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Properties

Label 1728.4.c.l
Level $1728$
Weight $4$
Character orbit 1728.c
Analytic conductor $101.955$
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] = [1728,4,Mod(1727,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1727");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), 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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 72 q^{13} - 264 q^{25} - 24 q^{37} + 456 q^{61} + 2184 q^{73} - 3072 q^{85} - 3672 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1 0 0 0 − 20.1103i 0 − 5.50399i 0 0 0
1727.2 0 0 0 − 20.1103i 0 5.50399i 0 0 0
1727.3 0 0 0 − 12.8736i 0 − 18.8903i 0 0 0
1727.4 0 0 0 − 12.8736i 0 18.8903i 0 0 0
1727.5 0 0 0 − 12.0582i 0 − 1.19407i 0 0 0
1727.6 0 0 0 − 12.0582i 0 1.19407i 0 0 0
1727.7 0 0 0 − 8.05443i 0 − 20.0504i 0 0 0
1727.8 0 0 0 − 8.05443i 0 20.0504i 0 0 0
1727.9 0 0 0 − 5.36372i 0 − 29.2067i 0 0 0
1727.10 0 0 0 − 5.36372i 0 29.2067i 0 0 0
1727.11 0 0 0 − 2.60859i 0 − 20.3565i 0 0 0
1727.12 0 0 0 − 2.60859i 0 20.3565i 0 0 0
1727.13 0 0 0 2.60859i 0 − 20.3565i 0 0 0
1727.14 0 0 0 2.60859i 0 20.3565i 0 0 0
1727.15 0 0 0 5.36372i 0 − 29.2067i 0 0 0
1727.16 0 0 0 5.36372i 0 29.2067i 0 0 0
1727.17 0 0 0 8.05443i 0 − 20.0504i 0 0 0
1727.18 0 0 0 8.05443i 0 20.0504i 0 0 0
1727.19 0 0 0 12.0582i 0 − 1.19407i 0 0 0
1727.20 0 0 0 12.0582i 0 1.19407i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.c.l 24
3.b odd 2 1 inner 1728.4.c.l 24
4.b odd 2 1 inner 1728.4.c.l 24
8.b even 2 1 864.4.c.a ✓ 24
8.d odd 2 1 864.4.c.a ✓ 24
12.b even 2 1 inner 1728.4.c.l 24
24.f even 2 1 864.4.c.a ✓ 24
24.h odd 2 1 864.4.c.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.4.c.a ✓ 24 8.b even 2 1
864.4.c.a ✓ 24 8.d odd 2 1
864.4.c.a ✓ 24 24.f even 2 1
864.4.c.a ✓ 24 24.h odd 2 1
1728.4.c.l 24 1.a even 1 1 trivial
1728.4.c.l 24 3.b odd 2 1 inner
1728.4.c.l 24 4.b odd 2 1 inner
1728.4.c.l 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{12} + 816T_{5}^{10} + 224304T_{5}^{8} + 26609152T_{5}^{6} + 1363342080T_{5}^{4} + 26302623744T_{5}^{2} + 123768868864 \) Copy content Toggle raw display
\( T_{7}^{12} + 2058 T_{7}^{10} + 1523055 T_{7}^{8} + 496425932 T_{7}^{6} + 65049243279 T_{7}^{4} + \cdots + 2190349680289 \) Copy content Toggle raw display
\( T_{11}^{12} - 8496 T_{11}^{10} + 22954032 T_{11}^{8} - 25669993984 T_{11}^{6} + 11675439330048 T_{11}^{4} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display