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Properties

Label 2240.2.e.g
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(2239,2240)]
 
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, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.2239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (of order \(2\), degree \(1\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 1120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{9} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} - 32 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1 0 − 2.21376i 0 −1.81856 − 1.30110i 0 1.09155 − 2.41009i 0 −1.90072 0
2239.2 0 2.21376i 0 −1.81856 + 1.30110i 0 1.09155 + 2.41009i 0 −1.90072 0
2239.3 0 − 3.06386i 0 −1.39003 + 1.75152i 0 2.46063 − 0.972270i 0 −6.38723 0
2239.4 0 3.06386i 0 −1.39003 − 1.75152i 0 2.46063 + 0.972270i 0 −6.38723 0
2239.5 0 − 1.86733i 0 2.07366 − 0.836614i 0 2.64040 + 0.168201i 0 −0.486908 0
2239.6 0 1.86733i 0 2.07366 + 0.836614i 0 2.64040 − 0.168201i 0 −0.486908 0
2239.7 0 − 3.06386i 0 −1.39003 − 1.75152i 0 −2.46063 − 0.972270i 0 −6.38723 0
2239.8 0 3.06386i 0 −1.39003 + 1.75152i 0 −2.46063 + 0.972270i 0 −6.38723 0
2239.9 0 − 1.86733i 0 −2.07366 + 0.836614i 0 −2.64040 + 0.168201i 0 −0.486908 0
2239.10 0 1.86733i 0 −2.07366 − 0.836614i 0 −2.64040 − 0.168201i 0 −0.486908 0
2239.11 0 − 0.326439i 0 −0.278349 − 2.21868i 0 −2.25431 − 1.38495i 0 2.89344 0
2239.12 0 0.326439i 0 −0.278349 + 2.21868i 0 −2.25431 + 1.38495i 0 2.89344 0
2239.13 0 − 0.326439i 0 0.278349 − 2.21868i 0 −2.25431 − 1.38495i 0 2.89344 0
2239.14 0 0.326439i 0 0.278349 + 2.21868i 0 −2.25431 + 1.38495i 0 2.89344 0
2239.15 0 − 3.06386i 0 1.39003 + 1.75152i 0 2.46063 − 0.972270i 0 −6.38723 0
2239.16 0 3.06386i 0 1.39003 − 1.75152i 0 2.46063 + 0.972270i 0 −6.38723 0
2239.17 0 − 2.40740i 0 0.475379 + 2.18495i 0 −0.283298 + 2.63054i 0 −2.79558 0
2239.18 0 2.40740i 0 0.475379 − 2.18495i 0 −0.283298 − 2.63054i 0 −2.79558 0
2239.19 0 − 3.06386i 0 1.39003 − 1.75152i 0 −2.46063 − 0.972270i 0 −6.38723 0
2239.20 0 3.06386i 0 1.39003 + 1.75152i 0 −2.46063 + 0.972270i 0 −6.38723 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2239.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.e.g 48
4.b odd 2 1 inner 2240.2.e.g 48
5.b even 2 1 inner 2240.2.e.g 48
7.b odd 2 1 inner 2240.2.e.g 48
8.b even 2 1 1120.2.e.a ✓ 48
8.d odd 2 1 1120.2.e.a ✓ 48
20.d odd 2 1 inner 2240.2.e.g 48
28.d even 2 1 inner 2240.2.e.g 48
35.c odd 2 1 inner 2240.2.e.g 48
40.e odd 2 1 1120.2.e.a ✓ 48
40.f even 2 1 1120.2.e.a ✓ 48
56.e even 2 1 1120.2.e.a ✓ 48
56.h odd 2 1 1120.2.e.a ✓ 48
140.c even 2 1 inner 2240.2.e.g 48
280.c odd 2 1 1120.2.e.a ✓ 48
280.n even 2 1 1120.2.e.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.e.a ✓ 48 8.b even 2 1
1120.2.e.a ✓ 48 8.d odd 2 1
1120.2.e.a ✓ 48 40.e odd 2 1
1120.2.e.a ✓ 48 40.f even 2 1
1120.2.e.a ✓ 48 56.e even 2 1
1120.2.e.a ✓ 48 56.h odd 2 1
1120.2.e.a ✓ 48 280.c odd 2 1
1120.2.e.a ✓ 48 280.n even 2 1
2240.2.e.g 48 1.a even 1 1 trivial
2240.2.e.g 48 4.b odd 2 1 inner
2240.2.e.g 48 5.b even 2 1 inner
2240.2.e.g 48 7.b odd 2 1 inner
2240.2.e.g 48 20.d odd 2 1 inner
2240.2.e.g 48 28.d even 2 1 inner
2240.2.e.g 48 35.c odd 2 1 inner
2240.2.e.g 48 140.c 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 \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{12} + 24T_{3}^{10} + 209T_{3}^{8} + 802T_{3}^{6} + 1244T_{3}^{4} + 424T_{3}^{2} + 32 \) Copy content Toggle raw display
\( T_{11}^{12} + 82T_{11}^{10} + 2417T_{11}^{8} + 30236T_{11}^{6} + 138480T_{11}^{4} + 76096T_{11}^{2} + 9216 \) Copy content Toggle raw display