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Properties

Label 539.2.e.m.67.1
Level $539$
Weight $2$
Character 539.67
Analytic conductor $4.304$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [539,2,Mod(67,539)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("539.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 539.67
Dual form 539.2.e.m.177.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.939693 - 1.62760i) q^{2} +(0.326352 - 0.565258i) q^{3} +(-0.766044 + 1.32683i) q^{4} +(1.76604 + 3.05888i) q^{5} -1.22668 q^{6} -0.879385 q^{8} +(1.28699 + 2.22913i) q^{9} +(3.31908 - 5.74881i) q^{10} +(-0.500000 + 0.866025i) q^{11} +(0.500000 + 0.866025i) q^{12} -4.41147 q^{13} +2.30541 q^{15} +(2.35844 + 4.08494i) q^{16} +(-2.62449 + 4.54574i) q^{17} +(2.41875 - 4.18939i) q^{18} +(0.907604 + 1.57202i) q^{19} -5.41147 q^{20} +1.87939 q^{22} +(3.16637 + 5.48432i) q^{23} +(-0.286989 + 0.497079i) q^{24} +(-3.73783 + 6.47410i) q^{25} +(4.14543 + 7.18009i) q^{26} +3.63816 q^{27} +1.92127 q^{29} +(-2.16637 - 3.75227i) q^{30} +(0.733956 - 1.27125i) q^{31} +(3.55303 - 6.15403i) q^{32} +(0.326352 + 0.565258i) q^{33} +9.86484 q^{34} -3.94356 q^{36} +(-2.22668 - 3.85673i) q^{37} +(1.70574 - 2.95442i) q^{38} +(-1.43969 + 2.49362i) q^{39} +(-1.55303 - 2.68993i) q^{40} -0.283119 q^{41} -3.41147 q^{43} +(-0.766044 - 1.32683i) q^{44} +(-4.54576 + 7.87349i) q^{45} +(5.95084 - 10.3072i) q^{46} +(-2.27719 - 3.94421i) q^{47} +3.07873 q^{48} +14.0496 q^{50} +(1.71301 + 2.96702i) q^{51} +(3.37939 - 5.85327i) q^{52} +(3.61721 - 6.26519i) q^{53} +(-3.41875 - 5.92145i) q^{54} -3.53209 q^{55} +1.18479 q^{57} +(-1.80541 - 3.12706i) q^{58} +(4.76991 - 8.26173i) q^{59} +(-1.76604 + 3.05888i) q^{60} +(-0.573978 - 0.994159i) q^{61} -2.75877 q^{62} -3.92127 q^{64} +(-7.79086 - 13.4942i) q^{65} +(0.613341 - 1.06234i) q^{66} +(0.347296 - 0.601535i) q^{67} +(-4.02094 - 6.96448i) q^{68} +4.13341 q^{69} +9.46110 q^{71} +(-1.13176 - 1.96026i) q^{72} +(1.17365 - 2.03282i) q^{73} +(-4.18479 + 7.24827i) q^{74} +(2.43969 + 4.22567i) q^{75} -2.78106 q^{76} +5.41147 q^{78} +(6.20961 + 10.7554i) q^{79} +(-8.33022 + 14.4284i) q^{80} +(-2.67365 + 4.63089i) q^{81} +(0.266044 + 0.460802i) q^{82} +11.3327 q^{83} -18.5398 q^{85} +(3.20574 + 5.55250i) q^{86} +(0.627011 - 1.08602i) q^{87} +(0.439693 - 0.761570i) q^{88} +(-1.73396 - 3.00330i) q^{89} +17.0865 q^{90} -9.70233 q^{92} +(-0.479055 - 0.829748i) q^{93} +(-4.27972 + 7.41268i) q^{94} +(-3.20574 + 5.55250i) q^{95} +(-2.31908 - 4.01676i) q^{96} -15.3473 q^{97} -2.57398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 6 q^{5} + 6 q^{6} + 6 q^{8} + 3 q^{10} - 3 q^{11} + 3 q^{12} - 6 q^{13} + 18 q^{15} + 6 q^{16} - 3 q^{17} + 12 q^{18} + 9 q^{19} - 12 q^{20} + 6 q^{24} - 3 q^{25} + 9 q^{26} - 12 q^{27}+ \cdots - 90 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.939693 − 1.62760i −0.664463 − 1.15088i −0.979431 − 0.201781i \(-0.935327\pi\)
