3.88209 q^{26} +4.48108 q^{27} +3.08007 q^{29} -4.48108 q^{30} +7.08007 q^{31} +5.73669 q^{32} +1.91223 q^{33} +1.08777 q^{34} -1.03014 q^{36} -4.51122 q^{37} -0.972507 q^{38} -11.3055 q^{39} -8.36317 q^{40} +1.28575 q^{41} +1.59899 q^{43} +1.56885 q^{44} +2.34338 q^{45} +2.19533 q^{46} +1.65662 q^{47} -3.05763 q^{48} +5.08007 q^{50} -3.16784 q^{51} -9.27540 q^{52} +9.22547 q^{53} +2.94237 q^{54} -3.56885 q^{55} +2.83216 q^{57} +2.02243 q^{58} -8.85195 q^{59} +10.7065 q^{60} +6.68676 q^{61} +4.64892 q^{62} +0.568850 q^{64} +21.0999 q^{65} +1.25561 q^{66} -9.82446 q^{67} -2.59899 q^{68} -6.39331 q^{69} -8.61878 q^{71} -1.53871 q^{72} -4.56115 q^{73} -2.96216 q^{74} -14.7943 q^{75} +2.32359 q^{76} -7.42345 q^{78} -6.39331 q^{79} +5.70655 q^{80} -10.5387 q^{81} +0.844248 q^{82} -0.167838 q^{83} +5.91223 q^{85} +1.04993 q^{86} -5.88979 q^{87} +2.34338 q^{88} +2.56885 q^{89} +1.53871 q^{90} -5.24526 q^{92} -13.5387 q^{93} +1.08777 q^{94} -5.28575 q^{95} -10.9699 q^{96} -9.73669 q^{97} -0.656620 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 4 q^{4} + 2 q^{5} + q^{6} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 4 q^{4} + 2 q^{5} + q^{6} - 9 q^{8} + 9 q^{10} - 3 q^{11} + 9 q^{12} + 11 q^{13} - 7 q^{15} + 2 q^{16} + 3 q^{17} + 10 q^{18} + 11 q^{19} - 14 q^{20} + 12 q^{23} - 2 q^{24} + 3 q^{25} - q^{26} - 2 q^{27} - 9 q^{29} + 2 q^{30} + 3 q^{31} - 3 q^{32} - q^{33} + 10 q^{34} - 9 q^{36} - 4 q^{37} - 8 q^{38} - 5 q^{39} + 3 q^{40} + 5 q^{41} + 2 q^{43} - 4 q^{44} + 9 q^{45} - 10 q^{46} + 3 q^{47} - 10 q^{48} - 3 q^{50} + 2 q^{51} + 7 q^{52} + 17 q^{53} + 8 q^{54} - 2 q^{55} + 20 q^{57} - 13 q^{58} - 8 q^{59} + 6 q^{60} + 24 q^{61} - 13 q^{62} - 7 q^{64} + 15 q^{65} - q^{66} - 16 q^{67} - 5 q^{68} + 3 q^{69} + 7 q^{71} + 10 q^{72} + 20 q^{73} + 22 q^{74} - 25 q^{75} + 39 q^{76} - 6 q^{78} + 3 q^{79} - 9 q^{80} - 17 q^{81} - 41 q^{82} + 11 q^{83} + 11 q^{85} - 21 q^{86} - 30 q^{87} + 9 q^{88} - q^{89} - 10 q^{90} + 25 q^{92} - 26 q^{93} + 10 q^{94} - 17 q^{95} - 27 q^{96} - 9 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.656620 0.464301 0.232150 0.972680i \(-0.425424\pi\)
0.232150 + 0.972680i \(0.425424\pi\)
\(3\) −1.91223 −1.10403 −0.552013 0.833835i \(-0.686140\pi\)
−0.552013 + 0.833835i \(0.686140\pi\)
\(4\) −1.56885 −0.784425
\(5\) 3.56885 1.59604 0.798019 0.602632i \(-0.205881\pi\)
0.798019 + 0.602632i \(0.205881\pi\)
\(6\) −1.25561 −0.512600
\(7\) 0 0
\(8\) −2.34338 −0.828510
\(9\) 0.656620 0.218873
\(10\) 2.34338 0.741042
\(11\) −1.00000 −0.301511
\(12\) 3.00000 0.866025
\(13\) 5.91223 1.63976 0.819879 0.572537i \(-0.194041\pi\)
0.819879 + 0.572537i \(0.194041\pi\)
\(14\) 0 0
\(15\) −6.82446 −1.76207
\(16\) 1.59899 0.399747
\(17\) 1.65662 0.401789 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(18\) 0.431150 0.101623
\(19\) −1.48108 −0.339783 −0.169891 0.985463i \(-0.554342\pi\)
−0.169891 + 0.985463i \(0.554342\pi\)
\(20\) −5.59899 −1.25197
\(21\) 0 0
\(22\) −0.656620 −0.139992
\(23\) 3.34338 0.697143 0.348571 0.937282i \(-0.386667\pi\)
0.348571 + 0.937282i \(0.386667\pi\)
\(24\) 4.48108 0.914696
\(25\) 7.73669 1.54734
\(26\) 3.88209 0.761341
\(27\) 4.48108 0.862384
\(28\) 0 0
\(29\) 3.08007 0.571954 0.285977 0.958236i \(-0.407682\pi\)
0.285977 + 0.958236i \(0.407682\pi\)
\(30\) −4.48108 −0.818129
\(31\) 7.08007 1.27162 0.635809 0.771847i \(-0.280667\pi\)
0.635809 + 0.771847i \(0.280667\pi\)
\(32\) 5.73669 1.01411
\(33\) 1.91223 0.332876
\(34\) 1.08777 0.186551
\(35\) 0 0
\(36\) −1.03014 −0.171690
\(37\) −4.51122 −0.741640 −0.370820 0.928705i \(-0.620923\pi\)
−0.370820 + 0.928705i \(0.620923\pi\)
\(38\) −0.972507 −0.157761
\(39\) −11.3055 −1.81033
\(40\) −8.36317 −1.32233
\(41\) 1.28575 0.200800 0.100400 0.994947i \(-0.467988\pi\)
0.100400 + 0.994947i \(0.467988\pi\)
\(42\) 0 0
\(43\) 1.59899 0.243843 0.121922 0.992540i \(-0.461094\pi\)
0.121922 + 0.992540i \(0.461094\pi\)
\(44\) 1.56885 0.236513
\(45\) 2.34338 0.349330
\(46\) 2.19533 0.323684
\(47\) 1.65662 0.241643 0.120821 0.992674i \(-0.461447\pi\)
0.120821 + 0.992674i \(0.461447\pi\)
\(48\) −3.05763 −0.441331
\(49\) 0 0
\(50\) 5.08007 0.718430
\(51\) −3.16784 −0.443586
\(52\) −9.27540 −1.28627
\(53\) 9.22547 1.26722 0.633608 0.773654i \(-0.281573\pi\)
0.633608 + 0.773654i \(0.281573\pi\)
\(54\) 2.94237 0.400406
\(55\) −3.56885 −0.481224
\(56\) 0 0
\(57\) 2.83216 0.375129
\(58\) 2.02243 0.265559
\(59\) −8.85195 −1.15243 −0.576213 0.817300i \(-0.695470\pi\)
−0.576213 + 0.817300i \(0.695470\pi\)
\(60\) 10.7065 1.38221
\(61\) 6.68676 0.856152 0.428076 0.903743i \(-0.359192\pi\)
0.428076 + 0.903743i \(0.359192\pi\)
\(62\) 4.64892 0.590413
\(63\) 0 0
\(64\) 0.568850 0.0711062
\(65\) 21.0999 2.61712
\(66\) 1.25561 0.154555
\(67\) −9.82446 −1.20025 −0.600124 0.799907i \(-0.704882\pi\)
−0.600124 + 0.799907i \(0.704882\pi\)
\(68\) −2.59899 −0.315174
\(69\) −6.39331 −0.769664
\(70\) 0 0
\(71\) −8.61878 −1.02286 −0.511430 0.859325i \(-0.670884\pi\)
−0.511430 + 0.859325i \(0.670884\pi\)
\(72\) −1.53871 −0.181339
\(73\) −4.56115 −0.533842 −0.266921 0.963718i \(-0.586006\pi\)
