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Properties

Label 927.2.f.f
Level $927$
Weight $2$
Character orbit 927.f
Analytic conductor $7.402$
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] = [927,2,Mod(46,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.f (of order \(3\), degree \(2\), 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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 18 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 18 q^{4} + 8 q^{7} - 32 q^{10} - 4 q^{13} - 18 q^{16} + 4 q^{19} - 40 q^{22} - 26 q^{25} + 8 q^{28} + 32 q^{31} + 8 q^{34} - 48 q^{37} + 22 q^{40} + 2 q^{43} + 18 q^{46} - 8 q^{49} - 68 q^{52} - 32 q^{55} - 24 q^{58} - 20 q^{61} + 68 q^{64} + 8 q^{67} + 38 q^{70} - 64 q^{73} - 188 q^{76} - 20 q^{79} + 60 q^{82} + 8 q^{85} + 6 q^{88} - 30 q^{91} - 92 q^{94} + 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.35501 − 2.34695i 0 −2.67212 + 4.62824i 1.61886 − 2.80395i 0 −0.805098 + 1.39447i 9.06295 0 −8.77431
46.2 −1.24321 − 2.15331i 0 −2.09117 + 3.62201i −1.45965 + 2.52819i 0 1.35416 − 2.34547i 5.42622 0 7.25864
46.3 −0.980806 − 1.69881i 0 −0.923962 + 1.60035i 0.348567 − 0.603736i 0 −1.12861 + 1.95480i −0.298316 0 −1.36751
46.4 −0.959463 − 1.66184i 0 −0.841140 + 1.45690i 1.43954 − 2.49335i 0 2.39836 − 4.15408i −0.609683 0 −5.52473
46.5 −0.711803 − 1.23288i 0 −0.0133274 + 0.0230838i 0.396502 − 0.686762i 0 −0.249615 + 0.432345i −2.80927 0 −1.12893
46.6 −0.609528 − 1.05573i 0 0.256951 − 0.445052i −1.48540 + 2.57279i 0 0.0501933 − 0.0869373i −3.06459 0 3.62157
46.7 −0.315463 − 0.546398i 0 0.800966 − 1.38731i 1.85440 − 3.21192i 0 −1.44599 + 2.50453i −2.27255 0 −2.33998
46.8 −0.0900175 − 0.155915i 0 0.983794 − 1.70398i −0.708871 + 1.22780i 0 1.82660 − 3.16376i −0.714305 0 0.255243
46.9 0.0900175 + 0.155915i 0 0.983794 − 1.70398i 0.708871 − 1.22780i 0 1.82660 − 3.16376i 0.714305 0 0.255243
46.10 0.315463 + 0.546398i 0 0.800966 − 1.38731i −1.85440 + 3.21192i 0 −1.44599 + 2.50453i 2.27255 0 −2.33998
46.11 0.609528 + 1.05573i 0 0.256951 − 0.445052i 1.48540 − 2.57279i 0 0.0501933 − 0.0869373i 3.06459 0 3.62157
46.12 0.711803 + 1.23288i 0 −0.0133274 + 0.0230838i −0.396502 + 0.686762i 0 −0.249615 + 0.432345i 2.80927 0 −1.12893
46.13 0.959463 + 1.66184i 0 −0.841140 + 1.45690i −1.43954 + 2.49335i 0 2.39836 − 4.15408i 0.609683 0 −5.52473
46.14 0.980806 + 1.69881i 0 −0.923962 + 1.60035i −0.348567 + 0.603736i 0 −1.12861 + 1.95480i 0.298316 0 −1.36751
46.15 1.24321 + 2.15331i 0 −2.09117 + 3.62201i 1.45965 − 2.52819i 0 1.35416 − 2.34547i −5.42622 0 7.25864
46.16 1.35501 + 2.34695i 0 −2.67212 + 4.62824i −1.61886 + 2.80395i 0 −0.805098 + 1.39447i −9.06295 0 −8.77431
262.1 −1.35501 + 2.34695i 0 −2.67212 − 4.62824i 1.61886 + 2.80395i 0 −0.805098 − 1.39447i 9.06295 0 −8.77431
262.2 −1.24321 + 2.15331i 0 −2.09117 − 3.62201i −1.45965 − 2.52819i 0 1.35416 + 2.34547i 5.42622 0 7.25864
262.3 −0.980806 + 1.69881i 0 −0.923962 − 1.60035i 0.348567 + 0.603736i 0 −1.12861 − 1.95480i −0.298316 0 −1.36751
262.4 −0.959463 + 1.66184i 0 −0.841140 − 1.45690i 1.43954 + 2.49335i 0 2.39836 + 4.15408i −0.609683 0 −5.52473
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.c even 3 1 inner
309.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.f.f ✓ 32
3.b odd 2 1 inner 927.2.f.f ✓ 32
103.c even 3 1 inner 927.2.f.f ✓ 32
309.h odd 6 1 inner 927.2.f.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.f.f ✓ 32 1.a even 1 1 trivial
927.2.f.f ✓ 32 3.b odd 2 1 inner
927.2.f.f ✓ 32 103.c even 3 1 inner
927.2.f.f ✓ 32 309.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 25 T_{2}^{30} + 376 T_{2}^{28} + 3691 T_{2}^{26} + 26818 T_{2}^{24} + 145258 T_{2}^{22} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display