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Properties

Label 1120.2.w.c
Level $1120$
Weight $2$
Character orbit 1120.w
Analytic conductor $8.943$
Analytic rank $0$
Dimension $72$
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(433,1120)]
 
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, 2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.w (of order \(4\), degree \(2\), 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: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 280)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{7} + 8 q^{15} - 40 q^{23} - 8 q^{25} + 32 q^{57} + 100 q^{63} - 56 q^{65} + 80 q^{71} + 72 q^{81} - 80 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1 0 −1.96562 − 1.96562i 0 0.686018 − 2.12823i 0 1.91879 + 1.82161i 0 4.72730i 0
433.2 0 −1.96562 − 1.96562i 0 0.686018 − 2.12823i 0 −1.82161 − 1.91879i 0 4.72730i 0
433.3 0 −1.92461 − 1.92461i 0 −2.22709 − 0.200181i 0 2.59487 − 0.516406i 0 4.40827i 0
433.4 0 −1.92461 − 1.92461i 0 −2.22709 − 0.200181i 0 0.516406 − 2.59487i 0 4.40827i 0
433.5 0 −1.88603 − 1.88603i 0 1.79374 + 1.33510i 0 −1.30671 − 2.30054i 0 4.11423i 0
433.6 0 −1.88603 − 1.88603i 0 1.79374 + 1.33510i 0 2.30054 + 1.30671i 0 4.11423i 0
433.7 0 −1.10062 − 1.10062i 0 0.557471 + 2.16546i 0 −1.45987 + 2.20653i 0 − 0.577263i 0
433.8 0 −1.10062 − 1.10062i 0 0.557471 + 2.16546i 0 −2.20653 + 1.45987i 0 − 0.577263i 0
433.9 0 −1.09655 − 1.09655i 0 −0.721255 + 2.11655i 0 2.63247 − 0.264726i 0 − 0.595144i 0
433.10 0 −1.09655 − 1.09655i 0 −0.721255 + 2.11655i 0 0.264726 − 2.63247i 0 − 0.595144i 0
433.11 0 −0.851048 − 0.851048i 0 −2.18999 − 0.451591i 0 −2.50752 − 0.844017i 0 − 1.55144i 0
433.12 0 −0.851048 − 0.851048i 0 −2.18999 − 0.451591i 0 0.844017 + 2.50752i 0 − 1.55144i 0
433.13 0 −0.639680 − 0.639680i 0 1.45663 − 1.69653i 0 2.64560 + 0.0280541i 0 − 2.18162i 0
433.14 0 −0.639680 − 0.639680i 0 1.45663 − 1.69653i 0 −0.0280541 − 2.64560i 0 − 2.18162i 0
433.15 0 −0.551796 − 0.551796i 0 −1.17738 − 1.90100i 0 −2.28135 − 1.33994i 0 − 2.39104i 0
433.16 0 −0.551796 − 0.551796i 0 −1.17738 − 1.90100i 0 1.33994 + 2.28135i 0 − 2.39104i 0
433.17 0 −0.152811 − 0.152811i 0 2.05351 − 0.884920i 0 −2.63870 − 0.192988i 0 − 2.95330i 0
433.18 0 −0.152811 − 0.152811i 0 2.05351 − 0.884920i 0 0.192988 + 2.63870i 0 − 2.95330i 0
433.19 0 0.152811 + 0.152811i 0 −2.05351 + 0.884920i 0 0.192988 + 2.63870i 0 − 2.95330i 0
433.20 0 0.152811 + 0.152811i 0 −2.05351 + 0.884920i 0 −2.63870 − 0.192988i 0 − 2.95330i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
56.h odd 2 1 inner
280.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.w.c 72
4.b odd 2 1 280.2.s.c ✓ 72
5.c odd 4 1 inner 1120.2.w.c 72
7.b odd 2 1 inner 1120.2.w.c 72
8.b even 2 1 inner 1120.2.w.c 72
8.d odd 2 1 280.2.s.c ✓ 72
20.e even 4 1 280.2.s.c ✓ 72
28.d even 2 1 280.2.s.c ✓ 72
35.f even 4 1 inner 1120.2.w.c 72
40.i odd 4 1 inner 1120.2.w.c 72
40.k even 4 1 280.2.s.c ✓ 72
56.e even 2 1 280.2.s.c ✓ 72
56.h odd 2 1 inner 1120.2.w.c 72
140.j odd 4 1 280.2.s.c ✓ 72
280.s even 4 1 inner 1120.2.w.c 72
280.y odd 4 1 280.2.s.c ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.s.c ✓ 72 4.b odd 2 1
280.2.s.c ✓ 72 8.d odd 2 1
280.2.s.c ✓ 72 20.e even 4 1
280.2.s.c ✓ 72 28.d even 2 1
280.2.s.c ✓ 72 40.k even 4 1
280.2.s.c ✓ 72 56.e even 2 1
280.2.s.c ✓ 72 140.j odd 4 1
280.2.s.c ✓ 72 280.y odd 4 1
1120.2.w.c 72 1.a even 1 1 trivial
1120.2.w.c 72 5.c odd 4 1 inner
1120.2.w.c 72 7.b odd 2 1 inner
1120.2.w.c 72 8.b even 2 1 inner
1120.2.w.c 72 35.f even 4 1 inner
1120.2.w.c 72 40.i odd 4 1 inner
1120.2.w.c 72 56.h odd 2 1 inner
1120.2.w.c 72 280.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 180 T_{3}^{32} + 11594 T_{3}^{28} + 312340 T_{3}^{24} + 3138761 T_{3}^{20} + 13351184 T_{3}^{16} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\). Copy content Toggle raw display