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Properties

Label 539.2.q.f
Level $539$
Weight $2$
Character orbit 539.q
Analytic conductor $4.304$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(214,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.q (of order \(15\), degree \(8\), not 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: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{2} - 2 q^{3} + 11 q^{4} - 5 q^{5} - 6 q^{6} - 10 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{2} - 2 q^{3} + 11 q^{4} - 5 q^{5} - 6 q^{6} - 10 q^{8} + 12 q^{9} + 12 q^{10} + 3 q^{11} + 18 q^{12} + 14 q^{13} - 36 q^{15} - 17 q^{16} - 5 q^{17} - 11 q^{18} + 19 q^{19} - 2 q^{20} - 66 q^{22} - 32 q^{23} - 35 q^{24} - 7 q^{25} - 27 q^{26} - 20 q^{27} + 6 q^{29} + 2 q^{30} - 7 q^{31} - 32 q^{32} - 26 q^{33} + 48 q^{34} + 104 q^{36} - 4 q^{37} - 5 q^{38} - 11 q^{39} - 10 q^{40} + 20 q^{41} - 16 q^{43} + 38 q^{44} + 70 q^{45} + 42 q^{46} - 23 q^{47} + 72 q^{48} + 104 q^{50} + 29 q^{51} + 33 q^{52} - 4 q^{53} + 60 q^{54} + 24 q^{55} - 22 q^{57} - 20 q^{58} + 17 q^{59} + 30 q^{60} - 7 q^{61} - 158 q^{62} + 14 q^{64} + 8 q^{65} + 8 q^{66} + 38 q^{67} - 2 q^{68} - 20 q^{69} - 28 q^{71} - 35 q^{73} + 29 q^{74} + 9 q^{75} - 104 q^{76} - 116 q^{78} - 15 q^{79} - 87 q^{80} + 14 q^{81} + 19 q^{82} - 10 q^{83} + 12 q^{85} + 52 q^{86} - 72 q^{87} - 55 q^{88} + 74 q^{89} + 28 q^{90} - 110 q^{92} - 32 q^{93} - 24 q^{94} - 32 q^{95} - 42 q^{96} - 40 q^{97} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −1.62629 + 1.80618i −1.31021 + 0.583344i −0.408403 − 3.88570i 1.23113 − 0.261684i 1.07716 − 3.31516i 0 3.74989 + 2.72445i −0.631025 + 0.700825i −1.52952 + 2.64921i
214.2 −0.942984 + 1.04729i 1.97635 − 0.879926i 0.00145969 + 0.0138880i −1.79137 + 0.380767i −0.942126 + 2.89957i 0 −2.29616 − 1.66826i 1.12429 − 1.24865i 1.29046 − 2.23514i
214.3 0.448146 − 0.497717i −2.86833 + 1.27706i 0.162170 + 1.54294i 2.09693 − 0.445717i −0.649815 + 1.99992i 0 1.92429 + 1.39808i 4.58902 − 5.09662i 0.717891 − 1.24342i
214.4 1.70758 − 1.89646i 0.375100 − 0.167005i −0.471675 − 4.48769i 3.35405 − 0.712924i 0.323795 − 0.996539i 0 −5.18703 − 3.76860i −1.89458 + 2.10415i 4.37528 − 7.57820i
312.1 −0.207438 + 1.97364i −1.86446 − 2.07069i −1.89594 − 0.402995i 0.0245966 + 0.0109511i 4.47357 − 3.25024i 0 −0.0378378 + 0.116453i −0.497972 + 4.73789i −0.0267159 + 0.0462732i
312.2 −0.116493 + 1.10836i 1.91292 + 2.12452i 0.741401 + 0.157590i −3.15728 − 1.40571i −2.57757 + 1.87272i 0 −0.949813 + 2.92322i −0.540709 + 5.14450i 1.92584 − 3.33566i
312.3 −0.0236455 + 0.224972i 0.146626 + 0.162845i 1.90624 + 0.405184i 2.27977 + 1.01502i −0.0401026 + 0.0291363i 0 −0.276036 + 0.849550i 0.308566 − 2.93581i −0.282258 + 0.488885i
312.4 0.178447 − 1.69781i −1.53335 − 1.70296i −0.894410 − 0.190113i −3.71481 − 1.65394i −3.16491 + 2.29944i 0 0.572703 − 1.76260i −0.235317 + 2.23889i −3.47097 + 6.01190i
324.1 −2.49618 − 0.530579i −0.0429192 + 0.408349i 4.12229 + 1.83536i −2.29443 − 2.54823i 0.323795 − 0.996539i 0 −5.18703 − 3.76860i 2.76954 + 0.588683i 4.37528 + 7.57820i
324.2 −0.655108 − 0.139248i 0.328196 − 3.12257i −1.41731 − 0.631029i −1.43447 − 1.59314i −0.649815 + 1.99992i 0 1.92429 + 1.39808i −6.70831 − 1.42589i 0.717891 + 1.24342i
324.3 1.37847 + 0.293003i −0.226135 + 2.15153i −0.0127572 − 0.00567988i 1.22544 + 1.36099i −0.942126 + 2.89957i 0 −2.29616 − 1.66826i −1.64350 − 0.349337i 1.29046 + 2.23514i
