↑

Properties

Label 480.2.v.d
Level $480$
Weight $2$
Character orbit 480.v
Analytic conductor $3.833$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(257,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.v (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: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{13} + 16 q^{21} + 16 q^{25} + 24 q^{33} + 32 q^{37} + 32 q^{45} + 16 q^{57} - 48 q^{61} - 56 q^{73} - 56 q^{81} - 64 q^{85} - 96 q^{93} - 88 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1 0 −1.68734 − 0.391016i 0 1.95188 + 1.09094i 0 0.246506 + 0.246506i 0 2.69421 + 1.31955i 0
257.2 0 −1.50010 − 0.865855i 0 0.167782 − 2.22976i 0 2.51912 + 2.51912i 0 1.50059 + 2.59774i 0
257.3 0 −1.36007 + 1.07247i 0 −2.15917 + 0.581371i 0 −1.61036 − 1.61036i 0 0.699602 − 2.91729i 0
257.4 0 −1.07247 + 1.36007i 0 2.15917 − 0.581371i 0 −1.61036 − 1.61036i 0 −0.699602 − 2.91729i 0
257.5 0 −0.865855 − 1.50010i 0 −0.167782 + 2.22976i 0 −2.51912 − 2.51912i 0 −1.50059 + 2.59774i 0
257.6 0 −0.391016 − 1.68734i 0 −1.95188 − 1.09094i 0 −0.246506 − 0.246506i 0 −2.69421 + 1.31955i 0
257.7 0 0.391016 + 1.68734i 0 −1.95188 − 1.09094i 0 0.246506 + 0.246506i 0 −2.69421 + 1.31955i 0
257.8 0 0.865855 + 1.50010i 0 −0.167782 + 2.22976i 0 2.51912 + 2.51912i 0 −1.50059 + 2.59774i 0
257.9 0 1.07247 − 1.36007i 0 2.15917 − 0.581371i 0 1.61036 + 1.61036i 0 −0.699602 − 2.91729i 0
257.10 0 1.36007 − 1.07247i 0 −2.15917 + 0.581371i 0 1.61036 + 1.61036i 0 0.699602 − 2.91729i 0
257.11 0 1.50010 + 0.865855i 0 0.167782 − 2.22976i 0 −2.51912 − 2.51912i 0 1.50059 + 2.59774i 0
257.12 0 1.68734 + 0.391016i 0 1.95188 + 1.09094i 0 −0.246506 − 0.246506i 0 2.69421 + 1.31955i 0
353.1 0 −1.68734 + 0.391016i 0 1.95188 − 1.09094i 0 0.246506 − 0.246506i 0 2.69421 − 1.31955i 0
353.2 0 −1.50010 + 0.865855i 0 0.167782 + 2.22976i 0 2.51912 − 2.51912i 0 1.50059 − 2.59774i 0
353.3 0 −1.36007 − 1.07247i 0 −2.15917 − 0.581371i 0 −1.61036 + 1.61036i 0 0.699602 + 2.91729i 0
353.4 0 −1.07247 − 1.36007i 0 2.15917 + 0.581371i 0 −1.61036 + 1.61036i 0 −0.699602 + 2.91729i 0
353.5 0 −0.865855 + 1.50010i 0 −0.167782 − 2.22976i 0 −2.51912 + 2.51912i 0 −1.50059 − 2.59774i 0
353.6 0 −0.391016 + 1.68734i 0 −1.95188 + 1.09094i 0 −0.246506 + 0.246506i 0 −2.69421 − 1.31955i 0
353.7 0 0.391016 − 1.68734i 0 −1.95188 + 1.09094i 0 0.246506 − 0.246506i 0 −2.69421 − 1.31955i 0
353.8 0 0.865855 − 1.50010i 0 −0.167782 − 2.22976i 0 2.51912 − 2.51912i 0 −1.50059 − 2.59774i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.v.d ✓ 24
3.b odd 2 1 inner 480.2.v.d ✓ 24
4.b odd 2 1 inner 480.2.v.d ✓ 24
5.c odd 4 1 inner 480.2.v.d ✓ 24
8.b even 2 1 960.2.v.n 24
8.d odd 2 1 960.2.v.n 24
12.b even 2 1 inner 480.2.v.d ✓ 24
15.e even 4 1 inner 480.2.v.d ✓ 24
20.e even 4 1 inner 480.2.v.d ✓ 24
24.f even 2 1 960.2.v.n 24
24.h odd 2 1 960.2.v.n 24
40.i odd 4 1 960.2.v.n 24
40.k even 4 1 960.2.v.n 24
60.l odd 4 1 inner 480.2.v.d ✓ 24
120.q odd 4 1 960.2.v.n 24
120.w even 4 1 960.2.v.n 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.v.d ✓ 24 1.a even 1 1 trivial
480.2.v.d ✓ 24 3.b odd 2 1 inner
480.2.v.d ✓ 24 4.b odd 2 1 inner
480.2.v.d ✓ 24 5.c odd 4 1 inner
480.2.v.d ✓ 24 12.b even 2 1 inner
480.2.v.d ✓ 24 15.e even 4 1 inner
480.2.v.d ✓ 24 20.e even 4 1 inner
480.2.v.d ✓ 24 60.l odd 4 1 inner
960.2.v.n 24 8.b even 2 1
960.2.v.n 24 8.d odd 2 1
960.2.v.n 24 24.f even 2 1
960.2.v.n 24 24.h odd 2 1
960.2.v.n 24 40.i odd 4 1
960.2.v.n 24 40.k even 4 1
960.2.v.n 24 120.q odd 4 1
960.2.v.n 24 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 188T_{7}^{8} + 4336T_{7}^{4} + 64 \) acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\). Copy content Toggle raw display