−0.266921 + 0.963718i \(0.586006\pi\)
\(74\) −2.96216 −0.344344
\(75\) −14.7943 −1.70830
\(76\) 2.32359 0.266534
\(77\) 0 0
\(78\) −7.42345 −0.840540
\(79\) −6.39331 −0.719303 −0.359652 0.933087i \(-0.617104\pi\)
−0.359652 + 0.933087i \(0.617104\pi\)
\(80\) 5.70655 0.638012
\(81\) −10.5387 −1.17097
\(82\) 0.844248 0.0932316
\(83\) −0.167838 −0.0184226 −0.00921130 0.999958i \(-0.502932\pi\)
−0.00921130 + 0.999958i \(0.502932\pi\)
\(84\) 0 0
\(85\) 5.91223 0.641271
\(86\) 1.04993 0.113217
\(87\) −5.88979 −0.631452
\(88\) 2.34338 0.249805
\(89\) 2.56885 0.272298 0.136149 0.990688i \(-0.456527\pi\)
0.136149 + 0.990688i \(0.456527\pi\)
\(90\) 1.53871 0.162194
\(91\) 0 0
\(92\) −5.24526 −0.546856
\(93\) −13.5387 −1.40390
\(94\) 1.08777 0.112195
\(95\) −5.28575 −0.542306
\(96\) −10.9699 −1.11961
\(97\) −9.73669 −0.988611 −0.494305 0.869288i \(-0.664578\pi\)
−0.494305 + 0.869288i \(0.664578\pi\)
\(98\) 0 0
\(99\) −0.656620 −0.0659928
\(100\) −12.1377 −1.21377
\(101\) −1.85460 −0.184539 −0.0922697 0.995734i \(-0.529412\pi\)
−0.0922697 + 0.995734i \(0.529412\pi\)
\(102\) −2.08007 −0.205957
\(103\) 3.16784 0.312136 0.156068 0.987746i \(-0.450118\pi\)
0.156068 + 0.987746i \(0.450118\pi\)
\(104\) −13.8546 −1.35856
\(105\) 0 0
\(106\) 6.05763 0.588369
\(107\) −4.76683 −0.460826 −0.230413 0.973093i \(-0.574008\pi\)
−0.230413 + 0.973093i \(0.574008\pi\)
\(108\) −7.03014 −0.676475
\(109\) −14.8821 −1.42545 −0.712723 0.701446i \(-0.752538\pi\)
−0.712723 + 0.701446i \(0.752538\pi\)
\(110\) −2.34338 −0.223432
\(111\) 8.62648 0.818789
\(112\) 0 0
\(113\) 12.4432 1.17056 0.585281 0.810831i \(-0.300984\pi\)
0.585281 + 0.810831i \(0.300984\pi\)
\(114\) 1.85966 0.174173
\(115\) 11.9320 1.11267
\(116\) −4.83216 −0.448655
\(117\) 3.88209 0.358899
\(118\) −5.81237 −0.535072
\(119\) 0 0
\(120\) 15.9923 1.45989
\(121\) 1.00000 0.0909091
\(122\) 4.39066 0.397512
\(123\) −2.45864 −0.221688
\(124\) −11.1076 −0.997488
\(125\) 9.76683 0.873571
\(126\) 0 0
\(127\) −6.62142 −0.587556 −0.293778 0.955874i \(-0.594913\pi\)
−0.293778 + 0.955874i \(0.594913\pi\)
\(128\) −11.0999 −0.981098
\(129\) −3.05763 −0.269209
\(130\) 13.8546 1.21513
\(131\) 6.05763 0.529258 0.264629 0.964350i \(-0.414751\pi\)
0.264629 + 0.964350i \(0.414751\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) −6.45094 −0.557276
\(135\) 15.9923 1.37640
\(136\) −3.88209 −0.332887
\(137\) −7.42345 −0.634228 −0.317114 0.948387i \(-0.602714\pi\)
−0.317114 + 0.948387i \(0.602714\pi\)
\(138\) −4.19798 −0.357356
\(139\) 10.8245 0.918119 0.459059 0.888406i \(-0.348187\pi\)
0.459059 + 0.888406i \(0.348187\pi\)
\(140\) 0 0
\(141\) −3.16784 −0.266780
\(142\) −5.65927 −0.474915
\(143\) −5.91223 −0.494405
\(144\) 1.04993 0.0874940
\(145\) 10.9923 0.912861
\(146\) −2.99494 −0.247863
\(147\) 0 0
\(148\) 7.07742 0.581760
\(149\) −1.00000 −0.0819232 −0.0409616 0.999161i \(-0.513042\pi\)
−0.0409616 + 0.999161i \(0.513042\pi\)
\(150\) −9.71425 −0.793165
\(151\) 16.4234 1.33652 0.668261 0.743927i \(-0.267039\pi\)
0.668261 + 0.743927i \(0.267039\pi\)
\(152\) 3.47073 0.281513
\(153\) 1.08777 0.0879411
\(154\) 0 0
\(155\) 25.2677 2.02955
\(156\) 17.7367 1.42007
\(157\) 11.4509 0.913885 0.456942 0.889496i \(-0.348945\pi\)
0.456942 + 0.889496i \(0.348945\pi\)
\(158\) −4.19798 −0.333973
\(159\) −17.6412 −1.39904
\(160\) 20.4734 1.61856
\(161\) 0 0
\(162\) −6.91993 −0.543681
\(163\) −8.93972 −0.700213 −0.350107 0.936710i \(-0.613855\pi\)
−0.350107 + 0.936710i \(0.613855\pi\)
\(164\) −2.01714 −0.157513
\(165\) 6.82446 0.531283
\(166\) −0.110206 −0.00855363
\(167\) 18.2178 1.40973 0.704867 0.709340i \(-0.251007\pi\)
0.704867 + 0.709340i \(0.251007\pi\)
\(168\) 0 0
\(169\) 21.9545 1.68880
\(170\) 3.88209 0.297743
\(171\) −0.972507 −0.0743694
\(172\) −2.50857 −0.191277
\(173\) −19.5611 −1.48721 −0.743603 0.668621i \(-0.766885\pi\)
−0.743603 + 0.668621i \(0.766885\pi\)
\(174\) −3.86736 −0.293184
\(175\) 0 0
\(176\) −1.59899 −0.120528
\(177\) 16.9270 1.27231
\(178\) 1.68676 0.126428
\(179\) 3.24791 0.242760 0.121380 0.992606i \(-0.461268\pi\)
0.121380 + 0.992606i \(0.461268\pi\)
\(180\) −3.67641 −0.274023
\(181\) 10.3407 0.768621 0.384310 0.923204i \(-0.374439\pi\)
0.384310 + 0.923204i \(0.374439\pi\)
\(182\) 0 0
\(183\) −12.7866 −0.945214
\(184\) −7.83481 −0.577590
\(185\) −16.0999 −1.18369
\(186\) −8.88979 −0.651831
\(187\) −1.65662 −0.121144
\(188\) −2.59899 −0.189551
\(189\) 0 0
\(190\) −3.47073 −0.251793
\(191\) 10.0224 0.725198 0.362599 0.931945i \(-0.381889\pi\)
0.362599 + 0.931945i \(0.381889\pi\)
\(192\) −1.08777 −0.0785031
\(193\) 25.3253 1.82296 0.911478 0.411348i \(-0.134942\pi\)
0.911478 + 0.411348i \(0.134942\pi\)
\(194\) −6.39331 −0.459013
\(195\) −40.3478 −2.88936
\(196\) 0 0
\(197\) −24.5809 −1.75132 −0.875660 0.482929i \(-0.839573\pi\)
−0.875660 + 0.482929i \(0.839573\pi\)
\(198\) −0.431150 −0.0306405
\(199\) −5.59128 −0.396356 −0.198178 0.980166i \(-0.563502\pi\)
−0.198178 + 0.980166i \(0.563502\pi\)
\(200\) −18.1300 −1.28198
\(201\) 18.7866 1.32511
\(202\) −1.21777 −0.0856817
\(203\) 0 0
\(204\) 4.96986 0.347960
\(205\) 4.58864 0.320484
