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Properties

Label 480.2.bi.c
Level $480$
Weight $2$
Character orbit 480.bi
Analytic conductor $3.833$
Analytic rank $0$
Dimension $32$
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(17,480)]
 
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, 2, 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("480.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.bi (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: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 28 q^{15} - 8 q^{25} + 32 q^{31} + 8 q^{33} - 8 q^{55} - 16 q^{57} - 48 q^{63} - 64 q^{73} + 48 q^{81} + 8 q^{87} + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −1.72368 + 0.170116i 0 −1.36575 + 1.77052i 0 −2.06963 − 2.06963i 0 2.94212 − 0.586449i 0
17.2 0 −1.68122 + 0.416519i 0 −1.62104 − 1.54020i 0 0.361989 + 0.361989i 0 2.65302 − 1.40052i 0
17.3 0 −1.59834 − 0.667305i 0 0.143028 − 2.23149i 0 −0.582772 − 0.582772i 0 2.10941 + 2.13317i 0
17.4 0 −1.40091 − 1.01856i 0 2.23305 − 0.116202i 0 2.29041 + 2.29041i 0 0.925085 + 2.85381i 0
17.5 0 −1.01856 − 1.40091i 0 −2.23305 + 0.116202i 0 2.29041 + 2.29041i 0 −0.925085 + 2.85381i 0
17.6 0 −0.667305 − 1.59834i 0 −0.143028 + 2.23149i 0 −0.582772 − 0.582772i 0 −2.10941 + 2.13317i 0
17.7 0 −0.416519 + 1.68122i 0 −1.62104 − 1.54020i 0 0.361989 + 0.361989i 0 −2.65302 − 1.40052i 0
17.8 0 −0.170116 + 1.72368i 0 −1.36575 + 1.77052i 0 −2.06963 − 2.06963i 0 −2.94212 − 0.586449i 0
17.9 0 0.170116 − 1.72368i 0 1.36575 − 1.77052i 0 −2.06963 − 2.06963i 0 −2.94212 − 0.586449i 0
17.10 0 0.416519 − 1.68122i 0 1.62104 + 1.54020i 0 0.361989 + 0.361989i 0 −2.65302 − 1.40052i 0
17.11 0 0.667305 + 1.59834i 0 0.143028 − 2.23149i 0 −0.582772 − 0.582772i 0 −2.10941 + 2.13317i 0
17.12 0 1.01856 + 1.40091i 0 2.23305 − 0.116202i 0 2.29041 + 2.29041i 0 −0.925085 + 2.85381i 0
17.13 0 1.40091 + 1.01856i 0 −2.23305 + 0.116202i 0 2.29041 + 2.29041i 0 0.925085 + 2.85381i 0
17.14 0 1.59834 + 0.667305i 0 −0.143028 + 2.23149i 0 −0.582772 − 0.582772i 0 2.10941 + 2.13317i 0
17.15 0 1.68122 − 0.416519i 0 1.62104 + 1.54020i 0 0.361989 + 0.361989i 0 2.65302 − 1.40052i 0
17.16 0 1.72368 − 0.170116i 0 1.36575 − 1.77052i 0 −2.06963 − 2.06963i 0 2.94212 − 0.586449i 0
113.1 0 −1.72368 − 0.170116i 0 −1.36575 − 1.77052i 0 −2.06963 + 2.06963i 0 2.94212 + 0.586449i 0
113.2 0 −1.68122 − 0.416519i 0 −1.62104 + 1.54020i 0 0.361989 − 0.361989i 0 2.65302 + 1.40052i 0
113.3 0 −1.59834 + 0.667305i 0 0.143028 + 2.23149i 0 −0.582772 + 0.582772i 0 2.10941 − 2.13317i 0
113.4 0 −1.40091 + 1.01856i 0 2.23305 + 0.116202i 0 2.29041 − 2.29041i 0 0.925085 − 2.85381i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
24.h odd 2 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.bi.c 32
3.b odd 2 1 inner 480.2.bi.c 32
4.b odd 2 1 120.2.w.c ✓ 32
5.c odd 4 1 inner 480.2.bi.c 32
8.b even 2 1 inner 480.2.bi.c 32
8.d odd 2 1 120.2.w.c ✓ 32
12.b even 2 1 120.2.w.c ✓ 32
15.e even 4 1 inner 480.2.bi.c 32
20.d odd 2 1 600.2.w.j 32
20.e even 4 1 120.2.w.c ✓ 32
20.e even 4 1 600.2.w.j 32
24.f even 2 1 120.2.w.c ✓ 32
24.h odd 2 1 inner 480.2.bi.c 32
40.e odd 2 1 600.2.w.j 32
40.i odd 4 1 inner 480.2.bi.c 32
40.k even 4 1 120.2.w.c ✓ 32
40.k even 4 1 600.2.w.j 32
60.h even 2 1 600.2.w.j 32
60.l odd 4 1 120.2.w.c ✓ 32
60.l odd 4 1 600.2.w.j 32
120.m even 2 1 600.2.w.j 32
120.q odd 4 1 120.2.w.c ✓ 32
120.q odd 4 1 600.2.w.j 32
120.w even 4 1 inner 480.2.bi.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.w.c ✓ 32 4.b odd 2 1
120.2.w.c ✓ 32 8.d odd 2 1
120.2.w.c ✓ 32 12.b even 2 1
120.2.w.c ✓ 32 20.e even 4 1
120.2.w.c ✓ 32 24.f even 2 1
120.2.w.c ✓ 32 40.k even 4 1
120.2.w.c ✓ 32 60.l odd 4 1
120.2.w.c ✓ 32 120.q odd 4 1
480.2.bi.c 32 1.a even 1 1 trivial
480.2.bi.c 32 3.b odd 2 1 inner
480.2.bi.c 32 5.c odd 4 1 inner
480.2.bi.c 32 8.b even 2 1 inner
480.2.bi.c 32 15.e even 4 1 inner
480.2.bi.c 32 24.h odd 2 1 inner
480.2.bi.c 32 40.i odd 4 1 inner
480.2.bi.c 32 120.w even 4 1 inner
600.2.w.j 32 20.d odd 2 1
600.2.w.j 32 20.e even 4 1
600.2.w.j 32 40.e odd 2 1
600.2.w.j 32 40.k even 4 1
600.2.w.j 32 60.h even 2 1
600.2.w.j 32 60.l odd 4 1
600.2.w.j 32 120.m even 2 1
600.2.w.j 32 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\):

\( T_{7}^{8} + 4T_{7}^{5} + 92T_{7}^{4} + 40T_{7}^{3} + 8T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 26T_{11}^{6} + 208T_{11}^{4} - 544T_{11}^{2} + 128 \) Copy content Toggle raw display