Properties

Label 1728.3.q.l
Level $1728$
Weight $3$
Character orbit 1728.q
Analytic conductor $47.085$
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] = [1728,3,Mod(449,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 288)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1 0 0 0 −7.38813 4.26554i 0 −1.36942 2.37191i 0 0 0
449.2 0 0 0 −7.38813 4.26554i 0 1.36942 + 2.37191i 0 0 0
449.3 0 0 0 −4.45884 2.57431i 0 −1.35076 2.33958i 0 0 0
449.4 0 0 0 −4.45884 2.57431i 0 1.35076 + 2.33958i 0 0 0
449.5 0 0 0 0.439631 + 0.253821i 0 6.44757 + 11.1675i 0 0 0
449.6 0 0 0 0.439631 + 0.253821i 0 −6.44757 11.1675i 0 0 0
449.7 0 0 0 1.15965 + 0.669525i 0 −0.328661 0.569258i 0 0 0
449.8 0 0 0 1.15965 + 0.669525i 0 0.328661 + 0.569258i 0 0 0
449.9 0 0 0 3.32266 + 1.91834i 0 −2.70775 4.68995i 0 0 0
449.10 0 0 0 3.32266 + 1.91834i 0 2.70775 + 4.68995i 0 0 0
449.11 0 0 0 6.92504 + 3.99817i 0 −6.10647 10.5767i 0 0 0
449.12 0 0 0 6.92504 + 3.99817i 0 6.10647 + 10.5767i 0 0 0
1601.1 0 0 0 −7.38813 + 4.26554i 0 −1.36942 + 2.37191i 0 0 0
1601.2 0 0 0 −7.38813 + 4.26554i 0 1.36942 2.37191i 0 0 0
1601.3 0 0 0 −4.45884 + 2.57431i 0 −1.35076 + 2.33958i 0 0 0
1601.4 0 0 0 −4.45884 + 2.57431i 0 1.35076 2.33958i 0 0 0
1601.5 0 0 0 0.439631 0.253821i 0 6.44757 11.1675i 0 0 0
1601.6 0 0 0 0.439631 0.253821i 0 −6.44757 + 11.1675i 0 0 0
1601.7 0 0 0 1.15965 0.669525i 0 −0.328661 + 0.569258i 0 0 0
1601.8 0 0 0 1.15965 0.669525i 0 0.328661 0.569258i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.q.l 24
3.b odd 2 1 576.3.q.k 24
4.b odd 2 1 inner 1728.3.q.l 24
8.b even 2 1 864.3.q.b 24
8.d odd 2 1 864.3.q.b 24
9.c even 3 1 576.3.q.k 24
9.d odd 6 1 inner 1728.3.q.l 24
12.b even 2 1 576.3.q.k 24
24.f even 2 1 288.3.q.a 24
24.h odd 2 1 288.3.q.a 24
36.f odd 6 1 576.3.q.k 24
36.h even 6 1 inner 1728.3.q.l 24
72.j odd 6 1 864.3.q.b 24
72.j odd 6 1 2592.3.e.j 24
72.l even 6 1 864.3.q.b 24
72.l even 6 1 2592.3.e.j 24
72.n even 6 1 288.3.q.a 24
72.n even 6 1 2592.3.e.j 24
72.p odd 6 1 288.3.q.a 24
72.p odd 6 1 2592.3.e.j 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.q.a 24 24.f even 2 1
288.3.q.a 24 24.h odd 2 1
288.3.q.a 24 72.n even 6 1
288.3.q.a 24 72.p odd 6 1
576.3.q.k 24 3.b odd 2 1
576.3.q.k 24 9.c even 3 1
576.3.q.k 24 12.b even 2 1
576.3.q.k 24 36.f odd 6 1
864.3.q.b 24 8.b even 2 1
864.3.q.b 24 8.d odd 2 1
864.3.q.b 24 72.j odd 6 1
864.3.q.b 24 72.l even 6 1
1728.3.q.l 24 1.a even 1 1 trivial
1728.3.q.l 24 4.b odd 2 1 inner
1728.3.q.l 24 9.d odd 6 1 inner
1728.3.q.l 24 36.h even 6 1 inner
2592.3.e.j 24 72.j odd 6 1
2592.3.e.j 24 72.l even 6 1
2592.3.e.j 24 72.n even 6 1
2592.3.e.j 24 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{12} - 90 T_{5}^{10} + 6627 T_{5}^{8} - 5292 T_{5}^{7} - 130514 T_{5}^{6} + 159084 T_{5}^{5} + 2092761 T_{5}^{4} - 6522444 T_{5}^{3} + 7884996 T_{5}^{2} - 4056048 T_{5} + 839056 \) Copy content Toggle raw display
\( T_{7}^{24} + 360 T_{7}^{22} + 90234 T_{7}^{20} + 11637392 T_{7}^{18} + 1080326187 T_{7}^{16} + 40447183032 T_{7}^{14} + 1071057738570 T_{7}^{12} + 13129538346000 T_{7}^{10} + \cdots + 296043132457216 \) Copy content Toggle raw display