\(206\) 2.08007 0.144925
\(207\) 2.19533 0.152586
\(208\) 9.45359 0.655488
\(209\) 1.48108 0.102448
\(210\) 0 0
\(211\) 12.0999 0.832988 0.416494 0.909138i \(-0.363259\pi\)
0.416494 + 0.909138i \(0.363259\pi\)
\(212\) −14.4734 −0.994035
\(213\) 16.4811 1.12926
\(214\) −3.13000 −0.213962
\(215\) 5.70655 0.389183
\(216\) −10.5009 −0.714494
\(217\) 0 0
\(218\) −9.77188 −0.661836
\(219\) 8.72196 0.589375
\(220\) 5.59899 0.377484
\(221\) 9.79432 0.658837
\(222\) 5.66432 0.380165
\(223\) −15.6265 −1.04643 −0.523213 0.852202i \(-0.675267\pi\)
−0.523213 + 0.852202i \(0.675267\pi\)
\(224\) 0 0
\(225\) 5.08007 0.338671
\(226\) 8.17048 0.543492
\(227\) 15.6687 1.03997 0.519984 0.854176i \(-0.325938\pi\)
0.519984 + 0.854176i \(0.325938\pi\)
\(228\) −4.44324 −0.294261
\(229\) 5.57149 0.368175 0.184087 0.982910i \(-0.441067\pi\)
0.184087 + 0.982910i \(0.441067\pi\)
\(230\) 7.83481 0.516612
\(231\) 0 0
\(232\) −7.21777 −0.473870
\(233\) −19.2754 −1.26277 −0.631387 0.775468i \(-0.717514\pi\)
−0.631387 + 0.775468i \(0.717514\pi\)
\(234\) 2.54906 0.166637
\(235\) 5.91223 0.385671
\(236\) 13.8874 0.903992
\(237\) 12.2255 0.794130
\(238\) 0 0
\(239\) −22.1575 −1.43325 −0.716624 0.697459i \(-0.754314\pi\)
−0.716624 + 0.697459i \(0.754314\pi\)
\(240\) −10.9122 −0.704381
\(241\) 19.8744 1.28022 0.640111 0.768283i \(-0.278888\pi\)
0.640111 + 0.768283i \(0.278888\pi\)
\(242\) 0.656620 0.0422092
\(243\) 6.70919 0.430395
\(244\) −10.4905 −0.671587
\(245\) 0 0
\(246\) −1.61440 −0.102930
\(247\) −8.75648 −0.557161
\(248\) −16.5913 −1.05355
\(249\) 0.320945 0.0203390
\(250\) 6.41310 0.405600
\(251\) −22.1076 −1.39542 −0.697708 0.716382i \(-0.745797\pi\)
−0.697708 + 0.716382i \(0.745797\pi\)
\(252\) 0 0
\(253\) −3.34338 −0.210196
\(254\) −4.34776 −0.272803
\(255\) −11.3055 −0.707980
\(256\) −8.42609 −0.526631
\(257\) −29.1196 −1.81643 −0.908217 0.418500i \(-0.862556\pi\)
−0.908217 + 0.418500i \(0.862556\pi\)
\(258\) −2.00770 −0.124994
\(259\) 0 0
\(260\) −33.1025 −2.05293
\(261\) 2.02243 0.125186
\(262\) 3.97757 0.245735
\(263\) −15.5035 −0.955988 −0.477994 0.878363i \(-0.658636\pi\)
−0.477994 + 0.878363i \(0.658636\pi\)
\(264\) −4.48108 −0.275791
\(265\) 32.9243 2.02252
\(266\) 0 0
\(267\) −4.91223 −0.300624
\(268\) 15.4131 0.941505
\(269\) −1.70655 −0.104050 −0.0520251 0.998646i \(-0.516568\pi\)
−0.0520251 + 0.998646i \(0.516568\pi\)
\(270\) 10.5009 0.639063
\(271\) 20.5284 1.24701 0.623505 0.781820i \(-0.285708\pi\)
0.623505 + 0.781820i \(0.285708\pi\)
\(272\) 2.64892 0.160614
\(273\) 0 0
\(274\) −4.87439 −0.294472
\(275\) −7.73669 −0.466540
\(276\) 10.0301 0.603743
\(277\) −26.6610 −1.60190 −0.800952 0.598728i \(-0.795673\pi\)
−0.800952 + 0.598728i \(0.795673\pi\)
\(278\) 7.10756 0.426283
\(279\) 4.64892 0.278323
\(280\) 0 0
\(281\) 15.7444 0.939232 0.469616 0.882871i \(-0.344392\pi\)
0.469616 + 0.882871i \(0.344392\pi\)
\(282\) −2.08007 −0.123866
\(283\) 16.0697 0.955246 0.477623 0.878565i \(-0.341499\pi\)
0.477623 + 0.878565i \(0.341499\pi\)
\(284\) 13.5216 0.802357
\(285\) 10.1076 0.598720
\(286\) −3.88209 −0.229553
\(287\) 0 0
\(288\) 3.76683 0.221962
\(289\) −14.2556 −0.838565
\(290\) 7.21777 0.423842
\(291\) 18.6188 1.09145
\(292\) 7.15575 0.418759
\(293\) −15.3357 −0.895920 −0.447960 0.894054i \(-0.647849\pi\)
−0.447960 + 0.894054i \(0.647849\pi\)
\(294\) 0 0
\(295\) −31.5913 −1.83932
\(296\) 10.5715 0.614456
\(297\) −4.48108 −0.260019
\(298\) −0.656620 −0.0380370
\(299\) 19.7668 1.14315
\(300\) 23.2101 1.34003
\(301\) 0 0
\(302\) 10.7840 0.620548
\(303\) 3.54641 0.203736
\(304\) −2.36823 −0.135827
\(305\) 23.8640 1.36645
\(306\) 0.714253 0.0408311
\(307\) 16.4707 0.940034 0.470017 0.882657i \(-0.344248\pi\)
0.470017 + 0.882657i \(0.344248\pi\)
\(308\) 0 0
\(309\) −6.05763 −0.344607
\(310\) 16.5913 0.942322
\(311\) −21.6291 −1.22648 −0.613238 0.789898i \(-0.710133\pi\)
−0.613238 + 0.789898i \(0.710133\pi\)
\(312\) 26.4932 1.49988
\(313\) −16.3882 −0.926319 −0.463159 0.886275i \(-0.653284\pi\)
−0.463159 + 0.886275i \(0.653284\pi\)
\(314\) 7.51892 0.424317
\(315\) 0 0
\(316\) 10.0301 0.564239
\(317\) 4.46129 0.250571 0.125285 0.992121i \(-0.460015\pi\)
0.125285 + 0.992121i \(0.460015\pi\)
\(318\) −11.5836 −0.649575
\(319\) −3.08007 −0.172451
\(320\) 2.03014 0.113488
\(321\) 9.11526 0.508764
\(322\) 0 0
\(323\) −2.45359 −0.136521
\(324\) 16.5337 0.918536
\(325\) 45.7411 2.53726
\(326\) −5.87000 −0.325109
\(327\) 28.4580 1.57373
\(328\) −3.01299 −0.166365
\(329\) 0 0
\(330\) 4.48108 0.246675
\(331\) 19.0396 1.04651 0.523255 0.852176i \(-0.324718\pi\)
0.523255 + 0.852176i \(0.324718\pi\)
\(332\) 0.263312 0.0144511
\(333\) −2.96216 −0.162325
\(334\) 11.9622 0.654540
\(335\) −35.0620 −1.91564
\(336\) 0 0
\(337\) 27.0147 1.47159 0.735793 0.677206i \(-0.236810\pi\)
0.735793 + 0.677206i \(0.236810\pi\)
\(338\) 14.4157 0.784113
\(339\) −23.7943 −1.29233
\(340\) −9.27540 −0.503029
\(341\) −7.08007 −0.383407
\(342\) −0.638568 −0.0345298
\(343\) 0 0
\(344\) −3.74704 −0.202027
\(345\) −22.8168 −1.22841
\(346\) −12.8442 −0.690511
\(347\) −20.2178 −1.08535 −0.542673 0.839944i \(-0.682588\pi\)
