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Properties

Label 1728.2.s.g
Level $1728$
Weight $2$
Character orbit 1728.s
Analytic conductor $13.798$
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] = [1728,2,Mod(575,1728)]
 
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([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.s (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: \(13.7981494693\)
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{25} + 24 q^{29} + 36 q^{41} + 12 q^{49} + 48 q^{65} + 24 q^{73} - 48 q^{77} + 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1 0 0 0 −3.01113 + 1.73848i 0 −3.12309 − 1.80312i 0 0 0
575.2 0 0 0 −3.01113 + 1.73848i 0 3.12309 + 1.80312i 0 0 0
575.3 0 0 0 −1.68236 + 0.971313i 0 −2.61432 − 1.50938i 0 0 0
575.4 0 0 0 −1.68236 + 0.971313i 0 2.61432 + 1.50938i 0 0 0
575.5 0 0 0 −0.398132 + 0.229862i 0 4.28309 + 2.47284i 0 0 0
575.6 0 0 0 −0.398132 + 0.229862i 0 −4.28309 − 2.47284i 0 0 0
575.7 0 0 0 −0.135038 + 0.0779642i 0 −0.349281 − 0.201658i 0 0 0
575.8 0 0 0 −0.135038 + 0.0779642i 0 0.349281 + 0.201658i 0 0 0
575.9 0 0 0 1.81740 − 1.04928i 0 −0.143714 − 0.0829731i 0 0 0
575.10 0 0 0 1.81740 − 1.04928i 0 0.143714 + 0.0829731i 0 0 0
575.11 0 0 0 3.40926 − 1.96834i 0 −0.961325 − 0.555021i 0 0 0
575.12 0 0 0 3.40926 − 1.96834i 0 0.961325 + 0.555021i 0 0 0
1151.1 0 0 0 −3.01113 − 1.73848i 0 −3.12309 + 1.80312i 0 0 0
1151.2 0 0 0 −3.01113 − 1.73848i 0 3.12309 − 1.80312i 0 0 0
1151.3 0 0 0 −1.68236 − 0.971313i 0 −2.61432 + 1.50938i 0 0 0
1151.4 0 0 0 −1.68236 − 0.971313i 0 2.61432 − 1.50938i 0 0 0
1151.5 0 0 0 −0.398132 − 0.229862i 0 4.28309 − 2.47284i 0 0 0
1151.6 0 0 0 −0.398132 − 0.229862i 0 −4.28309 + 2.47284i 0 0 0
1151.7 0 0 0 −0.135038 − 0.0779642i 0 −0.349281 + 0.201658i 0 0 0
1151.8 0 0 0 −0.135038 − 0.0779642i 0 0.349281 − 0.201658i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
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 1728.2.s.g 24
3.b odd 2 1 576.2.s.g 24
4.b odd 2 1 inner 1728.2.s.g 24
8.b even 2 1 864.2.s.a 24
8.d odd 2 1 864.2.s.a 24
9.c even 3 1 576.2.s.g 24
9.c even 3 1 5184.2.c.m 24
9.d odd 6 1 inner 1728.2.s.g 24
9.d odd 6 1 5184.2.c.m 24
12.b even 2 1 576.2.s.g 24
24.f even 2 1 288.2.s.a ✓ 24
24.h odd 2 1 288.2.s.a ✓ 24
36.f odd 6 1 576.2.s.g 24
36.f odd 6 1 5184.2.c.m 24
36.h even 6 1 inner 1728.2.s.g 24
36.h even 6 1 5184.2.c.m 24
72.j odd 6 1 864.2.s.a 24
72.j odd 6 1 2592.2.c.c 24
72.l even 6 1 864.2.s.a 24
72.l even 6 1 2592.2.c.c 24
72.n even 6 1 288.2.s.a ✓ 24
72.n even 6 1 2592.2.c.c 24
72.p odd 6 1 288.2.s.a ✓ 24
72.p odd 6 1 2592.2.c.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.s.a ✓ 24 24.f even 2 1
288.2.s.a ✓ 24 24.h odd 2 1
288.2.s.a ✓ 24 72.n even 6 1
288.2.s.a ✓ 24 72.p odd 6 1
576.2.s.g 24 3.b odd 2 1
576.2.s.g 24 9.c even 3 1
576.2.s.g 24 12.b even 2 1
576.2.s.g 24 36.f odd 6 1
864.2.s.a 24 8.b even 2 1
864.2.s.a 24 8.d odd 2 1
864.2.s.a 24 72.j odd 6 1
864.2.s.a 24 72.l even 6 1
1728.2.s.g 24 1.a even 1 1 trivial
1728.2.s.g 24 4.b odd 2 1 inner
1728.2.s.g 24 9.d odd 6 1 inner
1728.2.s.g 24 36.h even 6 1 inner
2592.2.c.c 24 72.j odd 6 1
2592.2.c.c 24 72.l even 6 1
2592.2.c.c 24 72.n even 6 1
2592.2.c.c 24 72.p odd 6 1
5184.2.c.m 24 9.c even 3 1
5184.2.c.m 24 9.d odd 6 1
5184.2.c.m 24 36.f odd 6 1
5184.2.c.m 24 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{12} - 18 T_{5}^{10} + 267 T_{5}^{8} + 156 T_{5}^{7} - 986 T_{5}^{6} - 684 T_{5}^{5} + 3081 T_{5}^{4} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{24} - 48 T_{7}^{22} + 1578 T_{7}^{20} - 27152 T_{7}^{18} + 338091 T_{7}^{16} - 2383368 T_{7}^{14} + \cdots + 256 \) Copy content Toggle raw display