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Properties

Label 576.7.h.b
Level $576$
Weight $7$
Character orbit 576.h
Analytic conductor $132.511$
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] = [576,7,Mod(161,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.h (of order \(2\), degree \(1\), 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: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 169888 q^{25} + 829792 q^{49} - 1493888 q^{73} - 15893248 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 0 0 −238.767 0 590.190 0 0 0
161.2 0 0 0 −238.767 0 −590.190 0 0 0
161.3 0 0 0 −238.767 0 −590.190 0 0 0
161.4 0 0 0 −238.767 0 590.190 0 0 0
161.5 0 0 0 −108.406 0 167.464 0 0 0
161.6 0 0 0 −108.406 0 −167.464 0 0 0
161.7 0 0 0 −108.406 0 −167.464 0 0 0
161.8 0 0 0 −108.406 0 167.464 0 0 0
161.9 0 0 0 −89.8672 0 443.362 0 0 0
161.10 0 0 0 −89.8672 0 −443.362 0 0 0
161.11 0 0 0 −89.8672 0 −443.362 0 0 0
161.12 0 0 0 −89.8672 0 443.362 0 0 0
161.13 0 0 0 −83.0572 0 −37.1638 0 0 0
161.14 0 0 0 −83.0572 0 37.1638 0 0 0
161.15 0 0 0 −83.0572 0 37.1638 0 0 0
161.16 0 0 0 −83.0572 0 −37.1638 0 0 0
161.17 0 0 0 83.0572 0 37.1638 0 0 0
161.18 0 0 0 83.0572 0 −37.1638 0 0 0
161.19 0 0 0 83.0572 0 −37.1638 0 0 0
161.20 0 0 0 83.0572 0 37.1638 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.h.b ✓ 32
3.b odd 2 1 inner 576.7.h.b ✓ 32
4.b odd 2 1 inner 576.7.h.b ✓ 32
8.b even 2 1 inner 576.7.h.b ✓ 32
8.d odd 2 1 inner 576.7.h.b ✓ 32
12.b even 2 1 inner 576.7.h.b ✓ 32
24.f even 2 1 inner 576.7.h.b ✓ 32
24.h odd 2 1 inner 576.7.h.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.7.h.b ✓ 32 1.a even 1 1 trivial
576.7.h.b ✓ 32 3.b odd 2 1 inner
576.7.h.b ✓ 32 4.b odd 2 1 inner
576.7.h.b ✓ 32 8.b even 2 1 inner
576.7.h.b ✓ 32 8.d odd 2 1 inner
576.7.h.b ✓ 32 12.b even 2 1 inner
576.7.h.b ✓ 32 24.f even 2 1 inner
576.7.h.b ✓ 32 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 83736T_{5}^{6} + 1755351000T_{5}^{4} - 13863338460000T_{5}^{2} + 37325660402250000 \) acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display