−0.542673 + 0.839944i \(0.682588\pi\)
\(348\) 9.24020 0.495327
\(349\) 12.0224 0.643546 0.321773 0.946817i \(-0.395721\pi\)
0.321773 + 0.946817i \(0.395721\pi\)
\(350\) 0 0
\(351\) 26.4932 1.41410
\(352\) −5.73669 −0.305766
\(353\) 10.7591 0.572650 0.286325 0.958133i \(-0.407566\pi\)
0.286325 + 0.958133i \(0.407566\pi\)
\(354\) 11.1146 0.590734
\(355\) −30.7591 −1.63252
\(356\) −4.03014 −0.213597
\(357\) 0 0
\(358\) 2.13264 0.112714
\(359\) 24.2901 1.28198 0.640992 0.767548i \(-0.278523\pi\)
0.640992 + 0.767548i \(0.278523\pi\)
\(360\) −5.49143 −0.289424
\(361\) −16.8064 −0.884548
\(362\) 6.78994 0.356871
\(363\) −1.91223 −0.100366
\(364\) 0 0
\(365\) −16.2780 −0.852032
\(366\) −8.39595 −0.438864
\(367\) −10.8442 −0.566065 −0.283033 0.959110i \(-0.591340\pi\)
−0.283033 + 0.959110i \(0.591340\pi\)
\(368\) 5.34602 0.278681
\(369\) 0.844248 0.0439498
\(370\) −10.5715 −0.549586
\(371\) 0 0
\(372\) 21.2402 1.10125
\(373\) −33.0242 −1.70993 −0.854963 0.518688i \(-0.826421\pi\)
−0.854963 + 0.518688i \(0.826421\pi\)
\(374\) −1.08777 −0.0562473
\(375\) −18.6764 −0.964446
\(376\) −3.88209 −0.200204
\(377\) 18.2101 0.937866
\(378\) 0 0
\(379\) −21.9320 −1.12657 −0.563286 0.826262i \(-0.690463\pi\)
−0.563286 + 0.826262i \(0.690463\pi\)
\(380\) 8.29254 0.425398
\(381\) 12.6617 0.648677
\(382\) 6.58094 0.336710
\(383\) −36.4630 −1.86317 −0.931587 0.363519i \(-0.881575\pi\)
−0.931587 + 0.363519i \(0.881575\pi\)
\(384\) 21.2255 1.08316
\(385\) 0 0
\(386\) 16.6291 0.846400
\(387\) 1.04993 0.0533709
\(388\) 15.2754 0.775491
\(389\) 19.4760 0.987473 0.493737 0.869611i \(-0.335631\pi\)
0.493737 + 0.869611i \(0.335631\pi\)
\(390\) −26.4932 −1.34153
\(391\) 5.53871 0.280105
\(392\) 0 0
\(393\) −11.5836 −0.584314
\(394\) −16.1403 −0.813139
\(395\) −22.8168 −1.14804
\(396\) 1.03014 0.0517664
\(397\) 7.83987 0.393472 0.196736 0.980457i \(-0.436966\pi\)
0.196736 + 0.980457i \(0.436966\pi\)
\(398\) −3.67135 −0.184028
\(399\) 0 0
\(400\) 12.3709 0.618544
\(401\) −24.4459 −1.22077 −0.610385 0.792105i \(-0.708985\pi\)
−0.610385 + 0.792105i \(0.708985\pi\)
\(402\) 12.3357 0.615248
\(403\) 41.8590 2.08514
\(404\) 2.90958 0.144757
\(405\) −37.6111 −1.86891
\(406\) 0 0
\(407\) 4.51122 0.223613
\(408\) 7.42345 0.367515
\(409\) 2.35373 0.116384 0.0581922 0.998305i \(-0.481466\pi\)
0.0581922 + 0.998305i \(0.481466\pi\)
\(410\) 3.01299 0.148801
\(411\) 14.1953 0.700204
\(412\) −4.96986 −0.244847
\(413\) 0 0
\(414\) 1.44150 0.0708458
\(415\) −0.598988 −0.0294032
\(416\) 33.9166 1.66290
\(417\) −20.6988 −1.01363
\(418\) 0.972507 0.0475669
\(419\) −9.29081 −0.453886 −0.226943 0.973908i \(-0.572873\pi\)
−0.226943 + 0.973908i \(0.572873\pi\)
\(420\) 0 0
\(421\) −39.0319 −1.90230 −0.951149 0.308733i \(-0.900095\pi\)
−0.951149 + 0.308733i \(0.900095\pi\)
\(422\) 7.94501 0.386757
\(423\) 1.08777 0.0528892
\(424\) −21.6188 −1.04990
\(425\) 12.8168 0.621704
\(426\) 10.8218 0.524319
\(427\) 0 0
\(428\) 7.47843 0.361484
\(429\) 11.3055 0.545836
\(430\) 3.74704 0.180698
\(431\) −3.80202 −0.183137 −0.0915685 0.995799i \(-0.529188\pi\)
−0.0915685 + 0.995799i \(0.529188\pi\)
\(432\) 7.16519 0.344735
\(433\) −8.22041 −0.395048 −0.197524 0.980298i \(-0.563290\pi\)
−0.197524 + 0.980298i \(0.563290\pi\)
\(434\) 0 0
\(435\) −21.0198 −1.00782
\(436\) 23.3478 1.11815
\(437\) −4.95181 −0.236877
\(438\) 5.72701 0.273647
\(439\) −4.54136 −0.216747 −0.108374 0.994110i \(-0.534564\pi\)
−0.108374 + 0.994110i \(0.534564\pi\)
\(440\) 8.36317 0.398698
\(441\) 0 0
\(442\) 6.43115 0.305899
\(443\) 13.7444 0.653016 0.326508 0.945194i \(-0.394128\pi\)
0.326508 + 0.945194i \(0.394128\pi\)
\(444\) −13.5337 −0.642279
\(445\) 9.16784 0.434597
\(446\) −10.2607 −0.485857
\(447\) 1.91223 0.0904453
\(448\) 0 0
\(449\) 21.5662 1.01777 0.508886 0.860834i \(-0.330057\pi\)
0.508886 + 0.860834i \(0.330057\pi\)
\(450\) 3.33568 0.157245
\(451\) −1.28575 −0.0605435
\(452\) −19.5216 −0.918217
\(453\) −31.4054 −1.47555
\(454\) 10.2884 0.482858
\(455\) 0 0
\(456\) −6.63683 −0.310798
\(457\) −3.30554 −0.154627 −0.0773133 0.997007i \(-0.524634\pi\)
−0.0773133 + 0.997007i \(0.524634\pi\)
\(458\) 3.65836 0.170944
\(459\) 7.42345 0.346497
\(460\) −18.7195 −0.872803
\(461\) 32.1524 1.49749 0.748744 0.662859i \(-0.230657\pi\)
0.748744 + 0.662859i \(0.230657\pi\)
\(462\) 0 0
\(463\) 5.82181 0.270563 0.135281 0.990807i \(-0.456806\pi\)
0.135281 + 0.990807i \(0.456806\pi\)
\(464\) 4.92499 0.228637
\(465\) −48.3176 −2.24068
\(466\) −12.6566 −0.586307
\(467\) 6.01473 0.278329 0.139164 0.990269i \(-0.455558\pi\)
0.139164 + 0.990269i \(0.455558\pi\)
\(468\) −6.09042 −0.281530
\(469\) 0 0
\(470\) 3.88209 0.179067
\(471\) −21.8968 −1.00895
\(472\) 20.7435 0.954796
\(473\) −1.59899 −0.0735216
\(474\) 8.02749 0.368715
\(475\) −11.4586 −0.525759
\(476\) 0 0
\(477\) 6.05763 0.277360
\(478\) −14.5491 −0.665459
\(479\) −17.1351 −0.782921 −0.391460 0.920195i \(-0.628030\pi\)
−0.391460 + 0.920195i \(0.628030\pi\)
\(480\) −39.1498 −1.78694
\(481\) −26.6714 −1.21611
\(482\) 13.0499 0.594408
\(483\) 0 0
\(484\) −1.56885 −0.0713113
\(485\) −34.7488 −1.57786
