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Properties

Label 3168.2.o.f
Level $3168$
Weight $2$
Character orbit 3168.o
Analytic conductor $25.297$
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] = [3168,2,Mod(703,3168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3168.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3168.o (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: \(25.2966073603\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 40 q^{25} + 40 q^{49} + 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1 0 0 0 −3.91261 0 −3.25594 0 0 0
703.2 0 0 0 −3.91261 0 3.25594 0 0 0
703.3 0 0 0 −3.91261 0 3.25594 0 0 0
703.4 0 0 0 −3.91261 0 −3.25594 0 0 0
703.5 0 0 0 −1.86960 0 −3.91787 0 0 0
703.6 0 0 0 −1.86960 0 3.91787 0 0 0
703.7 0 0 0 −1.86960 0 3.91787 0 0 0
703.8 0 0 0 −1.86960 0 −3.91787 0 0 0
703.9 0 0 0 −1.09364 0 −0.221727 0 0 0
703.10 0 0 0 −1.09364 0 0.221727 0 0 0
703.11 0 0 0 −1.09364 0 0.221727 0 0 0
703.12 0 0 0 −1.09364 0 −0.221727 0 0 0
703.13 0 0 0 1.09364 0 0.221727 0 0 0
703.14 0 0 0 1.09364 0 −0.221727 0 0 0
703.15 0 0 0 1.09364 0 −0.221727 0 0 0
703.16 0 0 0 1.09364 0 0.221727 0 0 0
703.17 0 0 0 1.86960 0 3.91787 0 0 0
703.18 0 0 0 1.86960 0 −3.91787 0 0 0
703.19 0 0 0 1.86960 0 −3.91787 0 0 0
703.20 0 0 0 1.86960 0 3.91787 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.24
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
11.b odd 2 1 inner
12.b even 2 1 inner
33.d even 2 1 inner
44.c even 2 1 inner
132.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3168.2.o.f ✓ 24
3.b odd 2 1 inner 3168.2.o.f ✓ 24
4.b odd 2 1 inner 3168.2.o.f ✓ 24
11.b odd 2 1 inner 3168.2.o.f ✓ 24
12.b even 2 1 inner 3168.2.o.f ✓ 24
33.d even 2 1 inner 3168.2.o.f ✓ 24
44.c even 2 1 inner 3168.2.o.f ✓ 24
132.d odd 2 1 inner 3168.2.o.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3168.2.o.f ✓ 24 1.a even 1 1 trivial
3168.2.o.f ✓ 24 3.b odd 2 1 inner
3168.2.o.f ✓ 24 4.b odd 2 1 inner
3168.2.o.f ✓ 24 11.b odd 2 1 inner
3168.2.o.f ✓ 24 12.b even 2 1 inner
3168.2.o.f ✓ 24 33.d even 2 1 inner
3168.2.o.f ✓ 24 44.c even 2 1 inner
3168.2.o.f ✓ 24 132.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3168, [\chi])\):

\( T_{5}^{6} - 20T_{5}^{4} + 76T_{5}^{2} - 64 \) Copy content Toggle raw display
\( T_{7}^{6} - 26T_{7}^{4} + 164T_{7}^{2} - 8 \) Copy content Toggle raw display