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Properties

Label 1248.2.bb.e
Level $1248$
Weight $2$
Character orbit 1248.bb
Analytic conductor $9.965$
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] = [1248,2,Mod(463,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 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("1248.463");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.bb (of order \(4\), degree \(2\), not 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: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{3} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{3} + 24 q^{9} - 16 q^{11} - 4 q^{19} - 24 q^{27} + 16 q^{33} + 48 q^{35} - 12 q^{41} + 4 q^{57} + 24 q^{59} + 4 q^{65} + 36 q^{67} + 24 q^{73} + 24 q^{81} - 20 q^{89} + 4 q^{91} - 8 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
463.1 0 −1.00000 0 −2.82280 + 2.82280i 0 −0.354259 − 0.354259i 0 1.00000 0
463.2 0 −1.00000 0 −1.81516 + 1.81516i 0 −0.550916 − 0.550916i 0 1.00000 0
463.3 0 −1.00000 0 −1.49858 + 1.49858i 0 −0.667296 − 0.667296i 0 1.00000 0
463.4 0 −1.00000 0 −1.12139 + 1.12139i 0 −0.891748 − 0.891748i 0 1.00000 0
463.5 0 −1.00000 0 −0.360527 + 0.360527i 0 −2.77372 − 2.77372i 0 1.00000 0
463.6 0 −1.00000 0 −0.322129 + 0.322129i 0 −3.10435 − 3.10435i 0 1.00000 0
463.7 0 −1.00000 0 0.322129 − 0.322129i 0 3.10435 + 3.10435i 0 1.00000 0
463.8 0 −1.00000 0 0.360527 − 0.360527i 0 2.77372 + 2.77372i 0 1.00000 0
463.9 0 −1.00000 0 1.12139 − 1.12139i 0 0.891748 + 0.891748i 0 1.00000 0
463.10 0 −1.00000 0 1.49858 − 1.49858i 0 0.667296 + 0.667296i 0 1.00000 0
463.11 0 −1.00000 0 1.81516 − 1.81516i 0 0.550916 + 0.550916i 0 1.00000 0
463.12 0 −1.00000 0 2.82280 − 2.82280i 0 0.354259 + 0.354259i 0 1.00000 0
655.1 0 −1.00000 0 −2.82280 − 2.82280i 0 −0.354259 + 0.354259i 0 1.00000 0
655.2 0 −1.00000 0 −1.81516 − 1.81516i 0 −0.550916 + 0.550916i 0 1.00000 0
655.3 0 −1.00000 0 −1.49858 − 1.49858i 0 −0.667296 + 0.667296i 0 1.00000 0
655.4 0 −1.00000 0 −1.12139 − 1.12139i 0 −0.891748 + 0.891748i 0 1.00000 0
655.5 0 −1.00000 0 −0.360527 − 0.360527i 0 −2.77372 + 2.77372i 0 1.00000 0
655.6 0 −1.00000 0 −0.322129 − 0.322129i 0 −3.10435 + 3.10435i 0 1.00000 0
655.7 0 −1.00000 0 0.322129 + 0.322129i 0 3.10435 − 3.10435i 0 1.00000 0
655.8 0 −1.00000 0 0.360527 + 0.360527i 0 2.77372 − 2.77372i 0 1.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 463.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.d odd 4 1 inner
104.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.bb.e 24
4.b odd 2 1 312.2.t.f ✓ 24
8.b even 2 1 312.2.t.f ✓ 24
8.d odd 2 1 inner 1248.2.bb.e 24
12.b even 2 1 936.2.w.i 24
13.d odd 4 1 inner 1248.2.bb.e 24
24.h odd 2 1 936.2.w.i 24
52.f even 4 1 312.2.t.f ✓ 24
104.j odd 4 1 312.2.t.f ✓ 24
104.m even 4 1 inner 1248.2.bb.e 24
156.l odd 4 1 936.2.w.i 24
312.y even 4 1 936.2.w.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.t.f ✓ 24 4.b odd 2 1
312.2.t.f ✓ 24 8.b even 2 1
312.2.t.f ✓ 24 52.f even 4 1
312.2.t.f ✓ 24 104.j odd 4 1
936.2.w.i 24 12.b even 2 1
936.2.w.i 24 24.h odd 2 1
936.2.w.i 24 156.l odd 4 1
936.2.w.i 24 312.y even 4 1
1248.2.bb.e 24 1.a even 1 1 trivial
1248.2.bb.e 24 8.d odd 2 1 inner
1248.2.bb.e 24 13.d odd 4 1 inner
1248.2.bb.e 24 104.m even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\):

\( T_{5}^{24} + 324T_{5}^{20} + 19072T_{5}^{16} + 332288T_{5}^{12} + 1443840T_{5}^{8} + 156672T_{5}^{4} + 4096 \) Copy content Toggle raw display
\( T_{7}^{24} + 612T_{7}^{20} + 90240T_{7}^{16} + 332288T_{7}^{12} + 305152T_{7}^{8} + 82944T_{7}^{4} + 4096 \) Copy content Toggle raw display