\(486\) 4.40539 0.199833
\(487\) 5.71425 0.258937 0.129469 0.991584i \(-0.458673\pi\)
0.129469 + 0.991584i \(0.458673\pi\)
\(488\) −15.6696 −0.709330
\(489\) 17.0948 0.773054
\(490\) 0 0
\(491\) 24.0673 1.08614 0.543071 0.839687i \(-0.317261\pi\)
0.543071 + 0.839687i \(0.317261\pi\)
\(492\) 3.85724 0.173898
\(493\) 5.10250 0.229805
\(494\) −5.74968 −0.258690
\(495\) −2.34338 −0.105327
\(496\) 11.3209 0.508325
\(497\) 0 0
\(498\) 0.210739 0.00944343
\(499\) 10.7893 0.482994 0.241497 0.970402i \(-0.422362\pi\)
0.241497 + 0.970402i \(0.422362\pi\)
\(500\) −15.3227 −0.685251
\(501\) −34.8365 −1.55638
\(502\) −14.5163 −0.647893
\(503\) −28.0121 −1.24900 −0.624499 0.781026i \(-0.714697\pi\)
−0.624499 + 0.781026i \(0.714697\pi\)
\(504\) 0 0
\(505\) −6.61878 −0.294532
\(506\) −2.19533 −0.0975944
\(507\) −41.9819 −1.86448
\(508\) 10.3880 0.460894
\(509\) 1.91487 0.0848753 0.0424377 0.999099i \(-0.486488\pi\)
0.0424377 + 0.999099i \(0.486488\pi\)
\(510\) −7.42345 −0.328716
\(511\) 0 0
\(512\) 16.6670 0.736583
\(513\) −6.63683 −0.293023
\(514\) −19.1206 −0.843372
\(515\) 11.3055 0.498181
\(516\) 4.79696 0.211175
\(517\) −1.65662 −0.0728581
\(518\) 0 0
\(519\) 37.4054 1.64191
\(520\) −49.4450 −2.16831
\(521\) −1.57920 −0.0691859 −0.0345930 0.999401i \(-0.511013\pi\)
−0.0345930 + 0.999401i \(0.511013\pi\)
\(522\) 1.32797 0.0581238
\(523\) 8.96986 0.392225 0.196112 0.980581i \(-0.437168\pi\)
0.196112 + 0.980581i \(0.437168\pi\)
\(524\) −9.50351 −0.415163
\(525\) 0 0
\(526\) −10.1799 −0.443866
\(527\) 11.7290 0.510923
\(528\) 3.05763 0.133066
\(529\) −11.8218 −0.513992
\(530\) 21.6188 0.939060
\(531\) −5.81237 −0.252235
\(532\) 0 0
\(533\) 7.60163 0.329263
\(534\) −3.22547 −0.139580
\(535\) −17.0121 −0.735497
\(536\) 23.0224 0.994418
\(537\) −6.21074 −0.268013
\(538\) −1.12055 −0.0483105
\(539\) 0 0
\(540\) −25.0895 −1.07968
\(541\) 18.1025 0.778287 0.389144 0.921177i \(-0.372771\pi\)
0.389144 + 0.921177i \(0.372771\pi\)
\(542\) 13.4793 0.578987
\(543\) −19.7739 −0.848577
\(544\) 9.50351 0.407460
\(545\) −53.1119 −2.27507
\(546\) 0 0
\(547\) 22.6885 0.970090 0.485045 0.874489i \(-0.338803\pi\)
0.485045 + 0.874489i \(0.338803\pi\)
\(548\) 11.6463 0.497504
\(549\) 4.39066 0.187389
\(550\) −5.08007 −0.216615
\(551\) −4.56182 −0.194340
\(552\) 14.9819 0.637674
\(553\) 0 0
\(554\) −17.5062 −0.743765
\(555\) 30.7866 1.30682
\(556\) −16.9819 −0.720195
\(557\) 38.3555 1.62517 0.812587 0.582840i \(-0.198059\pi\)
0.812587 + 0.582840i \(0.198059\pi\)
\(558\) 3.05257 0.129226
\(559\) 9.45359 0.399844
\(560\) 0 0
\(561\) 3.16784 0.133746
\(562\) 10.3381 0.436086
\(563\) 41.7739 1.76056 0.880279 0.474456i \(-0.157355\pi\)
0.880279 + 0.474456i \(0.157355\pi\)
\(564\) 4.96986 0.209269
\(565\) 44.4080 1.86826
\(566\) 10.5517 0.443521
\(567\) 0 0
\(568\) 20.1971 0.847450
\(569\) −11.8700 −0.497616 −0.248808 0.968553i \(-0.580039\pi\)
−0.248808 + 0.968553i \(0.580039\pi\)
\(570\) 6.63683 0.277986
\(571\) 19.8013 0.828661 0.414330 0.910127i \(-0.364016\pi\)
0.414330 + 0.910127i \(0.364016\pi\)
\(572\) 9.27540 0.387824
\(573\) −19.1652 −0.800637
\(574\) 0 0
\(575\) 25.8667 1.07872
\(576\) 0.373518 0.0155633
\(577\) −28.6791 −1.19392 −0.596962 0.802269i \(-0.703626\pi\)
−0.596962 + 0.802269i \(0.703626\pi\)
\(578\) −9.36052 −0.389346
\(579\) −48.4278 −2.01259
\(580\) −17.2453 −0.716070
\(581\) 0 0
\(582\) 12.2255 0.506762
\(583\) −9.22547 −0.382080
\(584\) 10.6885 0.442293
\(585\) 13.8546 0.572817
\(586\) −10.0697 −0.415976
\(587\) −1.01209 −0.0417733 −0.0208866 0.999782i \(-0.506649\pi\)
−0.0208866 + 0.999782i \(0.506649\pi\)
\(588\) 0 0
\(589\) −10.4861 −0.432074
\(590\) −20.7435 −0.853996
\(591\) 47.0044 1.93350
\(592\) −7.21338 −0.296468
\(593\) −14.2332 −0.584486 −0.292243 0.956344i \(-0.594402\pi\)
−0.292243 + 0.956344i \(0.594402\pi\)
\(594\) −2.94237 −0.120727
\(595\) 0 0
\(596\) 1.56885 0.0642626
\(597\) 10.6918 0.437587
\(598\) 12.9793 0.530763
\(599\) −26.5457 −1.08463 −0.542315 0.840175i \(-0.682452\pi\)
−0.542315 + 0.840175i \(0.682452\pi\)
\(600\) 34.6687 1.41534
\(601\) 12.1558 0.495843 0.247922 0.968780i \(-0.420252\pi\)
0.247922 + 0.968780i \(0.420252\pi\)
\(602\) 0 0
\(603\) −6.45094 −0.262703
\(604\) −25.7659 −1.04840
\(605\) 3.56885 0.145094
\(606\) 2.32865 0.0945949
\(607\) −13.9672 −0.566912 −0.283456 0.958985i \(-0.591481\pi\)
−0.283456 + 0.958985i \(0.591481\pi\)
\(608\) −8.49649 −0.344578
\(609\) 0 0
\(610\) 15.6696 0.634444
\(611\) 9.79432 0.396236
\(612\) −1.70655 −0.0689831
\(613\) 4.64189 0.187484 0.0937421 0.995597i \(-0.470117\pi\)
0.0937421 + 0.995597i \(0.470117\pi\)
\(614\) 10.8150 0.436459
\(615\) −8.77453 −0.353823
\(616\) 0 0
\(617\) −26.3960 −1.06266 −0.531331 0.847165i \(-0.678308\pi\)
−0.531331 + 0.847165i \(0.678308\pi\)
\(618\) −3.97757 −0.160001
\(619\) −14.6815 −0.590098 −0.295049 0.955482i \(-0.595336\pi\)
−0.295049 + 0.955482i \(0.595336\pi\)
\(620\) −39.6412 −1.59203
\(621\) 14.9819 0.601205
\(622\) −14.2021 −0.569453
\(623\) 0 0
\(624\) −18.0774 −0.723676
\(625\) −3.82710 −0.153084
\(626\) −10.7609 −0.430090
\(627\) −2.83216 −0.113106
