Properties

Label 912.2.bh.f
Level $912$
Weight $2$
Character orbit 912.bh
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(239,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bh (of order \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{9} + 12 q^{13} + 12 q^{21} + 8 q^{25} + 16 q^{37} + 20 q^{45} + 40 q^{49} + 18 q^{57} - 12 q^{61} + 28 q^{69} + 44 q^{73} - 22 q^{81} + 4 q^{85} + 8 q^{93} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −1.63593 + 0.568988i 0 0.164566 + 0.0950122i 0 0.883190i 0 2.35251 1.86164i 0
239.2 0 −1.63249 0.578768i 0 3.06808 + 1.77136i 0 0.587143i 0 2.33006 + 1.88967i 0
239.3 0 −1.31747 1.12440i 0 −3.06808 1.77136i 0 0.587143i 0 0.471472 + 2.96272i 0
239.4 0 −1.08081 + 1.35346i 0 −1.81928 1.05036i 0 3.85684i 0 −0.663700 2.92566i 0
239.5 0 −0.631725 + 1.61274i 0 1.81928 + 1.05036i 0 3.85684i 0 −2.20185 2.03761i 0
239.6 0 −0.325205 1.70125i 0 −0.164566 0.0950122i 0 0.883190i 0 −2.78848 + 1.10651i 0
239.7 0 0.325205 + 1.70125i 0 −0.164566 0.0950122i 0 0.883190i 0 −2.78848 + 1.10651i 0
239.8 0 0.631725 1.61274i 0 1.81928 + 1.05036i 0 3.85684i 0 −2.20185 2.03761i 0
239.9 0 1.08081 1.35346i 0 −1.81928 1.05036i 0 3.85684i 0 −0.663700 2.92566i 0
239.10 0 1.31747 + 1.12440i 0 −3.06808 1.77136i 0 0.587143i 0 0.471472 + 2.96272i 0
239.11 0 1.63249 + 0.578768i 0 3.06808 + 1.77136i 0 0.587143i 0 2.33006 + 1.88967i 0
239.12 0 1.63593 0.568988i 0 0.164566 + 0.0950122i 0 0.883190i 0 2.35251 1.86164i 0
767.1 0 −1.63593 0.568988i 0 0.164566 0.0950122i 0 0.883190i 0 2.35251 + 1.86164i 0
767.2 0 −1.63249 + 0.578768i 0 3.06808 1.77136i 0 0.587143i 0 2.33006 1.88967i 0
767.3 0 −1.31747 + 1.12440i 0 −3.06808 + 1.77136i 0 0.587143i 0 0.471472 2.96272i 0
767.4 0 −1.08081 1.35346i 0 −1.81928 + 1.05036i 0 3.85684i 0 −0.663700 + 2.92566i 0
767.5 0 −0.631725 1.61274i 0 1.81928 1.05036i 0 3.85684i 0 −2.20185 + 2.03761i 0
767.6 0 −0.325205 + 1.70125i 0 −0.164566 + 0.0950122i 0 0.883190i 0 −2.78848 1.10651i 0
767.7 0 0.325205 1.70125i 0 −0.164566 + 0.0950122i 0 0.883190i 0 −2.78848 1.10651i 0
767.8 0 0.631725 + 1.61274i 0 1.81928 1.05036i 0 3.85684i 0 −2.20185 + 2.03761i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner
76.g odd 6 1 inner
228.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bh.f 24
3.b odd 2 1 inner 912.2.bh.f 24
4.b odd 2 1 inner 912.2.bh.f 24
12.b even 2 1 inner 912.2.bh.f 24
19.c even 3 1 inner 912.2.bh.f 24
57.h odd 6 1 inner 912.2.bh.f 24
76.g odd 6 1 inner 912.2.bh.f 24
228.m even 6 1 inner 912.2.bh.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bh.f 24 1.a even 1 1 trivial
912.2.bh.f 24 3.b odd 2 1 inner
912.2.bh.f 24 4.b odd 2 1 inner
912.2.bh.f 24 12.b even 2 1 inner
912.2.bh.f 24 19.c even 3 1 inner
912.2.bh.f 24 57.h odd 6 1 inner
912.2.bh.f 24 76.g odd 6 1 inner
912.2.bh.f 24 228.m even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{12} - 17T_{5}^{10} + 233T_{5}^{8} - 948T_{5}^{6} + 3102T_{5}^{4} - 112T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 16T_{7}^{4} + 17T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{43}^{12} - 149T_{43}^{10} + 16204T_{43}^{8} - 773503T_{43}^{6} + 27020284T_{43}^{4} - 359969925T_{43}^{2} + 3603000625 \) Copy content Toggle raw display