Properties

Label 1120.2.bw.h
Level $1120$
Weight $2$
Character orbit 1120.bw
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(289,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1120.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bw (of order \(6\), 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: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{5} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{5} + 40 q^{9} - 4 q^{21} + 24 q^{25} + 24 q^{29} - 40 q^{41} - 8 q^{45} - 52 q^{49} - 44 q^{61} + 16 q^{65} - 56 q^{69} - 4 q^{81} - 72 q^{85} - 76 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 −2.73849 + 1.58107i 0 2.12487 + 0.696361i 0 −2.30440 1.29991i 0 3.49953 6.06137i 0
289.2 0 −2.73849 + 1.58107i 0 −1.66550 1.49201i 0 −2.30440 1.29991i 0 3.49953 6.06137i 0
289.3 0 −2.25275 + 1.30063i 0 −1.24022 + 1.86061i 0 2.63098 0.279192i 0 1.88326 3.26190i 0
289.4 0 −2.25275 + 1.30063i 0 −0.991222 + 2.00437i 0 2.63098 0.279192i 0 1.88326 3.26190i 0
289.5 0 −2.04145 + 1.17863i 0 −0.800879 2.08772i 0 0.745750 + 2.53848i 0 1.27835 2.21416i 0
289.6 0 −2.04145 + 1.17863i 0 2.20846 0.350280i 0 0.745750 + 2.53848i 0 1.27835 2.21416i 0
289.7 0 −1.21235 + 0.699953i 0 −1.99091 1.01797i 0 0.374974 2.61904i 0 −0.520132 + 0.900895i 0
289.8 0 −1.21235 + 0.699953i 0 1.87704 + 1.21520i 0 0.374974 2.61904i 0 −0.520132 + 0.900895i 0
289.9 0 −0.733819 + 0.423671i 0 −2.17978 + 0.498541i 0 −1.14933 + 2.38308i 0 −1.14101 + 1.97628i 0
289.10 0 −0.733819 + 0.423671i 0 0.658143 + 2.13702i 0 −1.14933 + 2.38308i 0 −1.14101 + 1.97628i 0
289.11 0 0.733819 0.423671i 0 0.658143 + 2.13702i 0 1.14933 2.38308i 0 −1.14101 + 1.97628i 0
289.12 0 0.733819 0.423671i 0 −2.17978 + 0.498541i 0 1.14933 2.38308i 0 −1.14101 + 1.97628i 0
289.13 0 1.21235 0.699953i 0 1.87704 + 1.21520i 0 −0.374974 + 2.61904i 0 −0.520132 + 0.900895i 0
289.14 0 1.21235 0.699953i 0 −1.99091 1.01797i 0 −0.374974 + 2.61904i 0 −0.520132 + 0.900895i 0
289.15 0 2.04145 1.17863i 0 2.20846 0.350280i 0 −0.745750 2.53848i 0 1.27835 2.21416i 0
289.16 0 2.04145 1.17863i 0 −0.800879 2.08772i 0 −0.745750 2.53848i 0 1.27835 2.21416i 0
289.17 0 2.25275 1.30063i 0 −0.991222 + 2.00437i 0 −2.63098 + 0.279192i 0 1.88326 3.26190i 0
289.18 0 2.25275 1.30063i 0 −1.24022 + 1.86061i 0 −2.63098 + 0.279192i 0 1.88326 3.26190i 0
289.19 0 2.73849 1.58107i 0 −1.66550 1.49201i 0 2.30440 + 1.29991i 0 3.49953 6.06137i 0
289.20 0 2.73849 1.58107i 0 2.12487 + 0.696361i 0 2.30440 + 1.29991i 0 3.49953 6.06137i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.20
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.c even 3 1 inner
20.d odd 2 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.bw.h 40
4.b odd 2 1 inner 1120.2.bw.h 40
5.b even 2 1 inner 1120.2.bw.h 40
7.c even 3 1 inner 1120.2.bw.h 40
20.d odd 2 1 inner 1120.2.bw.h 40
28.g odd 6 1 inner 1120.2.bw.h 40
35.j even 6 1 inner 1120.2.bw.h 40
140.p odd 6 1 inner 1120.2.bw.h 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.bw.h 40 1.a even 1 1 trivial
1120.2.bw.h 40 4.b odd 2 1 inner
1120.2.bw.h 40 5.b even 2 1 inner
1120.2.bw.h 40 7.c even 3 1 inner
1120.2.bw.h 40 20.d odd 2 1 inner
1120.2.bw.h 40 28.g odd 6 1 inner
1120.2.bw.h 40 35.j even 6 1 inner
1120.2.bw.h 40 140.p odd 6 1 inner

Hecke kernels

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

\( T_{3}^{20} - 25 T_{3}^{18} + 403 T_{3}^{16} - 3874 T_{3}^{14} + 27101 T_{3}^{12} - 124915 T_{3}^{10} + 415293 T_{3}^{8} - 798378 T_{3}^{6} + 1076987 T_{3}^{4} - 652257 T_{3}^{2} + 279841 \) Copy content Toggle raw display
\( T_{11}^{20} + 66 T_{11}^{18} + 2927 T_{11}^{16} + 72554 T_{11}^{14} + 1308857 T_{11}^{12} + 13557888 T_{11}^{10} + 96520448 T_{11}^{8} + 152625152 T_{11}^{6} + 183566336 T_{11}^{4} + 61865984 T_{11}^{2} + \cdots + 16777216 \) Copy content Toggle raw display