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Properties

Label 1120.2.n.b
Level $1120$
Weight $2$
Character orbit 1120.n
Analytic conductor $8.943$
Analytic rank $0$
Dimension $40$
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] = [1120,2,Mod(559,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("1120.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.n (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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 280)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1 0 −2.91053 0 −1.83654 + 1.27559i 0 −1.52252 − 2.16378i 0 5.47117 0
559.2 0 −2.91053 0 1.83654 − 1.27559i 0 1.52252 + 2.16378i 0 5.47117 0
559.3 0 −2.91053 0 1.83654 + 1.27559i 0 1.52252 − 2.16378i 0 5.47117 0
559.4 0 −2.91053 0 −1.83654 − 1.27559i 0 −1.52252 + 2.16378i 0 5.47117 0
559.5 0 −2.57969 0 0.460102 − 2.18822i 0 −2.31487 − 1.28116i 0 3.65479 0
559.6 0 −2.57969 0 −0.460102 + 2.18822i 0 2.31487 + 1.28116i 0 3.65479 0
559.7 0 −2.57969 0 −0.460102 − 2.18822i 0 2.31487 − 1.28116i 0 3.65479 0
559.8 0 −2.57969 0 0.460102 + 2.18822i 0 −2.31487 + 1.28116i 0 3.65479 0
559.9 0 −1.64952 0 −1.78913 − 1.34127i 0 −0.157160 − 2.64108i 0 −0.279099 0
559.10 0 −1.64952 0 1.78913 − 1.34127i 0 0.157160 − 2.64108i 0 −0.279099 0
559.11 0 −1.64952 0 1.78913 + 1.34127i 0 0.157160 + 2.64108i 0 −0.279099 0
559.12 0 −1.64952 0 −1.78913 + 1.34127i 0 −0.157160 + 2.64108i 0 −0.279099 0
559.13 0 −1.19038 0 2.22369 + 0.234978i 0 −2.11797 + 1.58562i 0 −1.58299 0
559.14 0 −1.19038 0 −2.22369 − 0.234978i 0 2.11797 − 1.58562i 0 −1.58299 0
559.15 0 −1.19038 0 −2.22369 + 0.234978i 0 2.11797 + 1.58562i 0 −1.58299 0
559.16 0 −1.19038 0 2.22369 − 0.234978i 0 −2.11797 − 1.58562i 0 −1.58299 0
559.17 0 −0.857983 0 1.33029 + 1.79731i 0 2.41097 − 1.08959i 0 −2.26387 0
559.18 0 −0.857983 0 −1.33029 − 1.79731i 0 −2.41097 + 1.08959i 0 −2.26387 0
559.19 0 −0.857983 0 −1.33029 + 1.79731i 0 −2.41097 − 1.08959i 0 −2.26387 0
559.20 0 −0.857983 0 1.33029 − 1.79731i 0 2.41097 + 1.08959i 0 −2.26387 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
40.e odd 2 1 inner
56.e even 2 1 inner
280.n even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.n.b 40
4.b odd 2 1 280.2.n.b ✓ 40
5.b even 2 1 inner 1120.2.n.b 40
7.b odd 2 1 inner 1120.2.n.b 40
8.b even 2 1 280.2.n.b ✓ 40
8.d odd 2 1 inner 1120.2.n.b 40
20.d odd 2 1 280.2.n.b ✓ 40
28.d even 2 1 280.2.n.b ✓ 40
35.c odd 2 1 inner 1120.2.n.b 40
40.e odd 2 1 inner 1120.2.n.b 40
40.f even 2 1 280.2.n.b ✓ 40
56.e even 2 1 inner 1120.2.n.b 40
56.h odd 2 1 280.2.n.b ✓ 40
140.c even 2 1 280.2.n.b ✓ 40
280.c odd 2 1 280.2.n.b ✓ 40
280.n even 2 1 inner 1120.2.n.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.n.b ✓ 40 4.b odd 2 1
280.2.n.b ✓ 40 8.b even 2 1
280.2.n.b ✓ 40 20.d odd 2 1
280.2.n.b ✓ 40 28.d even 2 1
280.2.n.b ✓ 40 40.f even 2 1
280.2.n.b ✓ 40 56.h odd 2 1
280.2.n.b ✓ 40 140.c even 2 1
280.2.n.b ✓ 40 280.c odd 2 1
1120.2.n.b 40 1.a even 1 1 trivial
1120.2.n.b 40 5.b even 2 1 inner
1120.2.n.b 40 7.b odd 2 1 inner
1120.2.n.b 40 8.d odd 2 1 inner
1120.2.n.b 40 35.c odd 2 1 inner
1120.2.n.b 40 40.e odd 2 1 inner
1120.2.n.b 40 56.e even 2 1 inner
1120.2.n.b 40 280.n even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 20T_{3}^{8} + 137T_{3}^{6} - 382T_{3}^{4} + 432T_{3}^{2} - 160 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\). Copy content Toggle raw display