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Properties

Label 2240.2.h.f
Level $2240$
Weight $2$
Character orbit 2240.h
Analytic conductor $17.886$
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] = [2240,2,Mod(671,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("2240.671");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(17.8864900528\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
671.1 0 − 2.84405i 0 1.00000 0 −0.712435 + 2.54803i 0 −5.08861 0
671.2 0 2.84405i 0 1.00000 0 −0.712435 − 2.54803i 0 −5.08861 0
671.3 0 − 0.237691i 0 1.00000 0 −2.63997 + 0.174795i 0 2.94350 0
671.4 0 0.237691i 0 1.00000 0 −2.63997 − 0.174795i 0 2.94350 0
671.5 0 − 2.59614i 0 1.00000 0 −2.28275 − 1.33755i 0 −3.73996 0
671.6 0 2.59614i 0 1.00000 0 −2.28275 + 1.33755i 0 −3.73996 0
671.7 0 − 2.91632i 0 1.00000 0 −2.36081 + 1.19440i 0 −5.50491 0
671.8 0 2.91632i 0 1.00000 0 −2.36081 − 1.19440i 0 −5.50491 0
671.9 0 − 1.85258i 0 1.00000 0 1.29892 − 2.30495i 0 −0.432061 0
671.10 0 1.85258i 0 1.00000 0 1.29892 + 2.30495i 0 −0.432061 0
671.11 0 − 0.421861i 0 1.00000 0 −1.02538 − 2.43897i 0 2.82203 0
671.12 0 0.421861i 0 1.00000 0 −1.02538 + 2.43897i 0 2.82203 0
671.13 0 − 0.421861i 0 1.00000 0 1.02538 − 2.43897i 0 2.82203 0
671.14 0 0.421861i 0 1.00000 0 1.02538 + 2.43897i 0 2.82203 0
671.15 0 − 1.85258i 0 1.00000 0 −1.29892 − 2.30495i 0 −0.432061 0
671.16 0 1.85258i 0 1.00000 0 −1.29892 + 2.30495i 0 −0.432061 0
671.17 0 − 2.91632i 0 1.00000 0 2.36081 + 1.19440i 0 −5.50491 0
671.18 0 2.91632i 0 1.00000 0 2.36081 − 1.19440i 0 −5.50491 0
671.19 0 − 2.59614i 0 1.00000 0 2.28275 − 1.33755i 0 −3.73996 0
671.20 0 2.59614i 0 1.00000 0 2.28275 + 1.33755i 0 −3.73996 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 671.24
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 2240.2.h.f yes 24
4.b odd 2 1 inner 2240.2.h.f yes 24
7.b odd 2 1 2240.2.h.e ✓ 24
8.b even 2 1 2240.2.h.e ✓ 24
8.d odd 2 1 2240.2.h.e ✓ 24
28.d even 2 1 2240.2.h.e ✓ 24
56.e even 2 1 inner 2240.2.h.f yes 24
56.h odd 2 1 inner 2240.2.h.f yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.h.e ✓ 24 7.b odd 2 1
2240.2.h.e ✓ 24 8.b even 2 1
2240.2.h.e ✓ 24 8.d odd 2 1
2240.2.h.e ✓ 24 28.d even 2 1
2240.2.h.f yes 24 1.a even 1 1 trivial
2240.2.h.f yes 24 4.b odd 2 1 inner
2240.2.h.f yes 24 56.e even 2 1 inner
2240.2.h.f yes 24 56.h 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{12} + 27T_{3}^{10} + 267T_{3}^{8} + 1145T_{3}^{6} + 1848T_{3}^{4} + 384T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{6} - T_{13}^{5} - 43T_{13}^{4} + 69T_{13}^{3} + 346T_{13}^{2} - 404T_{13} - 376 \) Copy content Toggle raw display