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Properties

Label 720.2.br.d
Level $720$
Weight $2$
Character orbit 720.br
Analytic conductor $5.749$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(239,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.br (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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{5} + 2 q^{9} + 6 q^{11} - 9 q^{15} - 18 q^{21} + 3 q^{25} + 12 q^{29} + 18 q^{31} + 30 q^{35} - 6 q^{39} - 12 q^{41} + q^{45} - 12 q^{49} + 36 q^{51} - 6 q^{59} + 3 q^{65} - 12 q^{69} - 96 q^{71} - 9 q^{75} + 18 q^{79} - 14 q^{81} - 24 q^{95} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −1.72238 − 0.182760i 0 −0.849949 + 2.06823i 0 −0.719872 − 1.24685i 0 2.93320 + 0.629566i 0
239.2 0 −1.56217 + 0.748070i 0 1.74305 − 1.40063i 0 2.45766 + 4.25679i 0 1.88078 − 2.33723i 0
239.3 0 −1.48223 − 0.896100i 0 −1.56073 − 1.60129i 0 1.17161 + 2.02929i 0 1.39401 + 2.65645i 0
239.4 0 −0.823761 − 1.52362i 0 2.18453 − 0.477294i 0 −1.34919 − 2.33687i 0 −1.64284 + 2.51020i 0
239.5 0 −0.802147 + 1.53511i 0 2.21199 + 0.327248i 0 −1.45945 − 2.52785i 0 −1.71312 − 2.46277i 0
239.6 0 −0.569196 + 1.63585i 0 −2.22594 + 0.212585i 0 0.344540 + 0.596760i 0 −2.35203 − 1.86224i 0
239.7 0 0.569196 − 1.63585i 0 −0.928866 − 2.03401i 0 −0.344540 − 0.596760i 0 −2.35203 − 1.86224i 0
239.8 0 0.802147 − 1.53511i 0 1.38940 + 1.75202i 0 1.45945 + 2.52785i 0 −1.71312 − 2.46277i 0
239.9 0 0.823761 + 1.52362i 0 0.678918 + 2.13051i 0 1.34919 + 2.33687i 0 −1.64284 + 2.51020i 0
239.10 0 1.48223 + 0.896100i 0 −2.16712 − 0.550987i 0 −1.17161 − 2.02929i 0 1.39401 + 2.65645i 0
239.11 0 1.56217 − 0.748070i 0 −0.341459 + 2.20984i 0 −2.45766 − 4.25679i 0 1.88078 − 2.33723i 0
239.12 0 1.72238 + 0.182760i 0 1.36617 − 1.77019i 0 0.719872 + 1.24685i 0 2.93320 + 0.629566i 0
479.1 0 −1.72238 + 0.182760i 0 −0.849949 − 2.06823i 0 −0.719872 + 1.24685i 0 2.93320 − 0.629566i 0
479.2 0 −1.56217 − 0.748070i 0 1.74305 + 1.40063i 0 2.45766 − 4.25679i 0 1.88078 + 2.33723i 0
479.3 0 −1.48223 + 0.896100i 0 −1.56073 + 1.60129i 0 1.17161 − 2.02929i 0 1.39401 − 2.65645i 0
479.4 0 −0.823761 + 1.52362i 0 2.18453 + 0.477294i 0 −1.34919 + 2.33687i 0 −1.64284 − 2.51020i 0
479.5 0 −0.802147 − 1.53511i 0 2.21199 − 0.327248i 0 −1.45945 + 2.52785i 0 −1.71312 + 2.46277i 0
479.6 0 −0.569196 − 1.63585i 0 −2.22594 − 0.212585i 0 0.344540 − 0.596760i 0 −2.35203 + 1.86224i 0
479.7 0 0.569196 + 1.63585i 0 −0.928866 + 2.03401i 0 −0.344540 + 0.596760i 0 −2.35203 + 1.86224i 0
479.8 0 0.802147 + 1.53511i 0 1.38940 − 1.75202i 0 1.45945 − 2.52785i 0 −1.71312 + 2.46277i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
36.h even 6 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.br.d yes 24
3.b odd 2 1 2160.2.br.c 24
4.b odd 2 1 720.2.br.c ✓ 24
5.b even 2 1 inner 720.2.br.d yes 24
9.c even 3 1 2160.2.br.d 24
9.d odd 6 1 720.2.br.c ✓ 24
12.b even 2 1 2160.2.br.d 24
15.d odd 2 1 2160.2.br.c 24
20.d odd 2 1 720.2.br.c ✓ 24
36.f odd 6 1 2160.2.br.c 24
36.h even 6 1 inner 720.2.br.d yes 24
45.h odd 6 1 720.2.br.c ✓ 24
45.j even 6 1 2160.2.br.d 24
60.h even 2 1 2160.2.br.d 24
180.n even 6 1 inner 720.2.br.d yes 24
180.p odd 6 1 2160.2.br.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.br.c ✓ 24 4.b odd 2 1
720.2.br.c ✓ 24 9.d odd 6 1
720.2.br.c ✓ 24 20.d odd 2 1
720.2.br.c ✓ 24 45.h odd 6 1
720.2.br.d yes 24 1.a even 1 1 trivial
720.2.br.d yes 24 5.b even 2 1 inner
720.2.br.d yes 24 36.h even 6 1 inner
720.2.br.d yes 24 180.n even 6 1 inner
2160.2.br.c 24 3.b odd 2 1
2160.2.br.c 24 15.d odd 2 1
2160.2.br.c 24 36.f odd 6 1
2160.2.br.c 24 180.p odd 6 1
2160.2.br.d 24 9.c even 3 1
2160.2.br.d 24 12.b even 2 1
2160.2.br.d 24 45.j even 6 1
2160.2.br.d 24 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{24} + 48 T_{7}^{22} + 1524 T_{7}^{20} + 26100 T_{7}^{18} + 317331 T_{7}^{16} + 2632176 T_{7}^{14} + \cdots + 65610000 \) Copy content Toggle raw display
\( T_{11}^{12} - 3 T_{11}^{11} + 42 T_{11}^{10} + 9 T_{11}^{9} + 972 T_{11}^{8} - 81 T_{11}^{7} + \cdots + 2916 \) Copy content Toggle raw display