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Properties

Label 1728.4.c.k
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 144 q^{13} - 768 q^{25} - 1536 q^{37} - 2304 q^{49} - 2784 q^{61} - 1416 q^{73} + 816 q^{85} + 648 q^{97}+O(q^{100}) \) 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 − 18.7846i 0 − 30.8989i 0 0 0
1727.2 0 0 0 − 18.7846i 0 30.8989i 0 0 0
1727.3 0 0 0 − 18.0256i 0 − 1.63267i 0 0 0
1727.4 0 0 0 − 18.0256i 0 1.63267i 0 0 0
1727.5 0 0 0 − 11.4141i 0 − 5.04782i 0 0 0
1727.6 0 0 0 − 11.4141i 0 5.04782i 0 0 0
1727.7 0 0 0 − 8.99651i 0 − 35.3962i 0 0 0
1727.8 0 0 0 − 8.99651i 0 35.3962i 0 0 0
1727.9 0 0 0 − 5.95128i 0 − 12.0727i 0 0 0
1727.10 0 0 0 − 5.95128i 0 12.0727i 0 0 0
1727.11 0 0 0 − 4.19264i 0 − 15.8895i 0 0 0
1727.12 0 0 0 − 4.19264i 0 15.8895i 0 0 0
1727.13 0 0 0 4.19264i 0 − 15.8895i 0 0 0
1727.14 0 0 0 4.19264i 0 15.8895i 0 0 0
1727.15 0 0 0 5.95128i 0 − 12.0727i 0 0 0
1727.16 0 0 0 5.95128i 0 12.0727i 0 0 0
1727.17 0 0 0 8.99651i 0 − 35.3962i 0 0 0
1727.18 0 0 0 8.99651i 0 35.3962i 0 0 0
1727.19 0 0 0 11.4141i 0 − 5.04782i 0 0 0
1727.20 0 0 0 11.4141i 0 5.04782i 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.k 24
3.b odd 2 1 inner 1728.4.c.k 24
4.b odd 2 1 inner 1728.4.c.k 24
8.b even 2 1 864.4.c.b ✓ 24
8.d odd 2 1 864.4.c.b ✓ 24
12.b even 2 1 inner 1728.4.c.k 24
24.f even 2 1 864.4.c.b ✓ 24
24.h odd 2 1 864.4.c.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.4.c.b ✓ 24 8.b even 2 1
864.4.c.b ✓ 24 8.d odd 2 1
864.4.c.b ✓ 24 24.f even 2 1
864.4.c.b ✓ 24 24.h odd 2 1
1728.4.c.k 24 1.a even 1 1 trivial
1728.4.c.k 24 3.b odd 2 1 inner
1728.4.c.k 24 4.b odd 2 1 inner
1728.4.c.k 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{12} + 942T_{5}^{10} + 316095T_{5}^{8} + 46139332T_{5}^{6} + 3038213199T_{5}^{4} + 83597472750T_{5}^{2} + 752686380625 \) Copy content Toggle raw display
\( T_{7}^{12} + 2634 T_{7}^{10} + 2185527 T_{7}^{8} + 617211692 T_{7}^{6} + 59854716159 T_{7}^{4} + \cdots + 2989700355625 \) Copy content Toggle raw display
\( T_{11}^{12} - 9522 T_{11}^{10} + 30504519 T_{11}^{8} - 37355540956 T_{11}^{6} + 15968841052095 T_{11}^{4} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display