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Properties

Label 1344.4.p.c
Level $1344$
Weight $4$
Character orbit 1344.p
Analytic conductor $79.299$
Analytic rank $0$
Dimension $32$
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] = [1344,4,Mod(223,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 0 − 3.00000i 0 −12.5168 0 18.2936 + 2.88837i 0 −9.00000 0
223.2 0 3.00000i 0 −12.5168 0 18.2936 − 2.88837i 0 −9.00000 0
223.3 0 − 3.00000i 0 −12.8225 0 6.32908 − 17.4053i 0 −9.00000 0
223.4 0 3.00000i 0 −12.8225 0 6.32908 + 17.4053i 0 −9.00000 0
223.5 0 − 3.00000i 0 20.7003 0 −11.8166 − 14.2607i 0 −9.00000 0
223.6 0 3.00000i 0 20.7003 0 −11.8166 + 14.2607i 0 −9.00000 0
223.7 0 − 3.00000i 0 −12.5168 0 −18.2936 + 2.88837i 0 −9.00000 0
223.8 0 3.00000i 0 −12.5168 0 −18.2936 − 2.88837i 0 −9.00000 0
223.9 0 − 3.00000i 0 20.7003 0 11.8166 − 14.2607i 0 −9.00000 0
223.10 0 3.00000i 0 20.7003 0 11.8166 + 14.2607i 0 −9.00000 0
223.11 0 − 3.00000i 0 −12.8225 0 −6.32908 − 17.4053i 0 −9.00000 0
223.12 0 3.00000i 0 −12.8225 0 −6.32908 + 17.4053i 0 −9.00000 0
223.13 0 − 3.00000i 0 −15.4018 0 −6.29285 + 17.4184i 0 −9.00000 0
223.14 0 3.00000i 0 −15.4018 0 −6.29285 − 17.4184i 0 −9.00000 0
223.15 0 − 3.00000i 0 1.64166 0 15.2747 + 10.4729i 0 −9.00000 0
223.16 0 3.00000i 0 1.64166 0 15.2747 − 10.4729i 0 −9.00000 0
223.17 0 − 3.00000i 0 5.86484 0 18.3297 − 2.64978i 0 −9.00000 0
223.18 0 3.00000i 0 5.86484 0 18.3297 + 2.64978i 0 −9.00000 0
223.19 0 − 3.00000i 0 −3.15485 0 6.11026 − 17.4833i 0 −9.00000 0
223.20 0 3.00000i 0 −3.15485 0 6.11026 + 17.4833i 0 −9.00000 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.p.c ✓ 32
4.b odd 2 1 inner 1344.4.p.c ✓ 32
7.b odd 2 1 1344.4.p.d yes 32
8.b even 2 1 1344.4.p.d yes 32
8.d odd 2 1 1344.4.p.d yes 32
28.d even 2 1 1344.4.p.d yes 32
56.e even 2 1 inner 1344.4.p.c ✓ 32
56.h odd 2 1 inner 1344.4.p.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.p.c ✓ 32 1.a even 1 1 trivial
1344.4.p.c ✓ 32 4.b odd 2 1 inner
1344.4.p.c ✓ 32 56.e even 2 1 inner
1344.4.p.c ✓ 32 56.h odd 2 1 inner
1344.4.p.d yes 32 7.b odd 2 1
1344.4.p.d yes 32 8.b even 2 1
1344.4.p.d yes 32 8.d odd 2 1
1344.4.p.d yes 32 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 640T_{5}^{6} - 1728T_{5}^{5} + 116464T_{5}^{4} + 458400T_{5}^{3} - 4875024T_{5}^{2} - 8581632T_{5} + 24385536 \) acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\). Copy content Toggle raw display