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Properties

Label 1120.2.l.a
Level $1120$
Weight $2$
Character orbit 1120.l
Analytic conductor $8.943$
Analytic rank $0$
Dimension $36$
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] = [1120,2,Mod(1009,1120)]
 
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([0, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.l (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: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 36 q^{9} + 4 q^{25} + 16 q^{31} + 32 q^{39} - 8 q^{41} - 36 q^{49} + 32 q^{55} - 24 q^{65} - 56 q^{71} - 24 q^{79} + 36 q^{81} - 40 q^{89} - 40 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1 0 −3.10393 0 −0.0152300 + 2.23602i 0 1.00000i 0 6.63436 0
1009.2 0 −3.10393 0 −0.0152300 − 2.23602i 0 − 1.00000i 0 6.63436 0
1009.3 0 −2.96656 0 −2.18846 + 0.458941i 0 − 1.00000i 0 5.80046 0
1009.4 0 −2.96656 0 −2.18846 − 0.458941i 0 1.00000i 0 5.80046 0
1009.5 0 −2.55593 0 0.790907 + 2.09152i 0 − 1.00000i 0 3.53279 0
1009.6 0 −2.55593 0 0.790907 − 2.09152i 0 1.00000i 0 3.53279 0
1009.7 0 −2.12140 0 2.02293 + 0.952769i 0 1.00000i 0 1.50036 0
1009.8 0 −2.12140 0 2.02293 − 0.952769i 0 − 1.00000i 0 1.50036 0
1009.9 0 −1.83171 0 −1.41945 + 1.72776i 0 − 1.00000i 0 0.355171 0
1009.10 0 −1.83171 0 −1.41945 − 1.72776i 0 1.00000i 0 0.355171 0
1009.11 0 −1.65329 0 2.21195 + 0.327549i 0 1.00000i 0 −0.266637 0
1009.12 0 −1.65329 0 2.21195 − 0.327549i 0 − 1.00000i 0 −0.266637 0
1009.13 0 −0.460762 0 −2.19557 + 0.423658i 0 1.00000i 0 −2.78770 0
1009.14 0 −0.460762 0 −2.19557 − 0.423658i 0 − 1.00000i 0 −2.78770 0
1009.15 0 −0.359051 0 −0.565870 + 2.16328i 0 1.00000i 0 −2.87108 0
1009.16 0 −0.359051 0 −0.565870 − 2.16328i 0 − 1.00000i 0 −2.87108 0
1009.17 0 −0.319826 0 1.20181 + 1.88564i 0 − 1.00000i 0 −2.89771 0
1009.18 0 −0.319826 0 1.20181 − 1.88564i 0 1.00000i 0 −2.89771 0
1009.19 0 0.319826 0 −1.20181 + 1.88564i 0 1.00000i 0 −2.89771 0
1009.20 0 0.319826 0 −1.20181 − 1.88564i 0 − 1.00000i 0 −2.89771 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1009.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f 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.l.a 36
4.b odd 2 1 280.2.l.a ✓ 36
5.b even 2 1 inner 1120.2.l.a 36
8.b even 2 1 inner 1120.2.l.a 36
8.d odd 2 1 280.2.l.a ✓ 36
20.d odd 2 1 280.2.l.a ✓ 36
40.e odd 2 1 280.2.l.a ✓ 36
40.f even 2 1 inner 1120.2.l.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.l.a ✓ 36 4.b odd 2 1
280.2.l.a ✓ 36 8.d odd 2 1
280.2.l.a ✓ 36 20.d odd 2 1
280.2.l.a ✓ 36 40.e odd 2 1
1120.2.l.a 36 1.a even 1 1 trivial
1120.2.l.a 36 5.b even 2 1 inner
1120.2.l.a 36 8.b even 2 1 inner
1120.2.l.a 36 40.f even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1120, [\chi])\).