Properties

Label 1764.2.l.j
Level $1764$
Weight $2$
Character orbit 1764.l
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(949,1764)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.949");
 
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.l (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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{9} + 8 q^{11} - 28 q^{15} + 16 q^{23} + 24 q^{25} - 32 q^{29} - 12 q^{37} + 32 q^{51} - 16 q^{53} + 52 q^{57} - 36 q^{65} + 12 q^{67} + 48 q^{71} + 12 q^{79} - 8 q^{81} + 12 q^{85} + 32 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1 0 −1.73160 + 0.0393104i 0 2.38486 0 0 0 2.99691 0.136140i 0
949.2 0 −1.52658 + 0.818256i 0 3.89246 0 0 0 1.66092 2.49827i 0
949.3 0 −1.36474 1.06653i 0 0.0223826 0 0 0 0.725035 + 2.91107i 0
949.4 0 −0.755620 + 1.55854i 0 −3.47961 0 0 0 −1.85808 2.35532i 0
949.5 0 −0.666148 1.59883i 0 −1.47387 0 0 0 −2.11249 + 2.13011i 0
949.6 0 −0.542084 + 1.64504i 0 0.938454 0 0 0 −2.41229 1.78350i 0
949.7 0 0.542084 1.64504i 0 −0.938454 0 0 0 −2.41229 1.78350i 0
949.8 0 0.666148 + 1.59883i 0 1.47387 0 0 0 −2.11249 + 2.13011i 0
949.9 0 0.755620 1.55854i 0 3.47961 0 0 0 −1.85808 2.35532i 0
949.10 0 1.36474 + 1.06653i 0 −0.0223826 0 0 0 0.725035 + 2.91107i 0
949.11 0 1.52658 0.818256i 0 −3.89246 0 0 0 1.66092 2.49827i 0
949.12 0 1.73160 0.0393104i 0 −2.38486 0 0 0 2.99691 0.136140i 0
961.1 0 −1.73160 0.0393104i 0 2.38486 0 0 0 2.99691 + 0.136140i 0
961.2 0 −1.52658 0.818256i 0 3.89246 0 0 0 1.66092 + 2.49827i 0
961.3 0 −1.36474 + 1.06653i 0 0.0223826 0 0 0 0.725035 2.91107i 0
961.4 0 −0.755620 1.55854i 0 −3.47961 0 0 0 −1.85808 + 2.35532i 0
961.5 0 −0.666148 + 1.59883i 0 −1.47387 0 0 0 −2.11249 2.13011i 0
961.6 0 −0.542084 1.64504i 0 0.938454 0 0 0 −2.41229 + 1.78350i 0
961.7 0 0.542084 + 1.64504i 0 −0.938454 0 0 0 −2.41229 + 1.78350i 0
961.8 0 0.666148 1.59883i 0 1.47387 0 0 0 −2.11249 2.13011i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 36T_{5}^{10} + 441T_{5}^{8} - 2140T_{5}^{6} + 3834T_{5}^{4} - 1998T_{5}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display