Properties

Label 576.4.l.a
Level $576$
Weight $4$
Character orbit 576.l
Analytic conductor $33.985$
Analytic rank $0$
Dimension $48$
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] = [576,4,Mod(143,576)]
 
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([2, 1, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.143");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.l (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: \(33.9851001633\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{19} - 864 q^{43} + 2352 q^{49} + 576 q^{55} + 1824 q^{61} - 816 q^{67} - 480 q^{85} + 3600 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1 0 0 0 −15.2065 + 15.2065i 0 −24.4971 0 0 0
143.2 0 0 0 −11.8474 + 11.8474i 0 12.7523 0 0 0
143.3 0 0 0 −11.7679 + 11.7679i 0 11.9978 0 0 0
143.4 0 0 0 −11.2962 + 11.2962i 0 19.1985 0 0 0
143.5 0 0 0 −9.40057 + 9.40057i 0 3.57327 0 0 0
143.6 0 0 0 −7.28730 + 7.28730i 0 −29.9574 0 0 0
143.7 0 0 0 −6.31601 + 6.31601i 0 −16.2772 0 0 0
143.8 0 0 0 −6.30986 + 6.30986i 0 −27.2034 0 0 0
143.9 0 0 0 −3.62348 + 3.62348i 0 33.7361 0 0 0
143.10 0 0 0 −3.43656 + 3.43656i 0 −8.14652 0 0 0
143.11 0 0 0 −3.22357 + 3.22357i 0 13.1030 0 0 0
143.12 0 0 0 −2.40838 + 2.40838i 0 11.7205 0 0 0
143.13 0 0 0 2.40838 2.40838i 0 11.7205 0 0 0
143.14 0 0 0 3.22357 3.22357i 0 13.1030 0 0 0
143.15 0 0 0 3.43656 3.43656i 0 −8.14652 0 0 0
143.16 0 0 0 3.62348 3.62348i 0 33.7361 0 0 0
143.17 0 0 0 6.30986 6.30986i 0 −27.2034 0 0 0
143.18 0 0 0 6.31601 6.31601i 0 −16.2772 0 0 0
143.19 0 0 0 7.28730 7.28730i 0 −29.9574 0 0 0
143.20 0 0 0 9.40057 9.40057i 0 3.57327 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.4.l.a 48
3.b odd 2 1 inner 576.4.l.a 48
4.b odd 2 1 144.4.l.a 48
8.b even 2 1 1152.4.l.a 48
8.d odd 2 1 1152.4.l.b 48
12.b even 2 1 144.4.l.a 48
16.e even 4 1 144.4.l.a 48
16.e even 4 1 1152.4.l.b 48
16.f odd 4 1 inner 576.4.l.a 48
16.f odd 4 1 1152.4.l.a 48
24.f even 2 1 1152.4.l.b 48
24.h odd 2 1 1152.4.l.a 48
48.i odd 4 1 144.4.l.a 48
48.i odd 4 1 1152.4.l.b 48
48.k even 4 1 inner 576.4.l.a 48
48.k even 4 1 1152.4.l.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.l.a 48 4.b odd 2 1
144.4.l.a 48 12.b even 2 1
144.4.l.a 48 16.e even 4 1
144.4.l.a 48 48.i odd 4 1
576.4.l.a 48 1.a even 1 1 trivial
576.4.l.a 48 3.b odd 2 1 inner
576.4.l.a 48 16.f odd 4 1 inner
576.4.l.a 48 48.k even 4 1 inner
1152.4.l.a 48 8.b even 2 1
1152.4.l.a 48 16.f odd 4 1
1152.4.l.a 48 24.h odd 2 1
1152.4.l.a 48 48.k even 4 1
1152.4.l.b 48 8.d odd 2 1
1152.4.l.b 48 16.e even 4 1
1152.4.l.b 48 24.f even 2 1
1152.4.l.b 48 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(576, [\chi])\).