\(628\) −17.9648 −0.716874
\(629\) −7.47338 −0.297983
\(630\) 0 0
\(631\) −30.1498 −1.20024 −0.600122 0.799908i \(-0.704881\pi\)
−0.600122 + 0.799908i \(0.704881\pi\)
\(632\) 14.9819 0.595950
\(633\) −23.1377 −0.919641
\(634\) 2.92937 0.116340
\(635\) −23.6309 −0.937762
\(636\) 27.6764 1.09744
\(637\) 0 0
\(638\) −2.02243 −0.0800690
\(639\) −5.65927 −0.223877
\(640\) −39.6137 −1.56587
\(641\) −16.1782 −0.639000 −0.319500 0.947586i \(-0.603515\pi\)
−0.319500 + 0.947586i \(0.603515\pi\)
\(642\) 5.98527 0.236220
\(643\) 2.33568 0.0921101 0.0460550 0.998939i \(-0.485335\pi\)
0.0460550 + 0.998939i \(0.485335\pi\)
\(644\) 0 0
\(645\) −10.9122 −0.429669
\(646\) −1.61107 −0.0633869
\(647\) −14.9622 −0.588223 −0.294112 0.955771i \(-0.595024\pi\)
−0.294112 + 0.955771i \(0.595024\pi\)
\(648\) 24.6962 0.970158
\(649\) 8.85195 0.347470
\(650\) 30.0345 1.17805
\(651\) 0 0
\(652\) 14.0251 0.549265
\(653\) −22.8898 −0.895747 −0.447873 0.894097i \(-0.647818\pi\)
−0.447873 + 0.894097i \(0.647818\pi\)
\(654\) 18.6861 0.730684
\(655\) 21.6188 0.844716
\(656\) 2.05590 0.0802692
\(657\) −2.99494 −0.116844
\(658\) 0 0
\(659\) −2.20568 −0.0859211 −0.0429606 0.999077i \(-0.513679\pi\)
−0.0429606 + 0.999077i \(0.513679\pi\)
\(660\) −10.7065 −0.416752
\(661\) 0.682377 0.0265414 0.0132707 0.999912i \(-0.495776\pi\)
0.0132707 + 0.999912i \(0.495776\pi\)
\(662\) 12.5018 0.485895
\(663\) −18.7290 −0.727373
\(664\) 0.393308 0.0152633
\(665\) 0 0
\(666\) −1.94501 −0.0753677
\(667\) 10.2978 0.398734
\(668\) −28.5809 −1.10583
\(669\) 29.8814 1.15528
\(670\) −23.0224 −0.889434
\(671\) −6.68676 −0.258139
\(672\) 0 0
\(673\) −10.8865 −0.419643 −0.209821 0.977740i \(-0.567288\pi\)
−0.209821 + 0.977740i \(0.567288\pi\)
\(674\) 17.7384 0.683259
\(675\) 34.6687 1.33440
\(676\) −34.4432 −1.32474
\(677\) −45.6654 −1.75506 −0.877532 0.479519i \(-0.840811\pi\)
−0.877532 + 0.479519i \(0.840811\pi\)
\(678\) −15.6238 −0.600030
\(679\) 0 0
\(680\) −13.8546 −0.531300
\(681\) −29.9622 −1.14815
\(682\) −4.64892 −0.178016
\(683\) 6.48372 0.248093 0.124046 0.992276i \(-0.460413\pi\)
0.124046 + 0.992276i \(0.460413\pi\)
\(684\) 1.52572 0.0583372
\(685\) −26.4932 −1.01225
\(686\) 0 0
\(687\) −10.6540 −0.406475
\(688\) 2.55676 0.0974757
\(689\) 54.5431 2.07793
\(690\) −14.9819 −0.570353
\(691\) −10.9468 −0.416434 −0.208217 0.978083i \(-0.566766\pi\)
−0.208217 + 0.978083i \(0.566766\pi\)
\(692\) 30.6885 1.16660
\(693\) 0 0
\(694\) −13.2754 −0.503927
\(695\) 38.6309 1.46535
\(696\) 13.8020 0.523164
\(697\) 2.13000 0.0806793
\(698\) 7.89418 0.298799
\(699\) 36.8590 1.39413
\(700\) 0 0
\(701\) −0.914874 −0.0345543 −0.0172772 0.999851i \(-0.505500\pi\)
−0.0172772 + 0.999851i \(0.505500\pi\)
\(702\) 17.3960 0.656568
\(703\) 6.68147 0.251996
\(704\) −0.568850 −0.0214393
\(705\) −11.3055 −0.425791
\(706\) 7.06466 0.265882
\(707\) 0 0
\(708\) −26.5559 −0.998030
\(709\) 44.3123 1.66418 0.832092 0.554637i \(-0.187143\pi\)
0.832092 + 0.554637i \(0.187143\pi\)
\(710\) −20.1971 −0.757982
\(711\) −4.19798 −0.157436
\(712\) −6.01979 −0.225601
\(713\) 23.6714 0.886499
\(714\) 0 0
\(715\) −21.0999 −0.789090
\(716\) −5.09547 −0.190427
\(717\) 42.3702 1.58234
\(718\) 15.9494 0.595226
\(719\) 5.20236 0.194015 0.0970076 0.995284i \(-0.469073\pi\)
0.0970076 + 0.995284i \(0.469073\pi\)
\(720\) 3.74704 0.139644
\(721\) 0 0
\(722\) −11.0354 −0.410696
\(723\) −38.0044 −1.41340
\(724\) −16.2231 −0.602925
\(725\) 23.8295 0.885006
\(726\) −1.25561 −0.0466000
\(727\) −50.0871 −1.85763 −0.928814 0.370547i \(-0.879170\pi\)
−0.928814 + 0.370547i \(0.879170\pi\)
\(728\) 0 0
\(729\) 18.7866 0.695801
\(730\) −10.6885 −0.395599
\(731\) 2.64892 0.0979737
\(732\) 20.0603 0.741449
\(733\) 47.0697 1.73856 0.869280 0.494320i \(-0.164583\pi\)
0.869280 + 0.494320i \(0.164583\pi\)
\(734\) −7.12055 −0.262824
\(735\) 0 0
\(736\) 19.1799 0.706981
\(737\) 9.82446 0.361889
\(738\) 0.554351 0.0204059
\(739\) 46.9914 1.72861 0.864303 0.502971i \(-0.167760\pi\)
0.864303 + 0.502971i \(0.167760\pi\)
\(740\) 25.2583 0.928512
\(741\) 16.7444 0.615121
\(742\) 0 0
\(743\) −5.19533 −0.190598 −0.0952991 0.995449i \(-0.530381\pi\)
−0.0952991 + 0.995449i \(0.530381\pi\)
\(744\) 31.7263 1.16314
\(745\) −3.56885 −0.130753
\(746\) −21.6843 −0.793920
\(747\) −0.110206 −0.00403222
\(748\) 2.59899 0.0950284
\(749\) 0 0
\(750\) −12.2633 −0.447793
\(751\) 33.4536 1.22074 0.610369 0.792117i \(-0.291021\pi\)
0.610369 + 0.792117i \(0.291021\pi\)
\(752\) 2.64892 0.0965961
\(753\) 42.2747 1.54058
\(754\) 11.9571 0.435452
\(755\) 58.6128 2.13314
\(756\) 0 0
\(757\) 40.0440 1.45542 0.727711 0.685884i \(-0.240584\pi\)
0.727711 + 0.685884i \(0.240584\pi\)
\(758\) −14.4010 −0.523068
\(759\) 6.39331 0.232062
\(760\) 12.3865 0.449306
\(761\) 7.55850 0.273995 0.136998 0.990571i \(-0.456255\pi\)
0.136998 + 0.990571i \(0.456255\pi\)
\(762\) 8.31392 0.301181
\(763\) 0 0
\(764\) −15.7237 −0.568863
\(765\) 3.88209 0.140357
\(766\) −23.9424 −0.865073
\(767\) −52.3348 −1.88970
\(768\) 16.1126 0.581414
\(769\) −51.5407 −1.85860 −0.929302 0.369320i \(-0.879591\pi\)
−0.929302 + 0.369320i \(0.879591\pi\)
