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Properties

Label 1764.2.j.i
Level $1764$
Weight $2$
Character orbit 1764.j
Analytic conductor $14.086$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(589,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.j (of order \(3\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{9} - 4 q^{11} - 28 q^{15} - 8 q^{23} - 12 q^{25} - 32 q^{29} + 24 q^{37} - 40 q^{51} + 32 q^{53} + 52 q^{57} - 36 q^{65} + 12 q^{67} + 48 q^{71} + 12 q^{79} + 16 q^{81} + 12 q^{85} + 48 q^{93} + 32 q^{95} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1 0 −1.72754 + 0.124883i 0 −1.73981 + 3.01343i 0 0 0 2.96881 − 0.431481i 0
589.2 0 −1.69569 + 0.353059i 0 0.469227 − 0.812725i 0 0 0 2.75070 − 1.19736i 0
589.3 0 −1.47192 − 0.912932i 0 1.94623 − 3.37097i 0 0 0 1.33311 + 2.68753i 0
589.4 0 −1.05155 + 1.37631i 0 0.736933 − 1.27640i 0 0 0 −0.788484 − 2.89453i 0
589.5 0 −0.899846 − 1.47996i 0 1.19243 − 2.06535i 0 0 0 −1.38055 + 2.66347i 0
589.6 0 −0.241270 + 1.71516i 0 −0.0111913 + 0.0193839i 0 0 0 −2.88358 − 0.827636i 0
589.7 0 0.241270 − 1.71516i 0 0.0111913 − 0.0193839i 0 0 0 −2.88358 − 0.827636i 0
589.8 0 0.899846 + 1.47996i 0 −1.19243 + 2.06535i 0 0 0 −1.38055 + 2.66347i 0
589.9 0 1.05155 − 1.37631i 0 −0.736933 + 1.27640i 0 0 0 −0.788484 − 2.89453i 0
589.10 0 1.47192 + 0.912932i 0 −1.94623 + 3.37097i 0 0 0 1.33311 + 2.68753i 0
589.11 0 1.69569 − 0.353059i 0 −0.469227 + 0.812725i 0 0 0 2.75070 − 1.19736i 0
589.12 0 1.72754 − 0.124883i 0 1.73981 − 3.01343i 0 0 0 2.96881 − 0.431481i 0
1177.1 0 −1.72754 − 0.124883i 0 −1.73981 − 3.01343i 0 0 0 2.96881 + 0.431481i 0
1177.2 0 −1.69569 − 0.353059i 0 0.469227 + 0.812725i 0 0 0 2.75070 + 1.19736i 0
1177.3 0 −1.47192 + 0.912932i 0 1.94623 + 3.37097i 0 0 0 1.33311 − 2.68753i 0
1177.4 0 −1.05155 − 1.37631i 0 0.736933 + 1.27640i 0 0 0 −0.788484 + 2.89453i 0
1177.5 0 −0.899846 + 1.47996i 0 1.19243 + 2.06535i 0 0 0 −1.38055 − 2.66347i 0
1177.6 0 −0.241270 − 1.71516i 0 −0.0111913 − 0.0193839i 0 0 0 −2.88358 + 0.827636i 0
1177.7 0 0.241270 + 1.71516i 0 0.0111913 + 0.0193839i 0 0 0 −2.88358 + 0.827636i 0
1177.8 0 0.899846 − 1.47996i 0 −1.19243 − 2.06535i 0 0 0 −1.38055 − 2.66347i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
63.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.j.i ✓ 24
3.b odd 2 1 5292.2.j.i 24
7.b odd 2 1 inner 1764.2.j.i ✓ 24
7.c even 3 1 1764.2.i.j 24
7.c even 3 1 1764.2.l.j 24
7.d odd 6 1 1764.2.i.j 24
7.d odd 6 1 1764.2.l.j 24
9.c even 3 1 inner 1764.2.j.i ✓ 24
9.d odd 6 1 5292.2.j.i 24
21.c even 2 1 5292.2.j.i 24
21.g even 6 1 5292.2.i.j 24
21.g even 6 1 5292.2.l.j 24
21.h odd 6 1 5292.2.i.j 24
21.h odd 6 1 5292.2.l.j 24
63.g even 3 1 1764.2.i.j 24
63.h even 3 1 1764.2.l.j 24
63.i even 6 1 5292.2.l.j 24
63.j odd 6 1 5292.2.l.j 24
63.k odd 6 1 1764.2.i.j 24
63.l odd 6 1 inner 1764.2.j.i ✓ 24
63.n odd 6 1 5292.2.i.j 24
63.o even 6 1 5292.2.j.i 24
63.s even 6 1 5292.2.i.j 24
63.t odd 6 1 1764.2.l.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.i.j 24 7.c even 3 1
1764.2.i.j 24 7.d odd 6 1
1764.2.i.j 24 63.g even 3 1
1764.2.i.j 24 63.k odd 6 1
1764.2.j.i ✓ 24 1.a even 1 1 trivial
1764.2.j.i ✓ 24 7.b odd 2 1 inner
1764.2.j.i ✓ 24 9.c even 3 1 inner
1764.2.j.i ✓ 24 63.l odd 6 1 inner
1764.2.l.j 24 7.c even 3 1
1764.2.l.j 24 7.d odd 6 1
1764.2.l.j 24 63.h even 3 1
1764.2.l.j 24 63.t odd 6 1
5292.2.i.j 24 21.g even 6 1
5292.2.i.j 24 21.h odd 6 1
5292.2.i.j 24 63.n odd 6 1
5292.2.i.j 24 63.s even 6 1
5292.2.j.i 24 3.b odd 2 1
5292.2.j.i 24 9.d odd 6 1
5292.2.j.i 24 21.c even 2 1
5292.2.j.i 24 63.o even 6 1
5292.2.l.j 24 21.g even 6 1
5292.2.l.j 24 21.h odd 6 1
5292.2.l.j 24 63.i even 6 1
5292.2.l.j 24 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 36 T_{5}^{22} + 855 T_{5}^{20} + 11596 T_{5}^{18} + 113607 T_{5}^{16} + 669690 T_{5}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display