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Properties

Label 1176.2.k.b
Level $1176$
Weight $2$
Character orbit 1176.k
Analytic conductor $9.390$
Analytic rank $0$
Dimension $24$
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] = [1176,2,Mod(881,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.k (of order \(2\), degree \(1\), 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: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{15} + 8 q^{25} + 16 q^{37} - 64 q^{39} + 16 q^{43} + 48 q^{51} + 48 q^{57} + 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1 0 −1.72171 − 0.188994i 0 4.01535 0 0 0 2.92856 + 0.650785i 0
881.2 0 −1.72171 + 0.188994i 0 4.01535 0 0 0 2.92856 − 0.650785i 0
881.3 0 −1.71558 − 0.238292i 0 −1.32383 0 0 0 2.88643 + 0.817619i 0
881.4 0 −1.71558 + 0.238292i 0 −1.32383 0 0 0 2.88643 − 0.817619i 0
881.5 0 −1.48931 − 0.884275i 0 −0.992103 0 0 0 1.43612 + 2.63393i 0
881.6 0 −1.48931 + 0.884275i 0 −0.992103 0 0 0 1.43612 − 2.63393i 0
881.7 0 −0.698115 − 1.58513i 0 0.489061 0 0 0 −2.02527 + 2.21321i 0
881.8 0 −0.698115 + 1.58513i 0 0.489061 0 0 0 −2.02527 − 2.21321i 0
881.9 0 −0.615077 − 1.61916i 0 1.31193 0 0 0 −2.24336 + 1.99182i 0
881.10 0 −0.615077 + 1.61916i 0 1.31193 0 0 0 −2.24336 − 1.99182i 0
881.11 0 −0.0935860 − 1.72952i 0 −3.34363 0 0 0 −2.98248 + 0.323718i 0
881.12 0 −0.0935860 + 1.72952i 0 −3.34363 0 0 0 −2.98248 − 0.323718i 0
881.13 0 0.0935860 − 1.72952i 0 3.34363 0 0 0 −2.98248 − 0.323718i 0
881.14 0 0.0935860 + 1.72952i 0 3.34363 0 0 0 −2.98248 + 0.323718i 0
881.15 0 0.615077 − 1.61916i 0 −1.31193 0 0 0 −2.24336 − 1.99182i 0
881.16 0 0.615077 + 1.61916i 0 −1.31193 0 0 0 −2.24336 + 1.99182i 0
881.17 0 0.698115 − 1.58513i 0 −0.489061 0 0 0 −2.02527 − 2.21321i 0
881.18 0 0.698115 + 1.58513i 0 −0.489061 0 0 0 −2.02527 + 2.21321i 0
881.19 0 1.48931 − 0.884275i 0 0.992103 0 0 0 1.43612 − 2.63393i 0
881.20 0 1.48931 + 0.884275i 0 0.992103 0 0 0 1.43612 + 2.63393i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.k.b ✓ 24
3.b odd 2 1 inner 1176.2.k.b ✓ 24
4.b odd 2 1 2352.2.k.j 24
7.b odd 2 1 inner 1176.2.k.b ✓ 24
7.c even 3 2 1176.2.u.c 48
7.d odd 6 2 1176.2.u.c 48
12.b even 2 1 2352.2.k.j 24
21.c even 2 1 inner 1176.2.k.b ✓ 24
21.g even 6 2 1176.2.u.c 48
21.h odd 6 2 1176.2.u.c 48
28.d even 2 1 2352.2.k.j 24
84.h odd 2 1 2352.2.k.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.k.b ✓ 24 1.a even 1 1 trivial
1176.2.k.b ✓ 24 3.b odd 2 1 inner
1176.2.k.b ✓ 24 7.b odd 2 1 inner
1176.2.k.b ✓ 24 21.c even 2 1 inner
1176.2.u.c 48 7.c even 3 2
1176.2.u.c 48 7.d odd 6 2
1176.2.u.c 48 21.g even 6 2
1176.2.u.c 48 21.h odd 6 2
2352.2.k.j 24 4.b odd 2 1
2352.2.k.j 24 12.b even 2 1
2352.2.k.j 24 28.d even 2 1
2352.2.k.j 24 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 32T_{5}^{10} + 316T_{5}^{8} - 1056T_{5}^{6} + 1476T_{5}^{4} - 832T_{5}^{2} + 128 \) acting on \(S_{2}^{\mathrm{new}}(1176, [\chi])\). Copy content Toggle raw display