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Properties

Label 6048.2.h.i
Level $6048$
Weight $2$
Character orbit 6048.h
Analytic conductor $48.294$
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] = [6048,2,Mod(2591,6048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6048, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6048.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6048.h (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: \(48.2935231425\)
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1 0 0 0 − 4.29730i 0 1.00000i 0 0 0
2591.2 0 0 0 4.29730i 0 − 1.00000i 0 0 0
2591.3 0 0 0 − 0.865222i 0 − 1.00000i 0 0 0
2591.4 0 0 0 0.865222i 0 1.00000i 0 0 0
2591.5 0 0 0 − 2.64388i 0 − 1.00000i 0 0 0
2591.6 0 0 0 2.64388i 0 1.00000i 0 0 0
2591.7 0 0 0 − 2.76574i 0 1.00000i 0 0 0
2591.8 0 0 0 2.76574i 0 − 1.00000i 0 0 0
2591.9 0 0 0 − 1.29686i 0 − 1.00000i 0 0 0
2591.10 0 0 0 1.29686i 0 1.00000i 0 0 0
2591.11 0 0 0 − 0.680677i 0 1.00000i 0 0 0
2591.12 0 0 0 0.680677i 0 − 1.00000i 0 0 0
2591.13 0 0 0 − 0.680677i 0 − 1.00000i 0 0 0
2591.14 0 0 0 0.680677i 0 1.00000i 0 0 0
2591.15 0 0 0 − 1.29686i 0 1.00000i 0 0 0
2591.16 0 0 0 1.29686i 0 − 1.00000i 0 0 0
2591.17 0 0 0 − 2.76574i 0 − 1.00000i 0 0 0
2591.18 0 0 0 2.76574i 0 1.00000i 0 0 0
2591.19 0 0 0 − 2.64388i 0 1.00000i 0 0 0
2591.20 0 0 0 2.64388i 0 − 1.00000i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.24
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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6048.2.h.i ✓ 24
3.b odd 2 1 inner 6048.2.h.i ✓ 24
4.b odd 2 1 inner 6048.2.h.i ✓ 24
12.b even 2 1 inner 6048.2.h.i ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6048.2.h.i ✓ 24 1.a even 1 1 trivial
6048.2.h.i ✓ 24 3.b odd 2 1 inner
6048.2.h.i ✓ 24 4.b odd 2 1 inner
6048.2.h.i ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6048, [\chi])\):

\( T_{5}^{12} + 36T_{5}^{10} + 422T_{5}^{8} + 2004T_{5}^{6} + 3649T_{5}^{4} + 2544T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{11}^{12} - 88T_{11}^{10} + 2856T_{11}^{8} - 42880T_{11}^{6} + 307984T_{11}^{4} - 1016448T_{11}^{2} + 1218816 \) Copy content Toggle raw display