Properties

Label 576.7.j.a
Level $576$
Weight $7$
Character orbit 576.j
Analytic conductor $132.511$
Analytic rank $0$
Dimension $96$
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] = [576,7,Mod(17,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 576.j (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: \(132.511152165\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 144)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 7872 q^{19} - 390528 q^{43} - 1613472 q^{49} + 652992 q^{61} + 603840 q^{67} + 1721856 q^{79} - 744000 q^{85} + 3700800 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 0 0 −168.179 168.179i 0 260.196i 0 0 0
17.2 0 0 0 −158.404 158.404i 0 133.106i 0 0 0
17.3 0 0 0 −143.326 143.326i 0 548.969i 0 0 0
17.4 0 0 0 −141.033 141.033i 0 48.8223i 0 0 0
17.5 0 0 0 −140.082 140.082i 0 123.966i 0 0 0
17.6 0 0 0 −128.928 128.928i 0 317.300i 0 0 0
17.7 0 0 0 −125.068 125.068i 0 645.212i 0 0 0
17.8 0 0 0 −109.398 109.398i 0 428.243i 0 0 0
17.9 0 0 0 −108.765 108.765i 0 659.213i 0 0 0
17.10 0 0 0 −102.459 102.459i 0 388.352i 0 0 0
17.11 0 0 0 −100.508 100.508i 0 48.6019i 0 0 0
17.12 0 0 0 −95.8031 95.8031i 0 356.832i 0 0 0
17.13 0 0 0 −78.6085 78.6085i 0 255.650i 0 0 0
17.14 0 0 0 −71.4717 71.4717i 0 315.658i 0 0 0
17.15 0 0 0 −68.0621 68.0621i 0 538.920i 0 0 0
17.16 0 0 0 −67.0992 67.0992i 0 212.431i 0 0 0
17.17 0 0 0 −43.8575 43.8575i 0 133.731i 0 0 0
17.18 0 0 0 −41.7984 41.7984i 0 240.685i 0 0 0
17.19 0 0 0 −33.4257 33.4257i 0 175.919i 0 0 0
17.20 0 0 0 −15.2729 15.2729i 0 498.002i 0 0 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.7.j.a 96
3.b odd 2 1 inner 576.7.j.a 96
4.b odd 2 1 144.7.j.a 96
12.b even 2 1 144.7.j.a 96
16.e even 4 1 inner 576.7.j.a 96
16.f odd 4 1 144.7.j.a 96
48.i odd 4 1 inner 576.7.j.a 96
48.k even 4 1 144.7.j.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.7.j.a 96 4.b odd 2 1
144.7.j.a 96 12.b even 2 1
144.7.j.a 96 16.f odd 4 1
144.7.j.a 96 48.k even 4 1
576.7.j.a 96 1.a even 1 1 trivial
576.7.j.a 96 3.b odd 2 1 inner
576.7.j.a 96 16.e even 4 1 inner
576.7.j.a 96 48.i odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(576, [\chi])\).