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Properties

Label 576.3.t.c
Level $576$
Weight $3$
Character orbit 576.t
Analytic conductor $15.695$
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] = [576,3,Mod(31,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, 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("576.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.t (of order \(6\), degree \(2\), 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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 18 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 18 q^{5} + 18 q^{9} + 30 q^{13} - 36 q^{17} - 102 q^{21} + 86 q^{25} + 162 q^{29} + 12 q^{33} - 36 q^{41} - 186 q^{45} + 138 q^{49} - 162 q^{57} + 42 q^{61} - 198 q^{65} + 474 q^{69} - 196 q^{73} - 666 q^{77} + 462 q^{81} - 180 q^{85} + 792 q^{89} - 174 q^{93} + 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −2.99901 + 0.0771699i 0 −5.86132 − 3.38403i 0 4.78016 − 2.75983i 0 8.98809 − 0.462866i 0
31.2 0 −2.88051 − 0.838238i 0 4.68159 + 2.70292i 0 −3.20912 + 1.85279i 0 7.59471 + 4.82911i 0
31.3 0 −2.79923 − 1.07903i 0 6.71523 + 3.87704i 0 11.5588 − 6.67346i 0 6.67137 + 6.04093i 0
31.4 0 −2.70112 + 1.30536i 0 −4.53220 − 2.61667i 0 1.71711 − 0.991374i 0 5.59208 − 7.05186i 0
31.5 0 −1.81830 − 2.38617i 0 −0.155865 − 0.0899885i 0 −9.17188 + 5.29539i 0 −2.38759 + 8.67752i 0
31.6 0 −1.06256 + 2.80553i 0 7.25986 + 4.19148i 0 −8.13959 + 4.69939i 0 −6.74195 − 5.96205i 0
31.7 0 −0.982091 + 2.83470i 0 −1.05990 − 0.611933i 0 −5.05685 + 2.91957i 0 −7.07100 − 5.56786i 0
31.8 0 −0.653563 − 2.92794i 0 −2.54739 − 1.47074i 0 0.316235 − 0.182579i 0 −8.14571 + 3.82719i 0
31.9 0 0.653563 + 2.92794i 0 −2.54739 − 1.47074i 0 −0.316235 + 0.182579i 0 −8.14571 + 3.82719i 0
31.10 0 0.982091 − 2.83470i 0 −1.05990 − 0.611933i 0 5.05685 − 2.91957i 0 −7.07100 − 5.56786i 0
31.11 0 1.06256 − 2.80553i 0 7.25986 + 4.19148i 0 8.13959 − 4.69939i 0 −6.74195 − 5.96205i 0
31.12 0 1.81830 + 2.38617i 0 −0.155865 − 0.0899885i 0 9.17188 − 5.29539i 0 −2.38759 + 8.67752i 0
31.13 0 2.70112 − 1.30536i 0 −4.53220 − 2.61667i 0 −1.71711 + 0.991374i 0 5.59208 − 7.05186i 0
31.14 0 2.79923 + 1.07903i 0 6.71523 + 3.87704i 0 −11.5588 + 6.67346i 0 6.67137 + 6.04093i 0
31.15 0 2.88051 + 0.838238i 0 4.68159 + 2.70292i 0 3.20912 − 1.85279i 0 7.59471 + 4.82911i 0
31.16 0 2.99901 − 0.0771699i 0 −5.86132 − 3.38403i 0 −4.78016 + 2.75983i 0 8.98809 − 0.462866i 0
223.1 0 −2.99901 − 0.0771699i 0 −5.86132 + 3.38403i 0 4.78016 + 2.75983i 0 8.98809 + 0.462866i 0
223.2 0 −2.88051 + 0.838238i 0 4.68159 − 2.70292i 0 −3.20912 − 1.85279i 0 7.59471 − 4.82911i 0
223.3 0 −2.79923 + 1.07903i 0 6.71523 − 3.87704i 0 11.5588 + 6.67346i 0 6.67137 − 6.04093i 0
223.4 0 −2.70112 − 1.30536i 0 −4.53220 + 2.61667i 0 1.71711 + 0.991374i 0 5.59208 + 7.05186i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.16
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 576.3.t.c yes 32
3.b odd 2 1 1728.3.t.a 32
4.b odd 2 1 inner 576.3.t.c yes 32
8.b even 2 1 576.3.t.a ✓ 32
8.d odd 2 1 576.3.t.a ✓ 32
9.c even 3 1 576.3.t.a ✓ 32
9.d odd 6 1 1728.3.t.c 32
12.b even 2 1 1728.3.t.a 32
24.f even 2 1 1728.3.t.c 32
24.h odd 2 1 1728.3.t.c 32
36.f odd 6 1 576.3.t.a ✓ 32
36.h even 6 1 1728.3.t.c 32
72.j odd 6 1 1728.3.t.a 32
72.l even 6 1 1728.3.t.a 32
72.n even 6 1 inner 576.3.t.c yes 32
72.p odd 6 1 inner 576.3.t.c yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.t.a ✓ 32 8.b even 2 1
576.3.t.a ✓ 32 8.d odd 2 1
576.3.t.a ✓ 32 9.c even 3 1
576.3.t.a ✓ 32 36.f odd 6 1
576.3.t.c yes 32 1.a even 1 1 trivial
576.3.t.c yes 32 4.b odd 2 1 inner
576.3.t.c yes 32 72.n even 6 1 inner
576.3.t.c yes 32 72.p odd 6 1 inner
1728.3.t.a 32 3.b odd 2 1
1728.3.t.a 32 12.b even 2 1
1728.3.t.a 32 72.j odd 6 1
1728.3.t.a 32 72.l even 6 1
1728.3.t.c 32 9.d odd 6 1
1728.3.t.c 32 24.f even 2 1
1728.3.t.c 32 24.h odd 2 1
1728.3.t.c 32 36.h even 6 1