0.314968 − 0.949102i \(-0.398006\pi\)
\(3\) 0.326352 − 0.565258i 0.188419 − 0.326352i −0.756304 − 0.654220i \(-0.772997\pi\)
0.944723 + 0.327868i \(0.106330\pi\)
\(4\) −0.766044 + 1.32683i −0.383022 + 0.663414i
\(5\) 1.76604 + 3.05888i 0.789799 + 1.36797i 0.926090 + 0.377303i \(0.123149\pi\)
−0.136291 + 0.990669i \(0.543518\pi\)
\(6\) −1.22668 −0.500791
\(7\) 0 0
\(8\) −0.879385 −0.310910
\(9\) 1.28699 + 2.22913i 0.428996 + 0.743043i
\(10\) 3.31908 − 5.74881i 1.04958 − 1.81793i
\(11\) −0.500000 + 0.866025i −0.150756 + 0.261116i
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) −4.41147 −1.22352 −0.611761 − 0.791042i \(-0.709539\pi\)
−0.611761 + 0.791042i \(0.709539\pi\)
\(14\) 0 0
\(15\) 2.30541 0.595254
\(16\) 2.35844 + 4.08494i 0.589610 + 1.02123i
\(17\) −2.62449 + 4.54574i −0.636531 + 1.10250i 0.349657 + 0.936878i \(0.386298\pi\)
−0.986189 + 0.165627i \(0.947035\pi\)
\(18\) 2.41875 − 4.18939i 0.570104 − 0.987450i
\(19\) 0.907604 + 1.57202i 0.208219 + 0.360645i 0.951153 − 0.308719i \(-0.0999001\pi\)
−0.742935 + 0.669364i \(0.766567\pi\)
\(20\) −5.41147 −1.21004
\(21\) 0 0
\(22\) 1.87939 0.400686
\(23\) 3.16637 + 5.48432i 0.660235 + 1.14356i 0.980554 + 0.196250i \(0.0628765\pi\)
−0.320319 + 0.947310i \(0.603790\pi\)
\(24\) −0.286989 + 0.497079i −0.0585814 + 0.101466i
\(25\) −3.73783 + 6.47410i −0.747565 + 1.29482i
\(26\) 4.14543 + 7.18009i 0.812986 + 1.40813i
\(27\) 3.63816 0.700163
\(28\) 0 0
\(29\) 1.92127 0.356772 0.178386 − 0.983961i \(-0.442912\pi\)
0.178386 + 0.983961i \(0.442912\pi\)
\(30\) −2.16637 − 3.75227i −0.395524 − 0.685068i
\(31\) 0.733956 − 1.27125i 0.131822 − 0.228323i −0.792557 − 0.609798i \(-0.791251\pi\)
0.924379 + 0.381475i \(0.124584\pi\)
\(32\) 3.55303 − 6.15403i 0.628094 − 1.08789i
\(33\) 0.326352 + 0.565258i 0.0568106 + 0.0983988i
\(34\) 9.86484 1.69181
\(35\) 0 0
\(36\) −3.94356 −0.657261
\(37\) −2.22668 − 3.85673i −0.366064 − 0.634042i 0.622882 − 0.782316i \(-0.285962\pi\)
−0.988946 + 0.148274i \(0.952628\pi\)
\(38\) 1.70574 − 2.95442i 0.276707 − 0.479271i
\(39\) −1.43969 + 2.49362i −0.230535 + 0.399299i
\(40\) −1.55303 − 2.68993i −0.245556 − 0.425316i
\(41\) −0.283119 −0.0442157 −0.0221078 − 0.999756i \(-0.507038\pi\)
−0.0221078 + 0.999756i \(0.507038\pi\)
\(42\) 0 0
