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Properties

Label 539.2.j.a
Level $539$
Weight $2$
Character orbit 539.j
Analytic conductor $4.304$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(78,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.78");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.j (of order \(7\), degree \(6\), minimal)

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: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 4 q^{3} - 30 q^{4} - 6 q^{5} - 4 q^{6} - 4 q^{7} - 6 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 4 q^{3} - 30 q^{4} - 6 q^{5} - 4 q^{6} - 4 q^{7} - 6 q^{8} - 16 q^{9} - 12 q^{10} - 24 q^{11} + 13 q^{12} - 5 q^{13} + 11 q^{14} - 6 q^{15} - 14 q^{16} - 18 q^{17} - 40 q^{18} + 50 q^{19} - 48 q^{20} + 7 q^{21} - 12 q^{23} - 54 q^{24} - 8 q^{25} - 30 q^{26} - 46 q^{27} - 46 q^{28} + 24 q^{29} - 52 q^{30} + 196 q^{31} + 85 q^{32} - 4 q^{33} + 4 q^{34} - 42 q^{35} - 34 q^{36} + 19 q^{37} - 60 q^{38} + 2 q^{39} - 40 q^{40} + 19 q^{41} - 22 q^{42} - 34 q^{43} - 16 q^{44} - 72 q^{45} - 22 q^{46} + 23 q^{47} + 66 q^{48} + 2 q^{49} + 218 q^{50} + 58 q^{51} - 12 q^{52} - 11 q^{53} - 54 q^{54} - 6 q^{55} + 67 q^{56} - 37 q^{57} - 32 q^{58} - 58 q^{59} - 108 q^{60} + 40 q^{61} + 20 q^{62} + 53 q^{63} - 104 q^{64} + 66 q^{65} + 17 q^{66} - 46 q^{67} - 44 q^{68} - 90 q^{69} + 78 q^{70} + 63 q^{71} - 15 q^{72} - 74 q^{73} + 88 q^{74} - 4 q^{75} - 35 q^{76} - 4 q^{77} - 33 q^{78} - 10 q^{79} + 408 q^{80} + 66 q^{81} - 29 q^{82} + 5 q^{83} - 118 q^{84} - 26 q^{85} - 97 q^{86} - 73 q^{87} - 6 q^{88} + 26 q^{89} + 222 q^{90} - 58 q^{91} + 19 q^{92} + 43 q^{93} - 39 q^{94} - 74 q^{95} + 139 q^{96} + 122 q^{97} + 63 q^{98} + 152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
78.1 −2.50813 − 1.20785i 0.0433708 + 0.190020i 3.58481 + 4.49521i 0.241452 + 1.05787i 0.120736 − 0.528979i 2.60198 + 0.479253i −2.32271 − 10.1764i 2.66868 − 1.28517i 0.672157 − 2.94491i
78.2 −2.41016 − 1.16067i −0.537081 − 2.35310i 3.21473 + 4.03114i −0.986700 − 4.32301i −1.43673 + 6.29473i −2.34625 + 1.22274i −1.87866 − 8.23093i −2.54574 + 1.22596i −2.63950 + 11.5644i
78.3 −2.12131 − 1.02157i 0.201166 + 0.881365i 2.20937 + 2.77046i −0.0120512 − 0.0528000i 0.473640 − 2.07515i −1.88065 − 1.86095i −0.808695 − 3.54312i 1.96657 − 0.947050i −0.0283743 + 0.124316i
78.4 −1.93793 − 0.933259i −0.713052 − 3.12408i 1.63763 + 2.05352i 0.353201 + 1.54747i −1.53373 + 6.71972i 2.02088 + 1.70765i −0.299887 − 1.31389i −6.54855 + 3.15361i 0.759714 − 3.32852i
78.5 −1.66233 − 0.800535i −0.456682 − 2.00085i 0.875500 + 1.09784i −0.541716 − 2.37341i −0.842599 + 3.69167i 0.996576 − 2.45088i 0.244616 + 1.07173i −1.09195 + 0.525858i −0.999491 + 4.37906i
78.6 −1.60042 − 0.770722i 0.101148 + 0.443157i 0.720354 + 0.903296i −0.344475 − 1.50924i 0.179672 − 0.787195i −1.05883 + 2.42464i 0.333863 + 1.46275i 2.51675 − 1.21200i −0.611903 + 2.68092i
78.7 −1.26716 − 0.610232i 0.640977 + 2.80830i −0.0136692 − 0.0171406i −0.783887 − 3.43443i 0.901497 − 3.94972i −2.61977 + 0.369854i 0.632786 + 2.77242i −4.77282 + 2.29847i −1.10249 + 4.83033i
