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Properties

Label 1120.2.bj.d
Level $1120$
Weight $2$
Character orbit 1120.bj
Analytic conductor $8.943$
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] = [1120,2,Mod(97,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bj (of order \(4\), degree \(2\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{25} - 16 q^{37} + 32 q^{53} - 64 q^{57} - 32 q^{65} - 16 q^{77} - 144 q^{81} + 32 q^{85} - 112 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −2.34594 − 2.34594i 0 −2.08821 − 0.799602i 0 −2.54394 − 0.726895i 0 8.00687i 0
97.2 0 −2.34594 − 2.34594i 0 2.08821 + 0.799602i 0 0.726895 + 2.54394i 0 8.00687i 0
97.3 0 −1.90095 − 1.90095i 0 2.07028 − 0.844958i 0 0.608743 − 2.57477i 0 4.22723i 0
97.4 0 −1.90095 − 1.90095i 0 −2.07028 + 0.844958i 0 2.57477 − 0.608743i 0 4.22723i 0
97.5 0 −1.46608 − 1.46608i 0 0.604967 − 2.15268i 0 −0.975674 + 2.45928i 0 1.29876i 0
97.6 0 −1.46608 − 1.46608i 0 −0.604967 + 2.15268i 0 −2.45928 + 0.975674i 0 1.29876i 0
97.7 0 −1.02680 − 1.02680i 0 −1.22369 − 1.87152i 0 1.76137 − 1.97423i 0 − 0.891351i 0
97.8 0 −1.02680 − 1.02680i 0 1.22369 + 1.87152i 0 1.97423 − 1.76137i 0 − 0.891351i 0
97.9 0 −0.805820 − 0.805820i 0 1.78958 − 1.34067i 0 −2.38028 − 1.15511i 0 − 1.70131i 0
97.10 0 −0.805820 − 0.805820i 0 −1.78958 + 1.34067i 0 1.15511 + 2.38028i 0 − 1.70131i 0
97.11 0 −0.172916 − 0.172916i 0 1.51239 + 1.64702i 0 −0.321965 − 2.62609i 0 − 2.94020i 0
97.12 0 −0.172916 − 0.172916i 0 −1.51239 − 1.64702i 0 2.62609 + 0.321965i 0 − 2.94020i 0
97.13 0 0.172916 + 0.172916i 0 −1.51239 − 1.64702i 0 −2.62609 − 0.321965i 0 − 2.94020i 0
97.14 0 0.172916 + 0.172916i 0 1.51239 + 1.64702i 0 0.321965 + 2.62609i 0 − 2.94020i 0
97.15 0 0.805820 + 0.805820i 0 −1.78958 + 1.34067i 0 −1.15511 − 2.38028i 0 − 1.70131i 0
97.16 0 0.805820 + 0.805820i 0 1.78958 − 1.34067i 0 2.38028 + 1.15511i 0 − 1.70131i 0
97.17 0 1.02680 + 1.02680i 0 1.22369 + 1.87152i 0 −1.97423 + 1.76137i 0 − 0.891351i 0
97.18 0 1.02680 + 1.02680i 0 −1.22369 − 1.87152i 0 −1.76137 + 1.97423i 0 − 0.891351i 0
97.19 0 1.46608 + 1.46608i 0 −0.604967 + 2.15268i 0 2.45928 − 0.975674i 0 1.29876i 0
97.20 0 1.46608 + 1.46608i 0 0.604967 − 2.15268i 0 0.975674 − 2.45928i 0 1.29876i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.bj.d ✓ 48
4.b odd 2 1 inner 1120.2.bj.d ✓ 48
5.c odd 4 1 inner 1120.2.bj.d ✓ 48
7.b odd 2 1 inner 1120.2.bj.d ✓ 48
20.e even 4 1 inner 1120.2.bj.d ✓ 48
28.d even 2 1 inner 1120.2.bj.d ✓ 48
35.f even 4 1 inner 1120.2.bj.d ✓ 48
140.j odd 4 1 inner 1120.2.bj.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.bj.d ✓ 48 1.a even 1 1 trivial
1120.2.bj.d ✓ 48 4.b odd 2 1 inner
1120.2.bj.d ✓ 48 5.c odd 4 1 inner
1120.2.bj.d ✓ 48 7.b odd 2 1 inner
1120.2.bj.d ✓ 48 20.e even 4 1 inner
1120.2.bj.d ✓ 48 28.d even 2 1 inner
1120.2.bj.d ✓ 48 35.f even 4 1 inner
1120.2.bj.d ✓ 48 140.j odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\):

\( T_{3}^{24} + 198T_{3}^{20} + 10717T_{3}^{16} + 176876T_{3}^{12} + 789300T_{3}^{8} + 879776T_{3}^{4} + 3136 \) Copy content Toggle raw display
\( T_{13}^{24} + 1382 T_{13}^{20} + 659981 T_{13}^{16} + 125672092 T_{13}^{12} + 7768988468 T_{13}^{8} + \cdots + 12834170944 \) Copy content Toggle raw display