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Properties

Label 1728.2.l.a
Level $1728$
Weight $2$
Character orbit 1728.l
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(431,1728)]
 
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([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.l (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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{19} - 32 q^{43} + 32 q^{49} - 64 q^{55} - 16 q^{61} + 32 q^{67} - 16 q^{85} - 24 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1 0 0 0 −2.82387 − 2.82387i 0 3.49224 0 0 0
431.2 0 0 0 −2.47363 − 2.47363i 0 0.311286 0 0 0
431.3 0 0 0 −2.29090 − 2.29090i 0 −3.92541 0 0 0
431.4 0 0 0 −1.39178 − 1.39178i 0 1.33599 0 0 0
431.5 0 0 0 −1.33778 − 1.33778i 0 −0.400629 0 0 0
431.6 0 0 0 −0.800188 − 0.800188i 0 −0.180707 0 0 0
431.7 0 0 0 −0.494369 − 0.494369i 0 −4.44678 0 0 0
431.8 0 0 0 −0.217410 − 0.217410i 0 3.81401 0 0 0
431.9 0 0 0 0.217410 + 0.217410i 0 3.81401 0 0 0
431.10 0 0 0 0.494369 + 0.494369i 0 −4.44678 0 0 0
431.11 0 0 0 0.800188 + 0.800188i 0 −0.180707 0 0 0
431.12 0 0 0 1.33778 + 1.33778i 0 −0.400629 0 0 0
431.13 0 0 0 1.39178 + 1.39178i 0 1.33599 0 0 0
431.14 0 0 0 2.29090 + 2.29090i 0 −3.92541 0 0 0
431.15 0 0 0 2.47363 + 2.47363i 0 0.311286 0 0 0
431.16 0 0 0 2.82387 + 2.82387i 0 3.49224 0 0 0
1295.1 0 0 0 −2.82387 + 2.82387i 0 3.49224 0 0 0
1295.2 0 0 0 −2.47363 + 2.47363i 0 0.311286 0 0 0
1295.3 0 0 0 −2.29090 + 2.29090i 0 −3.92541 0 0 0
1295.4 0 0 0 −1.39178 + 1.39178i 0 1.33599 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.16
Significant digits:
Format:

Inner twists

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

Twists

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