\(770\) 0 0
\(771\) 55.6834 2.00539
\(772\) −39.7316 −1.42997
\(773\) 15.4657 0.556262 0.278131 0.960543i \(-0.410285\pi\)
0.278131 + 0.960543i \(0.410285\pi\)
\(774\) 0.689404 0.0247801
\(775\) 54.7763 1.96762
\(776\) 22.8168 0.819074
\(777\) 0 0
\(778\) 12.7884 0.458485
\(779\) −1.90429 −0.0682284
\(780\) 63.2996 2.26649
\(781\) 8.61878 0.308404
\(782\) 3.63683 0.130053
\(783\) 13.8020 0.493244
\(784\) 0 0
\(785\) 40.8667 1.45859
\(786\) −7.60602 −0.271298
\(787\) −21.9518 −0.782497 −0.391249 0.920285i \(-0.627957\pi\)
−0.391249 + 0.920285i \(0.627957\pi\)
\(788\) 38.5638 1.37378
\(789\) 29.6463 1.05544
\(790\) −14.9819 −0.533034
\(791\) 0 0
\(792\) 1.53871 0.0546757
\(793\) 39.5337 1.40388
\(794\) 5.14782 0.182689
\(795\) −62.9588 −2.23292
\(796\) 8.77188 0.310911
\(797\) −42.6258 −1.50988 −0.754942 0.655792i \(-0.772335\pi\)
−0.754942 + 0.655792i \(0.772335\pi\)
\(798\) 0 0
\(799\) 2.74439 0.0970896
\(800\) 44.3830 1.56917
\(801\) 1.68676 0.0595987
\(802\) −16.0517 −0.566804
\(803\) 4.56115 0.160959
\(804\) −29.4734 −1.03945
\(805\) 0 0
\(806\) 27.4855 0.968134
\(807\) 3.26331 0.114874
\(808\) 4.34602 0.152893
\(809\) 3.96745 0.139488 0.0697440 0.997565i \(-0.477782\pi\)
0.0697440 + 0.997565i \(0.477782\pi\)
\(810\) −24.6962 −0.867736
\(811\) 46.9217 1.64764 0.823821 0.566850i \(-0.191838\pi\)
0.823821 + 0.566850i \(0.191838\pi\)
\(812\) 0 0
\(813\) −39.2549 −1.37673
\(814\) 2.96216 0.103824
\(815\) −31.9045 −1.11757
\(816\) −5.06534 −0.177322
\(817\) −2.36823 −0.0828538
\(818\) 1.54551 0.0540374
\(819\) 0 0
\(820\) −7.19888 −0.251396
\(821\) −50.6000 −1.76595 −0.882977 0.469416i \(-0.844464\pi\)
−0.882977 + 0.469416i \(0.844464\pi\)
\(822\) 9.32094 0.325105
\(823\) 0.398599 0.0138943 0.00694714 0.999976i \(-0.497789\pi\)
0.00694714 + 0.999976i \(0.497789\pi\)
\(824\) −7.42345 −0.258608
\(825\) 14.7943 0.515072
\(826\) 0 0
\(827\) −20.4234 −0.710193 −0.355096 0.934830i \(-0.615552\pi\)
−0.355096 + 0.934830i \(0.615552\pi\)
\(828\) −3.44414 −0.119692
\(829\) 23.2633 0.807968 0.403984 0.914766i \(-0.367625\pi\)
0.403984 + 0.914766i \(0.367625\pi\)
\(830\) −0.393308 −0.0136519
\(831\) 50.9819 1.76854
\(832\) 3.36317 0.116597
\(833\) 0 0
\(834\) −13.5913 −0.470628
\(835\) 65.0165 2.24999
\(836\) −2.32359 −0.0803630
\(837\) 31.7263 1.09662
\(838\) −6.10053 −0.210739
\(839\) 16.4861 0.569165 0.284582 0.958652i \(-0.408145\pi\)
0.284582 + 0.958652i \(0.408145\pi\)
\(840\) 0 0
\(841\) −19.5132 −0.672869
\(842\) −25.6291 −0.883238
\(843\) −30.1069 −1.03694
\(844\) −18.9829 −0.653417
\(845\) 78.3521 2.69540
\(846\) 0.714253 0.0245565
\(847\) 0 0
\(848\) 14.7514 0.506566
\(849\) −30.7290 −1.05462
\(850\) 8.41574 0.288658
\(851\) −15.0827 −0.517029
\(852\) −25.8563 −0.885823
\(853\) 14.8315 0.507820 0.253910 0.967228i \(-0.418283\pi\)
0.253910 + 0.967228i \(0.418283\pi\)
\(854\) 0 0
\(855\) −3.47073 −0.118696
\(856\) 11.1705 0.381799
\(857\) 5.03188 0.171886 0.0859428 0.996300i \(-0.472610\pi\)
0.0859428 + 0.996300i \(0.472610\pi\)
\(858\) 7.42345 0.253432
\(859\) 56.0363 1.91193 0.955966 0.293477i \(-0.0948123\pi\)
0.955966 + 0.293477i \(0.0948123\pi\)
\(860\) −8.95272 −0.305285
\(861\) 0 0
\(862\) −2.49649 −0.0850307
\(863\) −36.4760 −1.24166 −0.620829 0.783946i \(-0.713204\pi\)
−0.620829 + 0.783946i \(0.713204\pi\)
\(864\) 25.7065 0.874555
\(865\) −69.8108 −2.37364
\(866\) −5.39769 −0.183421
\(867\) 27.2600 0.925798
\(868\) 0 0
\(869\) 6.39331 0.216878
\(870\) −13.8020 −0.467932
\(871\) −58.0844 −1.96812
\(872\) 34.8744 1.18100
\(873\) −6.39331 −0.216381
\(874\) −3.25146 −0.109982
\(875\) 0 0
\(876\) −13.6834 −0.462321
\(877\) −6.61613 −0.223411 −0.111705 0.993741i \(-0.535631\pi\)
−0.111705 + 0.993741i \(0.535631\pi\)
\(878\) −2.98195 −0.100636
\(879\) 29.3253 0.989119
\(880\) −5.70655 −0.192368
\(881\) 22.5286 0.759008 0.379504 0.925190i \(-0.376095\pi\)
0.379504 + 0.925190i \(0.376095\pi\)
\(882\) 0 0
\(883\) −51.1652 −1.72185 −0.860923 0.508735i \(-0.830113\pi\)
−0.860923 + 0.508735i \(0.830113\pi\)
\(884\) −15.3658 −0.516808
\(885\) 60.4098 2.03065
\(886\) 9.02485 0.303196
\(887\) 6.33809 0.212812 0.106406 0.994323i \(-0.466066\pi\)
0.106406 + 0.994323i \(0.466066\pi\)
\(888\) −20.2151 −0.678375
\(889\) 0 0
\(890\) 6.01979 0.201784
\(891\) 10.5387 0.353060
\(892\) 24.5156 0.820843
\(893\) −2.45359 −0.0821061
\(894\) 1.25561 0.0419938
\(895\) 11.5913 0.387454
\(896\) 0 0
\(897\) −37.7987 −1.26206
\(898\) 14.1608 0.472552
\(899\) 21.8071 0.727307
\(900\) −7.96986 −0.265662
\(901\) 15.2831 0.509154
\(902\) −0.844248 −0.0281104
\(903\) 0 0
\(904\) −29.1592 −0.969821
\(905\) 36.9045 1.22675
\(906\) −20.6214 −0.685101
\(907\) −6.88474 −0.228604 −0.114302 0.993446i \(-0.536463\pi\)
−0.114302 + 0.993446i \(0.536463\pi\)
\(908\) −24.5818 −0.815777
\(909\) −1.21777 −0.0403908
\(910\) 0 0
\(911\) −41.0818 −1.36110 −0.680550 0.732701i \(-0.738259\pi\)
−0.680550 + 0.732701i \(0.738259\pi\)
\(912\) 4.52859 0.149957
\(913\) 0.167838 0.00555462
\(914\) −2.17048 −0.0717932
\(915\) −45.6335 −1.50860
\(916\) −8.74084 −0.288805
\(917\) 0 0
\(918\) 4.87439 0.160879