324.4 2.37734 + 0.505320i 0.149915 − 1.42635i 3.56931 + 1.58916i −0.842189 − 0.935345i 1.07716 − 3.31516i 0 3.74989 + 2.72445i 0.922445 + 0.196072i −1.52952 − 2.64921i
361.1 −2.49618 + 0.530579i −0.0429192 − 0.408349i 4.12229 − 1.83536i −2.29443 + 2.54823i 0.323795 + 0.996539i 0 −5.18703 + 3.76860i 2.76954 − 0.588683i 4.37528 − 7.57820i
361.2 −0.655108 + 0.139248i 0.328196 + 3.12257i −1.41731 + 0.631029i −1.43447 + 1.59314i −0.649815 − 1.99992i 0 1.92429 − 1.39808i −6.70831 + 1.42589i 0.717891 − 1.24342i
361.3 1.37847 − 0.293003i −0.226135 − 2.15153i −0.0127572 + 0.00567988i 1.22544 − 1.36099i −0.942126 − 2.89957i 0 −2.29616 + 1.66826i −1.64350 + 0.349337i 1.29046 − 2.23514i
361.4 2.37734 − 0.505320i 0.149915 + 1.42635i 3.56931 − 1.58916i −0.842189 + 0.935345i 1.07716 + 3.31516i 0 3.74989 − 2.72445i 0.922445 − 0.196072i −1.52952 + 2.64921i
410.1 −1.55957 − 0.694364i 2.24148 + 0.476441i 0.611847 + 0.679525i 0.425051 − 4.04409i −3.16491 − 2.29944i 0 0.572703 + 1.76260i 2.05659 + 0.915654i −3.47097 + 6.01190i
410.2 0.206654 + 0.0920084i −0.214341 − 0.0455596i −1.30402 − 1.44826i −0.260853 + 2.48185i −0.0401026 − 0.0291363i 0 −0.276036 − 0.849550i −2.69677 − 1.20068i −0.282258 + 0.488885i
410.3 1.01812 + 0.453294i −2.79635 − 0.594382i −0.507177 − 0.563277i 0.361259 − 3.43715i −2.57757 − 1.87272i 0 −0.949813 − 2.92322i 4.72563 + 2.10398i 1.92584 − 3.33566i
410.4 1.81294 + 0.807175i 2.72550 + 0.579324i 1.29698 + 1.44044i −0.00281436 + 0.0267768i 4.47357 + 3.25024i 0 −0.0378378 − 0.116453i 4.35212 + 1.93769i −0.0267159 + 0.0462732i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.q.f 32
7.b odd 2 1 539.2.q.g 32
7.c even 3 1 539.2.f.e 16
7.c even 3 1 inner 539.2.q.f 32
7.d odd 6 1 77.2.f.b ✓ 16
7.d odd 6 1 539.2.q.g 32
11.c even 5 1 inner 539.2.q.f 32
21.g even 6 1 693.2.m.i 16
77.i even 6 1 847.2.f.x 16
77.j odd 10 1 539.2.q.g 32
77.m even 15 1 539.2.f.e 16
77.m even 15 1 inner 539.2.q.f 32
77.m even 15 1 5929.2.a.bt 8
77.n even 30 1 847.2.a.o 8
77.n even 30 2 847.2.f.v 16
77.n even 30 1 847.2.f.x 16
77.o odd 30 1 5929.2.a.bs 8
77.p odd 30 1 77.2.f.b ✓ 16
77.p odd 30 1 539.2.q.g 32
77.p odd 30 1 847.2.a.p 8
77.p odd 30 2 847.2.f.w 16
231.bc even 30 1 693.2.m.i 16
231.bc even 30 1 7623.2.a.ct 8
231.bf odd 30 1 7623.2.a.cw 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b ✓ 16 7.d odd 6 1
77.2.f.b ✓ 16 77.p odd 30 1
539.2.f.e 16 7.c even 3 1
539.2.f.e 16 77.m even 15 1
539.2.q.f 32 1.a even 1 1 trivial
539.2.q.f 32 7.c even 3 1 inner
539.2.q.f 32 11.c even 5 1 inner
539.2.q.f 32 77.m even 15 1 inner
539.2.q.g 32 7.b odd 2 1
539.2.q.g 32 7.d odd 6 1
539.2.q.g 32 77.j odd 10 1
539.2.q.g 32 77.p odd 30 1
693.2.m.i 16 21.g even 6 1
693.2.m.i 16 231.bc even 30 1
847.2.a.o 8 77.n even 30 1
847.2.a.p 8 77.p odd 30 1
847.2.f.v 16 77.n even 30 2
847.2.f.w 16 77.p odd 30 2
847.2.f.x 16 77.i even 6 1
847.2.f.x 16 77.n even 30 1
5929.2.a.bs 8 77.o odd 30 1
5929.2.a.bt 8 77.m even 15 1
7623.2.a.ct 8 231.bc even 30 1
7623.2.a.cw 8 231.bf odd 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{32} - 3 T_{2}^{31} - 5 T_{2}^{30} + 22 T_{2}^{29} + 14 T_{2}^{28} - 77 T_{2}^{27} - 125 T_{2}^{26} + \cdots + 625 \) Copy content Toggle raw display
\( T_{3}^{32} + 2 T_{3}^{31} - 10 T_{3}^{30} - 24 T_{3}^{29} + 20 T_{3}^{28} - 32 T_{3}^{27} + 234 T_{3}^{26} + \cdots + 65536 \) Copy content Toggle raw display