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Properties

Label 1728.2.k.d
Level $1728$
Weight $2$
Character orbit 1728.k
Analytic conductor $13.798$
Analytic rank $0$
Dimension $32$
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(433,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.k (of order \(4\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{19} + 16 q^{43} - 32 q^{49} - 32 q^{61} - 32 q^{67} + 32 q^{79} + 32 q^{85} + 48 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1 0 0 0 −2.86290 − 2.86290i 0 − 1.20181i 0 0 0
433.2 0 0 0 −2.51567 − 2.51567i 0 4.16864i 0 0 0
433.3 0 0 0 −1.82760 − 1.82760i 0 − 1.81881i 0 0 0
433.4 0 0 0 −1.45562 − 1.45562i 0 − 2.37837i 0 0 0
433.5 0 0 0 −1.27540 − 1.27540i 0 0.0285435i 0 0 0
433.6 0 0 0 −1.14284 − 1.14284i 0 4.40079i 0 0 0
433.7 0 0 0 −0.920303 − 0.920303i 0 − 4.02115i 0 0 0
433.8 0 0 0 −0.486359 − 0.486359i 0 0.822162i 0 0 0
433.9 0 0 0 0.486359 + 0.486359i 0 0.822162i 0 0 0
433.10 0 0 0 0.920303 + 0.920303i 0 − 4.02115i 0 0 0
433.11 0 0 0 1.14284 + 1.14284i 0 4.40079i 0 0 0
433.12 0 0 0 1.27540 + 1.27540i 0 0.0285435i 0 0 0
433.13 0 0 0 1.45562 + 1.45562i 0 − 2.37837i 0 0 0
433.14 0 0 0 1.82760 + 1.82760i 0 − 1.81881i 0 0 0
433.15 0 0 0 2.51567 + 2.51567i 0 4.16864i 0 0 0
433.16 0 0 0 2.86290 + 2.86290i 0 − 1.20181i 0 0 0
1297.1 0 0 0 −2.86290 + 2.86290i 0 1.20181i 0 0 0
1297.2 0 0 0 −2.51567 + 2.51567i 0 − 4.16864i 0 0 0
1297.3 0 0 0 −1.82760 + 1.82760i 0 1.81881i 0 0 0
1297.4 0 0 0 −1.45562 + 1.45562i 0 2.37837i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.k.d 32
3.b odd 2 1 inner 1728.2.k.d 32
4.b odd 2 1 432.2.k.d ✓ 32
12.b even 2 1 432.2.k.d ✓ 32
16.e even 4 1 inner 1728.2.k.d 32
16.f odd 4 1 432.2.k.d ✓ 32
48.i odd 4 1 inner 1728.2.k.d 32
48.k even 4 1 432.2.k.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.k.d ✓ 32 4.b odd 2 1
432.2.k.d ✓ 32 12.b even 2 1
432.2.k.d ✓ 32 16.f odd 4 1
432.2.k.d ✓ 32 48.k even 4 1
1728.2.k.d 32 1.a even 1 1 trivial
1728.2.k.d 32 3.b odd 2 1 inner
1728.2.k.d 32 16.e even 4 1 inner
1728.2.k.d 32 48.i odd 4 1 inner

Hecke kernels

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

\( T_{5}^{32} + 512 T_{5}^{28} + 80896 T_{5}^{24} + 4549632 T_{5}^{20} + 105848960 T_{5}^{16} + \cdots + 1600000000 \) Copy content Toggle raw display
\( T_{11}^{32} + 2048 T_{11}^{28} + 1408768 T_{11}^{24} + 431806464 T_{11}^{20} + 60248647808 T_{11}^{16} + \cdots + 4096 \) Copy content Toggle raw display