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Properties

Label 480.2.o.a
Level $480$
Weight $2$
Character orbit 480.o
Analytic conductor $3.833$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(479,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.479");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.o (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{21} + 8 q^{25} - 24 q^{45} + 40 q^{49} + 32 q^{61} - 56 q^{69} + 8 q^{81} - 32 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
479.1 0 −1.69977 − 0.332823i 0 −1.59038 − 1.57184i 0 −1.60011 0 2.77846 + 1.13145i 0
479.2 0 −1.69977 − 0.332823i 0 1.59038 − 1.57184i 0 −1.60011 0 2.77846 + 1.13145i 0
479.3 0 −1.69977 + 0.332823i 0 −1.59038 + 1.57184i 0 −1.60011 0 2.77846 − 1.13145i 0
479.4 0 −1.69977 + 0.332823i 0 1.59038 + 1.57184i 0 −1.60011 0 2.77846 − 1.13145i 0
479.5 0 −1.16422 − 1.28241i 0 −2.19399 + 0.431733i 0 4.22289 0 −0.289169 + 2.98603i 0
479.6 0 −1.16422 − 1.28241i 0 2.19399 + 0.431733i 0 4.22289 0 −0.289169 + 2.98603i 0
479.7 0 −1.16422 + 1.28241i 0 −2.19399 − 0.431733i 0 4.22289 0 −0.289169 − 2.98603i 0
479.8 0 −1.16422 + 1.28241i 0 2.19399 − 0.431733i 0 4.22289 0 −0.289169 − 2.98603i 0
479.9 0 −0.505327 − 1.65670i 0 −0.810603 + 2.08397i 0 −2.36789 0 −2.48929 + 1.67435i 0
479.10 0 −0.505327 − 1.65670i 0 0.810603 + 2.08397i 0 −2.36789 0 −2.48929 + 1.67435i 0
479.11 0 −0.505327 + 1.65670i 0 −0.810603 − 2.08397i 0 −2.36789 0 −2.48929 − 1.67435i 0
479.12 0 −0.505327 + 1.65670i 0 0.810603 − 2.08397i 0 −2.36789 0 −2.48929 − 1.67435i 0
479.13 0 0.505327 − 1.65670i 0 −0.810603 − 2.08397i 0 2.36789 0 −2.48929 − 1.67435i 0
479.14 0 0.505327 − 1.65670i 0 0.810603 − 2.08397i 0 2.36789 0 −2.48929 − 1.67435i 0
479.15 0 0.505327 + 1.65670i 0 −0.810603 + 2.08397i 0 2.36789 0 −2.48929 + 1.67435i 0
479.16 0 0.505327 + 1.65670i 0 0.810603 + 2.08397i 0 2.36789 0 −2.48929 + 1.67435i 0
479.17 0 1.16422 − 1.28241i 0 −2.19399 − 0.431733i 0 −4.22289 0 −0.289169 − 2.98603i 0
479.18 0 1.16422 − 1.28241i 0 2.19399 − 0.431733i 0 −4.22289 0 −0.289169 − 2.98603i 0
479.19 0 1.16422 + 1.28241i 0 −2.19399 + 0.431733i 0 −4.22289 0 −0.289169 + 2.98603i 0
479.20 0 1.16422 + 1.28241i 0 2.19399 + 0.431733i 0 −4.22289 0 −0.289169 + 2.98603i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 479.24
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.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.o.a ✓ 24
3.b odd 2 1 inner 480.2.o.a ✓ 24
4.b odd 2 1 inner 480.2.o.a ✓ 24
5.b even 2 1 inner 480.2.o.a ✓ 24
5.c odd 4 2 2400.2.h.h 24
8.b even 2 1 960.2.o.e 24
8.d odd 2 1 960.2.o.e 24
12.b even 2 1 inner 480.2.o.a ✓ 24
15.d odd 2 1 inner 480.2.o.a ✓ 24
15.e even 4 2 2400.2.h.h 24
20.d odd 2 1 inner 480.2.o.a ✓ 24
20.e even 4 2 2400.2.h.h 24
24.f even 2 1 960.2.o.e 24
24.h odd 2 1 960.2.o.e 24
40.e odd 2 1 960.2.o.e 24
40.f even 2 1 960.2.o.e 24
60.h even 2 1 inner 480.2.o.a ✓ 24
60.l odd 4 2 2400.2.h.h 24
120.i odd 2 1 960.2.o.e 24
120.m even 2 1 960.2.o.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.o.a ✓ 24 1.a even 1 1 trivial
480.2.o.a ✓ 24 3.b odd 2 1 inner
480.2.o.a ✓ 24 4.b odd 2 1 inner
480.2.o.a ✓ 24 5.b even 2 1 inner
480.2.o.a ✓ 24 12.b even 2 1 inner
480.2.o.a ✓ 24 15.d odd 2 1 inner
480.2.o.a ✓ 24 20.d odd 2 1 inner
480.2.o.a ✓ 24 60.h even 2 1 inner
960.2.o.e 24 8.b even 2 1
960.2.o.e 24 8.d odd 2 1
960.2.o.e 24 24.f even 2 1
960.2.o.e 24 24.h odd 2 1
960.2.o.e 24 40.e odd 2 1
960.2.o.e 24 40.f even 2 1
960.2.o.e 24 120.i odd 2 1
960.2.o.e 24 120.m even 2 1
2400.2.h.h 24 5.c odd 4 2
2400.2.h.h 24 15.e even 4 2
2400.2.h.h 24 20.e even 4 2
2400.2.h.h 24 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(480, [\chi])\).