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Properties

Label 2160.2.br.c
Level $2160$
Weight $2$
Character orbit 2160.br
Analytic conductor $17.248$
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] = [2160,2,Mod(719,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, 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("2160.719");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.br (of order \(6\), 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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 720)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{5} - 6 q^{11} + 3 q^{25} - 12 q^{29} + 18 q^{31} - 30 q^{35} + 12 q^{41} - 12 q^{49} + 6 q^{59} - 3 q^{65} + 96 q^{71} + 18 q^{79} + 24 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1 0 0 0 −2.21199 − 0.327248i 0 −1.45945 − 2.52785i 0 0 0
719.2 0 0 0 −2.18453 + 0.477294i 0 −1.34919 − 2.33687i 0 0 0
719.3 0 0 0 −1.74305 + 1.40063i 0 2.45766 + 4.25679i 0 0 0
719.4 0 0 0 −1.38940 − 1.75202i 0 1.45945 + 2.52785i 0 0 0
719.5 0 0 0 −1.36617 + 1.77019i 0 0.719872 + 1.24685i 0 0 0
719.6 0 0 0 −0.678918 − 2.13051i 0 1.34919 + 2.33687i 0 0 0
719.7 0 0 0 0.341459 − 2.20984i 0 −2.45766 − 4.25679i 0 0 0
719.8 0 0 0 0.849949 − 2.06823i 0 −0.719872 − 1.24685i 0 0 0
719.9 0 0 0 0.928866 + 2.03401i 0 −0.344540 − 0.596760i 0 0 0
719.10 0 0 0 1.56073 + 1.60129i 0 1.17161 + 2.02929i 0 0 0
719.11 0 0 0 2.16712 + 0.550987i 0 −1.17161 − 2.02929i 0 0 0
719.12 0 0 0 2.22594 − 0.212585i 0 0.344540 + 0.596760i 0 0 0
1439.1 0 0 0 −2.21199 + 0.327248i 0 −1.45945 + 2.52785i 0 0 0
1439.2 0 0 0 −2.18453 − 0.477294i 0 −1.34919 + 2.33687i 0 0 0
1439.3 0 0 0 −1.74305 − 1.40063i 0 2.45766 − 4.25679i 0 0 0
1439.4 0 0 0 −1.38940 + 1.75202i 0 1.45945 − 2.52785i 0 0 0
1439.5 0 0 0 −1.36617 − 1.77019i 0 0.719872 − 1.24685i 0 0 0
1439.6 0 0 0 −0.678918 + 2.13051i 0 1.34919 − 2.33687i 0 0 0
1439.7 0 0 0 0.341459 + 2.20984i 0 −2.45766 + 4.25679i 0 0 0
1439.8 0 0 0 0.849949 + 2.06823i 0 −0.719872 + 1.24685i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.12
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 2160.2.br.c 24
3.b odd 2 1 720.2.br.d yes 24
4.b odd 2 1 2160.2.br.d 24
5.b even 2 1 inner 2160.2.br.c 24
9.c even 3 1 720.2.br.c ✓ 24
9.d odd 6 1 2160.2.br.d 24
12.b even 2 1 720.2.br.c ✓ 24
15.d odd 2 1 720.2.br.d yes 24
20.d odd 2 1 2160.2.br.d 24
36.f odd 6 1 720.2.br.d yes 24
36.h even 6 1 inner 2160.2.br.c 24
45.h odd 6 1 2160.2.br.d 24
45.j even 6 1 720.2.br.c ✓ 24
60.h even 2 1 720.2.br.c ✓ 24
180.n even 6 1 inner 2160.2.br.c 24
180.p odd 6 1 720.2.br.d yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.br.c ✓ 24 9.c even 3 1
720.2.br.c ✓ 24 12.b even 2 1
720.2.br.c ✓ 24 45.j even 6 1
720.2.br.c ✓ 24 60.h even 2 1
720.2.br.d yes 24 3.b odd 2 1
720.2.br.d yes 24 15.d odd 2 1
720.2.br.d yes 24 36.f odd 6 1
720.2.br.d yes 24 180.p odd 6 1
2160.2.br.c 24 1.a even 1 1 trivial
2160.2.br.c 24 5.b even 2 1 inner
2160.2.br.c 24 36.h even 6 1 inner
2160.2.br.c 24 180.n even 6 1 inner
2160.2.br.d 24 4.b odd 2 1
2160.2.br.d 24 9.d odd 6 1
2160.2.br.d 24 20.d odd 2 1
2160.2.br.d 24 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{24} + 48 T_{7}^{22} + 1524 T_{7}^{20} + 26100 T_{7}^{18} + 317331 T_{7}^{16} + 2632176 T_{7}^{14} + \cdots + 65610000 \) 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