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Properties

Label 1728.3.t.a
Level $1728$
Weight $3$
Character orbit 1728.t
Analytic conductor $47.085$
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] = [1728,3,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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 18 q^{5} + 30 q^{13} + 36 q^{17} + 86 q^{25} - 162 q^{29} + 36 q^{41} + 138 q^{49} + 42 q^{61} + 198 q^{65} - 196 q^{73} + 666 q^{77} - 180 q^{85} - 792 q^{89} + 64 q^{97}+O(q^{100}) \) 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 −7.25986 − 4.19148i 0 8.13959 − 4.69939i 0 0 0
415.2 0 0 0 −7.25986 − 4.19148i 0 −8.13959 + 4.69939i 0 0 0
415.3 0 0 0 −6.71523 − 3.87704i 0 11.5588 − 6.67346i 0 0 0
415.4 0 0 0 −6.71523 − 3.87704i 0 −11.5588 + 6.67346i 0 0 0
415.5 0 0 0 −4.68159 − 2.70292i 0 −3.20912 + 1.85279i 0 0 0
415.6 0 0 0 −4.68159 − 2.70292i 0 3.20912 − 1.85279i 0 0 0
415.7 0 0 0 0.155865 + 0.0899885i 0 9.17188 − 5.29539i 0 0 0
415.8 0 0 0 0.155865 + 0.0899885i 0 −9.17188 + 5.29539i 0 0 0
415.9 0 0 0 1.05990 + 0.611933i 0 −5.05685 + 2.91957i 0 0 0
415.10 0 0 0 1.05990 + 0.611933i 0 5.05685 − 2.91957i 0 0 0
415.11 0 0 0 2.54739 + 1.47074i 0 −0.316235 + 0.182579i 0 0 0
415.12 0 0 0 2.54739 + 1.47074i 0 0.316235 − 0.182579i 0 0 0
415.13 0 0 0 4.53220 + 2.61667i 0 1.71711 − 0.991374i 0 0 0
415.14 0 0 0 4.53220 + 2.61667i 0 −1.71711 + 0.991374i 0 0 0
415.15 0 0 0 5.86132 + 3.38403i 0 4.78016 − 2.75983i 0 0 0
415.16 0 0 0 5.86132 + 3.38403i 0 −4.78016 + 2.75983i 0 0 0
991.1 0 0 0 −7.25986 + 4.19148i 0 8.13959 + 4.69939i 0 0 0
991.2 0 0 0 −7.25986 + 4.19148i 0 −8.13959 − 4.69939i 0 0 0
991.3 0 0 0 −6.71523 + 3.87704i 0 11.5588 + 6.67346i 0 0 0
991.4 0 0 0 −6.71523 + 3.87704i 0 −11.5588 − 6.67346i 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
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.a 32
3.b odd 2 1 576.3.t.c yes 32
4.b odd 2 1 inner 1728.3.t.a 32
8.b even 2 1 1728.3.t.c 32
8.d odd 2 1 1728.3.t.c 32
9.c even 3 1 1728.3.t.c 32
9.d odd 6 1 576.3.t.a ✓ 32
12.b even 2 1 576.3.t.c yes 32
24.f even 2 1 576.3.t.a ✓ 32
24.h odd 2 1 576.3.t.a ✓ 32
36.f odd 6 1 1728.3.t.c 32
36.h even 6 1 576.3.t.a ✓ 32
72.j odd 6 1 576.3.t.c yes 32
72.l even 6 1 576.3.t.c yes 32
72.n even 6 1 inner 1728.3.t.a 32
72.p odd 6 1 inner 1728.3.t.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.t.a ✓ 32 9.d odd 6 1
576.3.t.a ✓ 32 24.f even 2 1
576.3.t.a ✓ 32 24.h odd 2 1
576.3.t.a ✓ 32 36.h even 6 1
576.3.t.c yes 32 3.b odd 2 1
576.3.t.c yes 32 12.b even 2 1
576.3.t.c yes 32 72.j odd 6 1
576.3.t.c yes 32 72.l even 6 1
1728.3.t.a 32 1.a even 1 1 trivial
1728.3.t.a 32 4.b odd 2 1 inner
1728.3.t.a 32 72.n even 6 1 inner
1728.3.t.a 32 72.p odd 6 1 inner
1728.3.t.c 32 8.b even 2 1
1728.3.t.c 32 8.d odd 2 1
1728.3.t.c 32 9.c even 3 1
1728.3.t.c 32 36.f odd 6 1