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Properties

Label 1728.3.t.b
Level $1728$
Weight $3$
Character orbit 1728.t
Analytic conductor $47.085$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,3,Mod(415,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.t (of order \(6\), 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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 72 q^{17} + 68 q^{25} - 216 q^{41} + 60 q^{49} - 396 q^{65} + 392 q^{73} + 1008 q^{89} - 128 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1 0 0 0 −7.88649 − 4.55327i 0 2.78986 − 1.61073i 0 0 0
415.2 0 0 0 −7.88649 − 4.55327i 0 −2.78986 + 1.61073i 0 0 0
415.3 0 0 0 −3.79512 − 2.19111i 0 −2.44739 + 1.41300i 0 0 0
415.4 0 0 0 −3.79512 − 2.19111i 0 2.44739 − 1.41300i 0 0 0
415.5 0 0 0 −3.33903 − 1.92779i 0 9.08206 − 5.24353i 0 0 0
415.6 0 0 0 −3.33903 − 1.92779i 0 −9.08206 + 5.24353i 0 0 0
415.7 0 0 0 −0.0337710 − 0.0194977i 0 −7.87357 + 4.54581i 0 0 0
415.8 0 0 0 −0.0337710 − 0.0194977i 0 7.87357 − 4.54581i 0 0 0
415.9 0 0 0 0.0337710 + 0.0194977i 0 −7.87357 + 4.54581i 0 0 0
415.10 0 0 0 0.0337710 + 0.0194977i 0 7.87357 − 4.54581i 0 0 0
415.11 0 0 0 3.33903 + 1.92779i 0 9.08206 − 5.24353i 0 0 0
415.12 0 0 0 3.33903 + 1.92779i 0 −9.08206 + 5.24353i 0 0 0
415.13 0 0 0 3.79512 + 2.19111i 0 −2.44739 + 1.41300i 0 0 0
415.14 0 0 0 3.79512 + 2.19111i 0 2.44739 − 1.41300i 0 0 0
415.15 0 0 0 7.88649 + 4.55327i 0 2.78986 − 1.61073i 0 0 0
415.16 0 0 0 7.88649 + 4.55327i 0 −2.78986 + 1.61073i 0 0 0
991.1 0 0 0 −7.88649 + 4.55327i 0 2.78986 + 1.61073i 0 0 0
991.2 0 0 0 −7.88649 + 4.55327i 0 −2.78986 − 1.61073i 0 0 0
991.3 0 0 0 −3.79512 + 2.19111i 0 −2.44739 − 1.41300i 0 0 0
991.4 0 0 0 −3.79512 + 2.19111i 0 2.44739 + 1.41300i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.t.b 32
3.b odd 2 1 576.3.t.b ✓ 32
4.b odd 2 1 inner 1728.3.t.b 32
8.b even 2 1 inner 1728.3.t.b 32
8.d odd 2 1 inner 1728.3.t.b 32
9.c even 3 1 inner 1728.3.t.b 32
9.d odd 6 1 576.3.t.b ✓ 32
12.b even 2 1 576.3.t.b ✓ 32
24.f even 2 1 576.3.t.b ✓ 32
24.h odd 2 1 576.3.t.b ✓ 32
36.f odd 6 1 inner 1728.3.t.b 32
36.h even 6 1 576.3.t.b ✓ 32
72.j odd 6 1 576.3.t.b ✓ 32
72.l even 6 1 576.3.t.b ✓ 32
72.n even 6 1 inner 1728.3.t.b 32
72.p odd 6 1 inner 1728.3.t.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.t.b ✓ 32 3.b odd 2 1
576.3.t.b ✓ 32 9.d odd 6 1
576.3.t.b ✓ 32 12.b even 2 1
576.3.t.b ✓ 32 24.f even 2 1
576.3.t.b ✓ 32 24.h odd 2 1
576.3.t.b ✓ 32 36.h even 6 1
576.3.t.b ✓ 32 72.j odd 6 1
576.3.t.b ✓ 32 72.l even 6 1
1728.3.t.b 32 1.a even 1 1 trivial
1728.3.t.b 32 4.b odd 2 1 inner
1728.3.t.b 32 8.b even 2 1 inner
1728.3.t.b 32 8.d odd 2 1 inner
1728.3.t.b 32 9.c even 3 1 inner
1728.3.t.b 32 36.f odd 6 1 inner
1728.3.t.b 32 72.n even 6 1 inner
1728.3.t.b 32 72.p odd 6 1 inner