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Properties

Label 1764.2.i.j
Level $1764$
Weight $2$
Character orbit 1764.i
Analytic conductor $14.086$
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] = [1764,2,Mod(373,1764)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.373");
 
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.i (of order \(3\), 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: \(14.0856109166\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{9} - 4 q^{11} - 28 q^{15} - 8 q^{23} - 12 q^{25} - 32 q^{29} - 12 q^{37} + 8 q^{51} - 16 q^{53} + 52 q^{57} + 72 q^{65} - 24 q^{67} + 48 q^{71} - 24 q^{79} - 8 q^{81} + 12 q^{85} - 48 q^{93} - 64 q^{95} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1 0 −1.71770 − 0.222512i 0 −0.736933 − 1.27640i 0 0 0 2.90098 + 0.764417i 0
373.2 0 −1.60601 + 0.648636i 0 0.0111913 + 0.0193839i 0 0 0 2.15854 − 2.08343i 0
373.3 0 −1.15360 − 1.29198i 0 −0.469227 − 0.812725i 0 0 0 −0.338409 + 2.98085i 0
373.4 0 −0.971923 − 1.43365i 0 1.73981 + 3.01343i 0 0 0 −1.11073 + 2.78680i 0
373.5 0 −0.831759 + 1.51927i 0 1.19243 + 2.06535i 0 0 0 −1.61636 − 2.52733i 0
373.6 0 −0.0546616 + 1.73119i 0 1.94623 + 3.37097i 0 0 0 −2.99402 − 0.189259i 0
373.7 0 0.0546616 − 1.73119i 0 −1.94623 − 3.37097i 0 0 0 −2.99402 − 0.189259i 0
373.8 0 0.831759 − 1.51927i 0 −1.19243 − 2.06535i 0 0 0 −1.61636 − 2.52733i 0
373.9 0 0.971923 + 1.43365i 0 −1.73981 − 3.01343i 0 0 0 −1.11073 + 2.78680i 0
373.10 0 1.15360 + 1.29198i 0 0.469227 + 0.812725i 0 0 0 −0.338409 + 2.98085i 0
373.11 0 1.60601 − 0.648636i 0 −0.0111913 − 0.0193839i 0 0 0 2.15854 − 2.08343i 0
373.12 0 1.71770 + 0.222512i 0 0.736933 + 1.27640i 0 0 0 2.90098 + 0.764417i 0
1537.1 0 −1.71770 + 0.222512i 0 −0.736933 + 1.27640i 0 0 0 2.90098 − 0.764417i 0
1537.2 0 −1.60601 − 0.648636i 0 0.0111913 − 0.0193839i 0 0 0 2.15854 + 2.08343i 0
1537.3 0 −1.15360 + 1.29198i 0 −0.469227 + 0.812725i 0 0 0 −0.338409 − 2.98085i 0
1537.4 0 −0.971923 + 1.43365i 0 1.73981 − 3.01343i 0 0 0 −1.11073 − 2.78680i 0
1537.5 0 −0.831759 − 1.51927i 0 1.19243 − 2.06535i 0 0 0 −1.61636 + 2.52733i 0
1537.6 0 −0.0546616 − 1.73119i 0 1.94623 − 3.37097i 0 0 0 −2.99402 + 0.189259i 0
1537.7 0 0.0546616 + 1.73119i 0 −1.94623 + 3.37097i 0 0 0 −2.99402 + 0.189259i 0
1537.8 0 0.831759 + 1.51927i 0 −1.19243 + 2.06535i 0 0 0 −1.61636 + 2.52733i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.12
Significant digits:
Format:

Inner twists

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