\(919\) −11.6265 −0.383522 −0.191761 0.981442i \(-0.561420\pi\)
−0.191761 + 0.981442i \(0.561420\pi\)
\(920\) −27.9612 −0.921855
\(921\) −31.4958 −1.03782
\(922\) 21.1119 0.695285
\(923\) −50.9562 −1.67724
\(924\) 0 0
\(925\) −34.9019 −1.14757
\(926\) 3.82272 0.125622
\(927\) 2.08007 0.0683184
\(928\) 17.6694 0.580026
\(929\) −24.7668 −0.812573 −0.406287 0.913746i \(-0.633177\pi\)
−0.406287 + 0.913746i \(0.633177\pi\)
\(930\) −31.7263 −1.04035
\(931\) 0 0
\(932\) 30.2402 0.990551
\(933\) 41.3598 1.35406
\(934\) 3.94940 0.129228
\(935\) −5.91223 −0.193351
\(936\) −9.09721 −0.297352
\(937\) −1.15046 −0.0375839 −0.0187920 0.999823i \(-0.505982\pi\)
−0.0187920 + 0.999823i \(0.505982\pi\)
\(938\) 0 0
\(939\) 31.3381 1.02268
\(940\) −9.27540 −0.302530
\(941\) −10.1102 −0.329583 −0.164792 0.986328i \(-0.552695\pi\)
−0.164792 + 0.986328i \(0.552695\pi\)
\(942\) −14.3779 −0.468457
\(943\) 4.29874 0.139986
\(944\) −14.1542 −0.460679
\(945\) 0 0
\(946\) −1.04993 −0.0341361
\(947\) −24.4553 −0.794691 −0.397346 0.917669i \(-0.630069\pi\)
−0.397346 + 0.917669i \(0.630069\pi\)
\(948\) −19.1799 −0.622935
\(949\) −26.9665 −0.875371
\(950\) −7.52398 −0.244110
\(951\) −8.53101 −0.276637
\(952\) 0 0
\(953\) −16.1696 −0.523784 −0.261892 0.965097i \(-0.584346\pi\)
−0.261892 + 0.965097i \(0.584346\pi\)
\(954\) 3.97757 0.128778
\(955\) 35.7686 1.15744
\(956\) 34.7618 1.12428
\(957\) 5.88979 0.190390
\(958\) −11.2512 −0.363511
\(959\) 0 0
\(960\) −3.88209 −0.125294
\(961\) 19.1274 0.617011
\(962\) −17.5130 −0.564640
\(963\) −3.13000 −0.100863
\(964\) −31.1799 −1.00424
\(965\) 90.3823 2.90951
\(966\) 0 0
\(967\) −1.55941 −0.0501472 −0.0250736 0.999686i \(-0.507982\pi\)
−0.0250736 + 0.999686i \(0.507982\pi\)
\(968\) −2.34338 −0.0753191
\(969\) 4.69182 0.150723
\(970\) −22.8168 −0.732602
\(971\) 9.04728 0.290341 0.145171 0.989407i \(-0.453627\pi\)
0.145171 + 0.989407i \(0.453627\pi\)
\(972\) −10.5257 −0.337613
\(973\) 0 0
\(974\) 3.75209 0.120225
\(975\) −87.4674 −2.80120
\(976\) 10.6920 0.342244
\(977\) −6.75648 −0.216159 −0.108079 0.994142i \(-0.534470\pi\)
−0.108079 + 0.994142i \(0.534470\pi\)
\(978\) 11.2248 0.358929
\(979\) −2.56885 −0.0821008
\(980\) 0 0
\(981\) −9.77188 −0.311992
\(982\) 15.8031 0.504297
\(983\) −26.4305 −0.843001 −0.421501 0.906828i \(-0.638496\pi\)
−0.421501 + 0.906828i \(0.638496\pi\)
\(984\) 5.76154 0.183671
\(985\) −87.7257 −2.79517
\(986\) 3.35041 0.106699
\(987\) 0 0
\(988\) 13.7376 0.437051
\(989\) 5.34602 0.169994
\(990\) −1.53871 −0.0489034
\(991\) −0.634185 −0.0201456 −0.0100728 0.999949i \(-0.503206\pi\)
−0.0100728 + 0.999949i \(0.503206\pi\)
\(992\) 40.6161 1.28956
\(993\) −36.4080 −1.15537
\(994\) 0 0
\(995\) −19.9545 −0.632599
\(996\) −0.503514 −0.0159544
\(997\) −10.1274 −0.320736 −0.160368 0.987057i \(-0.551268\pi\)
−0.160368 + 0.987057i \(0.551268\pi\)
\(998\) 7.08445 0.224254
\(999\) −20.2151 −0.639578
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 539.2.a.i.1.2 3
3.2 odd 2 4851.2.a.bn.1.2 3
4.3 odd 2 8624.2.a.ck.1.3 3
7.2 even 3 539.2.e.l.67.2 6
7.3 odd 6 77.2.e.b.23.2 6
7.4 even 3 539.2.e.l.177.2 6
7.5 odd 6 77.2.e.b.67.2 yes 6
7.6 odd 2 539.2.a.h.1.2 3
11.10 odd 2 5929.2.a.w.1.2 3
21.5 even 6 693.2.i.g.298.2 6
21.17 even 6 693.2.i.g.100.2 6
21.20 even 2 4851.2.a.bo.1.2 3
28.3 even 6 1232.2.q.k.177.3 6
28.19 even 6 1232.2.q.k.529.3 6
28.27 even 2 8624.2.a.cl.1.1 3
77.3 odd 30 847.2.n.e.9.2 24
77.5 odd 30 847.2.n.e.487.2 24
77.10 even 6 847.2.e.d.485.2 6
77.17 even 30 847.2.n.d.366.2 24
77.19 even 30 847.2.n.d.130.2 24
77.24 even 30 847.2.n.d.807.2 24
77.26 odd 30 847.2.n.e.753.2 24
77.31 odd 30 847.2.n.e.807.2 24
77.38 odd 30 847.2.n.e.366.2 24
77.40 even 30 847.2.n.d.753.2 24
77.47 odd 30 847.2.n.e.130.2 24
77.52 even 30 847.2.n.d.9.2 24
77.54 even 6 847.2.e.d.606.2 6
77.59 odd 30 847.2.n.e.632.2 24
77.61 even 30 847.2.n.d.487.2 24
77.68 even 30 847.2.n.d.81.2 24
77.73 even 30 847.2.n.d.632.2 24
77.75 odd 30 847.2.n.e.81.2 24
77.76 even 2 5929.2.a.v.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.e.b.23.2 6 7.3 odd 6
77.2.e.b.67.2 yes 6 7.5 odd 6
539.2.a.h.1.2 3 7.6 odd 2
539.2.a.i.1.2 3 1.1 even 1 trivial
539.2.e.l.67.2 6 7.2 even 3
539.2.e.l.177.2 6 7.4 even 3
693.2.i.g.100.2 6 21.17 even 6
693.2.i.g.298.2 6 21.5 even 6
847.2.e.d.485.2 6 77.10 even 6
847.2.e.d.606.2 6 77.54 even 6
847.2.n.d.9.2 24 77.52 even 30
847.2.n.d.81.2 24 77.68 even 30
847.2.n.d.130.2 24 77.19 even 30
847.2.n.d.366.2 24 77.17 even 30
847.2.n.d.487.2 24 77.61 even 30
847.2.n.d.632.2 24 77.73 even 30
847.2.n.d.753.2 24 77.40 even 30
847.2.n.d.807.2 24 77.24 even 30
847.2.n.e.9.2 24 77.3 odd 30
847.2.n.e.81.2 24 77.75 odd 30
847.2.n.e.130.2 24 77.47 odd 30
847.2.n.e.366.2 24 77.38 odd 30
847.2.n.e.487.2 24 77.5 odd 30
847.2.n.e.632.2 24 77.59 odd 30
847.2.n.e.753.2 24 77.26 odd 30
847.2.n.e.807.2 24 77.31 odd 30
1232.2.q.k.177.3 6 28.3 even 6
1232.2.q.k.529.3 6 28.19 even 6
4851.2.a.bn.1.2 3 3.2 odd 2
4851.2.a.bo.1.2 3 21.20 even 2
5929.2.a.v.1.2 3 77.76 even 2
5929.2.a.w.1.2 3 11.10 odd 2
8624.2.a.ck.1.3 3 4.3 odd 2
8624.2.a.cl.1.1 3 28.27 even 2