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Properties

Label 2736.3.m.g
Level $2736$
Weight $3$
Character orbit 2736.m
Analytic conductor $74.551$
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] = [2736,3,Mod(1711,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.1711");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.m (of order \(2\), degree \(1\), 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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 72 q^{13} + 236 q^{25} - 128 q^{37} + 52 q^{49} + 84 q^{61} - 364 q^{73} - 44 q^{85} + 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1711.1 0 0 0 −9.31996 0 − 8.94955i 0 0 0
1711.2 0 0 0 −9.31996 0 8.94955i 0 0 0
1711.3 0 0 0 −8.04061 0 − 5.39854i 0 0 0
1711.4 0 0 0 −8.04061 0 5.39854i 0 0 0
1711.5 0 0 0 −5.26229 0 − 3.79905i 0 0 0
1711.6 0 0 0 −5.26229 0 3.79905i 0 0 0
1711.7 0 0 0 −3.99147 0 3.17803i 0 0 0
1711.8 0 0 0 −3.99147 0 − 3.17803i 0 0 0
1711.9 0 0 0 −3.60475 0 11.8725i 0 0 0
1711.10 0 0 0 −3.60475 0 − 11.8725i 0 0 0
1711.11 0 0 0 −0.932326 0 2.50436i 0 0 0
1711.12 0 0 0 −0.932326 0 − 2.50436i 0 0 0
1711.13 0 0 0 0.932326 0 2.50436i 0 0 0
1711.14 0 0 0 0.932326 0 − 2.50436i 0 0 0
1711.15 0 0 0 3.60475 0 11.8725i 0 0 0
1711.16 0 0 0 3.60475 0 − 11.8725i 0 0 0
1711.17 0 0 0 3.99147 0 3.17803i 0 0 0
1711.18 0 0 0 3.99147 0 − 3.17803i 0 0 0
1711.19 0 0 0 5.26229 0 − 3.79905i 0 0 0
1711.20 0 0 0 5.26229 0 3.79905i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1711.24
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 2736.3.m.g ✓ 24
3.b odd 2 1 inner 2736.3.m.g ✓ 24
4.b odd 2 1 inner 2736.3.m.g ✓ 24
12.b even 2 1 inner 2736.3.m.g ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.3.m.g ✓ 24 1.a even 1 1 trivial
2736.3.m.g ✓ 24 3.b odd 2 1 inner
2736.3.m.g ✓ 24 4.b odd 2 1 inner
2736.3.m.g ✓ 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 209T_{5}^{10} + 15383T_{5}^{8} - 489627T_{5}^{6} + 6943548T_{5}^{4} - 37869376T_{5}^{2} + 27983872 \) acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\). Copy content Toggle raw display