78.8 −1.05781 − 0.509416i −0.0998048 − 0.437273i −0.387516 − 0.485930i −0.435810 − 1.90941i −0.117179 + 0.513396i 2.23808 + 1.41103i 0.684896 + 3.00072i 2.52166 − 1.21437i −0.511677 + 2.24180i
78.9 −0.833672 − 0.401475i −0.119354 − 0.522922i −0.713153 − 0.894266i 0.700265 + 3.06806i −0.110439 + 0.483863i −2.36032 − 1.19536i 0.647310 + 2.83605i 2.44370 − 1.17683i 0.647959 − 2.83889i
78.10 −0.682941 − 0.328887i 0.566928 + 2.48387i −0.888738 − 1.11444i 0.642314 + 2.81416i 0.429735 − 1.88279i −0.0787885 + 2.64458i 0.577775 + 2.53140i −3.14531 + 1.51470i 0.486879 − 2.13316i
78.11 −0.601232 − 0.289538i −0.477265 − 2.09103i −0.969332 − 1.21550i 0.374398 + 1.64034i −0.318487 + 1.39538i −2.24160 + 1.40543i 0.527843 + 2.31263i −1.44173 + 0.694302i 0.249842 − 1.09463i
78.12 −0.00104427 0.000502894i −0.314778 − 1.37913i −1.24698 − 1.56366i −0.571617 − 2.50442i −0.000364844 0.00159848i −0.354877 − 2.62184i 0.00103165 + 0.00451997i 0.899989 − 0.433412i −0.000662533 0.00290275i
78.13 0.0921948 + 0.0443987i 0.569695 + 2.49600i −1.24045 − 1.55548i −0.103073 − 0.451591i −0.0582961 + 0.255412i −1.74698 − 1.98697i −0.0908425 − 0.398007i −3.20255 + 1.54227i 0.0105473 − 0.0462106i
78.14 0.430439 + 0.207288i 0.236124 + 1.03453i −1.10467 − 1.38521i 0.729186 + 3.19477i −0.112808 + 0.494246i 2.12199 − 1.58025i −0.400974 − 1.75678i 1.68842 − 0.813099i −0.348369 + 1.52631i
78.15 0.830079 + 0.399745i −0.720518 − 3.15679i −0.717745 − 0.900024i 0.810701 + 3.55191i 0.663826 − 2.90841i −0.672872 − 2.55876i −0.646030 − 2.83044i −6.74330 + 3.24740i −0.746913 + 3.27244i
78.16 0.847855 + 0.408305i −0.283654 − 1.24277i −0.694835 − 0.871296i −0.587296 − 2.57311i 0.266932 − 1.16950i 1.66852 + 2.05330i −0.652170 − 2.85734i 1.23889 − 0.596620i 0.552674 − 2.42142i
78.17 0.924936 + 0.445426i 0.433733 + 1.90031i −0.589877 − 0.739683i −0.663416 − 2.90661i −0.445271 + 1.95086i 2.64131 − 0.153209i −0.673006 − 2.94863i −0.720137 + 0.346800i 0.681063 − 2.98393i
78.18 1.34094 + 0.645761i −0.139138 − 0.609605i 0.134125 + 0.168188i −0.0379204 − 0.166140i 0.207084 − 0.907293i −2.52510 − 0.789845i −0.591124 − 2.58988i 2.35065 − 1.13201i 0.0564380 − 0.247271i
78.19 1.72954 + 0.832903i −0.423087 − 1.85367i 1.05060 + 1.31742i 0.248692 + 1.08959i 0.812178 − 3.55838i 2.64255 − 0.130128i −0.134539 − 0.589456i −0.554170 + 0.266874i −0.477401 + 2.09163i
78.20 1.91923 + 0.924253i 0.741919 + 3.25056i 1.58222 + 1.98405i 0.129274 + 0.566385i −1.58042 + 6.92429i 0.423216 − 2.61168i 0.254872 + 1.11667i −7.31277 + 3.52165i −0.275377 + 1.20651i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 78.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.j.a ✓ 144
49.e even 7 1 inner 539.2.j.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.j.a ✓ 144 1.a even 1 1 trivial
539.2.j.a ✓ 144 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} + 39 T_{2}^{142} + 2 T_{2}^{141} + 836 T_{2}^{140} + 37 T_{2}^{139} + 13237 T_{2}^{138} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display