\(43\) −3.41147 −0.520245 −0.260122 − 0.965576i \(-0.583763\pi\)
−0.260122 + 0.965576i \(0.583763\pi\)
\(44\) −0.766044 − 1.32683i −0.115486 − 0.200027i
\(45\) −4.54576 + 7.87349i −0.677642 + 1.17371i
\(46\) 5.95084 − 10.3072i 0.877403 − 1.51971i
\(47\) −2.27719 − 3.94421i −0.332162 − 0.575322i 0.650773 − 0.759272i \(-0.274445\pi\)
−0.982936 + 0.183950i \(0.941111\pi\)
\(48\) 3.07873 0.444376
\(49\) 0 0
\(50\) 14.0496 1.98692
\(51\) 1.71301 + 2.96702i 0.239870 + 0.415466i
\(52\) 3.37939 − 5.85327i 0.468636 − 0.811702i
\(53\) 3.61721 − 6.26519i 0.496862 − 0.860591i −0.503131 − 0.864210i \(-0.667819\pi\)
0.999993 + 0.00361947i \(0.00115212\pi\)
\(54\) −3.41875 − 5.92145i −0.465233 − 0.805807i
\(55\) −3.53209 −0.476267
\(56\) 0 0
\(57\) 1.18479 0.156930
\(58\) −1.80541 − 3.12706i −0.237062 − 0.410603i
\(59\) 4.76991 − 8.26173i 0.620990 − 1.07559i −0.368312 − 0.929702i \(-0.620064\pi\)
0.989302 − 0.145884i \(-0.0466026\pi\)
\(60\) −1.76604 + 3.05888i −0.227995 + 0.394900i
\(61\) −0.573978 − 0.994159i −0.0734903 − 0.127289i 0.826938 − 0.562292i \(-0.190080\pi\)
−0.900429 + 0.435003i \(0.856747\pi\)
\(62\) −2.75877 −0.350364
\(63\) 0 0
\(64\) −3.92127 −0.490159
\(65\) −7.79086 − 13.4942i −0.966337 − 1.67375i
\(66\) 0.613341 − 1.06234i 0.0754970 − 0.130765i
\(67\) 0.347296 − 0.601535i 0.0424290 − 0.0734892i −0.844031 − 0.536294i \(-0.819824\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(68\) −4.02094 − 6.96448i −0.487611 − 0.844567i
\(69\) 4.13341 0.497604
\(70\) 0 0
\(71\) 9.46110 1.12283 0.561413 − 0.827536i \(-0.310258\pi\)
0.561413 + 0.827536i \(0.310258\pi\)
\(72\) −1.13176 − 1.96026i −0.133379 − 0.231019i
\(73\) 1.17365 − 2.03282i 0.137365 − 0.237923i −0.789133 − 0.614222i \(-0.789470\pi\)
0.926498 + 0.376299i \(0.122803\pi\)
\(74\) −4.18479 + 7.24827i −0.486472 + 0.842595i
\(75\) 2.43969 + 4.22567i 0.281711 + 0.487939i
\(76\) −2.78106 −0.319009
\(77\) 0 0
\(78\) 5.41147 0.612729
\(79\) 6.20961 + 10.7554i 0.698635 + 1.21007i 0.968940 + 0.247297i \(0.0795424\pi\)
−0.270304 + 0.962775i \(0.587124\pi\)
\(80\) −8.33022 + 14.4284i −0.931347 + 1.61314i
\(81\) −2.67365 + 4.63089i −0.297072 + 0.514544i
\(82\) 0.266044 + 0.460802i 0.0293797 + 0.0508871i
\(83\) 11.3327 1.24393 0.621965 − 0.783045i \(-0.286334\pi\)
0.621965 + 0.783045i \(0.286334\pi\)
\(84\) 0 0
\(85\) −18.5398 −2.01093
\(86\) 3.20574 + 5.55250i 0.345684 + 0.598741i
\(87\) 0.627011 − 1.08602i 0.0672227 − 0.116433i
\(88\) 0.439693 − 0.761570i 0.0468714 − 0.0811836i
\(89\) −1.73396 − 3.00330i −0.183799 − 0.318349i 0.759372 − 0.650656i \(-0.225506\pi\)
−0.943171 + 0.332307i \(0.892173\pi\)
\(90\) 17.0865 1.80107
