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Properties

Label 684.3.bj.a
Level $684$
Weight $3$
Character orbit 684.bj
Analytic conductor $18.638$
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] = [684,3,Mod(125,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.125");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.bj (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: \(18.6376500822\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{7} + 4 q^{13} - 60 q^{19} + 24 q^{25} + 40 q^{31} + 224 q^{37} + 52 q^{43} + 144 q^{49} + 156 q^{55} - 72 q^{61} + 124 q^{67} - 88 q^{73} - 28 q^{79} - 192 q^{85} + 236 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 0 0 0 −6.16613 − 3.56002i 0 −4.91760 0 0 0
125.2 0 0 0 −6.10124 − 3.52255i 0 −10.8174 0 0 0
125.3 0 0 0 −4.70066 − 2.71393i 0 3.44919 0 0 0
125.4 0 0 0 −3.59674 − 2.07658i 0 4.23001 0 0 0
125.5 0 0 0 −3.24084 − 1.87110i 0 10.7108 0 0 0
125.6 0 0 0 −0.847283 − 0.489179i 0 −6.65498 0 0 0
125.7 0 0 0 0.847283 + 0.489179i 0 −6.65498 0 0 0
125.8 0 0 0 3.24084 + 1.87110i 0 10.7108 0 0 0
125.9 0 0 0 3.59674 + 2.07658i 0 4.23001 0 0 0
125.10 0 0 0 4.70066 + 2.71393i 0 3.44919 0 0 0
125.11 0 0 0 6.10124 + 3.52255i 0 −10.8174 0 0 0
125.12 0 0 0 6.16613 + 3.56002i 0 −4.91760 0 0 0
197.1 0 0 0 −6.16613 + 3.56002i 0 −4.91760 0 0 0
197.2 0 0 0 −6.10124 + 3.52255i 0 −10.8174 0 0 0
197.3 0 0 0 −4.70066 + 2.71393i 0 3.44919 0 0 0
197.4 0 0 0 −3.59674 + 2.07658i 0 4.23001 0 0 0
197.5 0 0 0 −3.24084 + 1.87110i 0 10.7108 0 0 0
197.6 0 0 0 −0.847283 + 0.489179i 0 −6.65498 0 0 0
197.7 0 0 0 0.847283 − 0.489179i 0 −6.65498 0 0 0
197.8 0 0 0 3.24084 − 1.87110i 0 10.7108 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.bj.a ✓ 24
3.b odd 2 1 inner 684.3.bj.a ✓ 24
19.c even 3 1 inner 684.3.bj.a ✓ 24
57.h odd 6 1 inner 684.3.bj.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.bj.a ✓ 24 1.a even 1 1 trivial
684.3.bj.a ✓ 24 3.b odd 2 1 inner
684.3.bj.a ✓ 24 19.c even 3 1 inner
684.3.bj.a ✓ 24 57.h odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).