\(91\) 0 0
\(92\) −9.70233 −1.01154
\(93\) −0.479055 − 0.829748i −0.0496757 − 0.0860409i
\(94\) −4.27972 + 7.41268i −0.441419 + 0.764560i
\(95\) −3.20574 + 5.55250i −0.328902 + 0.569674i
\(96\) −2.31908 − 4.01676i −0.236690 − 0.409959i
\(97\) −15.3473 −1.55828 −0.779141 − 0.626849i \(-0.784344\pi\)
−0.779141 + 0.626849i \(0.784344\pi\)
\(98\) 0 0
\(99\) −2.57398 −0.258695
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 539.2.e.m.67.1 6
7.2 even 3 inner 539.2.e.m.177.1 6
7.3 odd 6 539.2.a.j.1.3 3
7.4 even 3 539.2.a.g.1.3 3
7.5 odd 6 77.2.e.a.23.1 ✓ 6
7.6 odd 2 77.2.e.a.67.1 yes 6
21.5 even 6 693.2.i.h.100.3 6
21.11 odd 6 4851.2.a.bk.1.1 3
21.17 even 6 4851.2.a.bj.1.1 3
21.20 even 2 693.2.i.h.298.3 6
28.3 even 6 8624.2.a.ch.1.2 3
28.11 odd 6 8624.2.a.co.1.2 3
28.19 even 6 1232.2.q.m.177.2 6
28.27 even 2 1232.2.q.m.529.2 6
77.5 odd 30 847.2.n.g.366.1 24
77.6 even 10 847.2.n.f.487.1 24
77.10 even 6 5929.2.a.x.1.1 3
77.13 even 10 847.2.n.f.81.3 24
77.19 even 30 847.2.n.f.9.1 24
77.20 odd 10 847.2.n.g.81.1 24
77.26 odd 30 847.2.n.g.632.1 24
77.27 odd 10 847.2.n.g.487.3 24
77.32 odd 6 5929.2.a.u.1.1 3
77.40 even 30 847.2.n.f.632.3 24
77.41 even 10 847.2.n.f.130.3 24
77.47 odd 30 847.2.n.g.9.3 24
77.48 odd 10 847.2.n.g.753.3 24
77.54 even 6 847.2.e.c.485.3 6
77.61 even 30 847.2.n.f.366.3 24
77.62 even 10 847.2.n.f.753.1 24
77.68 even 30 847.2.n.f.807.1 24
77.69 odd 10 847.2.n.g.130.1 24
77.75 odd 30 847.2.n.g.807.3 24
77.76 even 2 847.2.e.c.606.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.e.a.23.1 ✓ 6 7.5 odd 6
77.2.e.a.67.1 yes 6 7.6 odd 2
539.2.a.g.1.3 3 7.4 even 3
539.2.a.j.1.3 3 7.3 odd 6
539.2.e.m.67.1 6 1.1 even 1 trivial
539.2.e.m.177.1 6 7.2 even 3 inner
693.2.i.h.100.3 6 21.5 even 6
693.2.i.h.298.3 6 21.20 even 2
847.2.e.c.485.3 6 77.54 even 6
847.2.e.c.606.3 6 77.76 even 2
847.2.n.f.9.1 24 77.19 even 30
847.2.n.f.81.3 24 77.13 even 10
847.2.n.f.130.3 24 77.41 even 10
847.2.n.f.366.3 24 77.61 even 30
847.2.n.f.487.1 24 77.6 even 10
847.2.n.f.632.3 24 77.40 even 30
847.2.n.f.753.1 24 77.62 even 10
847.2.n.f.807.1 24 77.68 even 30
847.2.n.g.9.3 24 77.47 odd 30
847.2.n.g.81.1 24 77.20 odd 10
847.2.n.g.130.1 24 77.69 odd 10
847.2.n.g.366.1 24 77.5 odd 30
847.2.n.g.487.3 24 77.27 odd 10
847.2.n.g.632.1 24 77.26 odd 30
847.2.n.g.753.3 24 77.48 odd 10
847.2.n.g.807.3 24 77.75 odd 30
1232.2.q.m.177.2 6 28.19 even 6
1232.2.q.m.529.2 6 28.27 even 2
4851.2.a.bj.1.1 3 21.17 even 6
4851.2.a.bk.1.1 3 21.11 odd 6
5929.2.a.u.1.1 3 77.32 odd 6
5929.2.a.x.1.1 3 77.10 even 6
8624.2.a.ch.1.2 3 28.3 even 6
8624.2.a.co.1.2 3 28.11 odd 6