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Properties

Label 576.4.k.b
Level $576$
Weight $4$
Character orbit 576.k
Analytic conductor $33.985$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,4,Mod(145,576)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.145");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.k (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: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79} + 2680 q^{83} - 240 q^{85} + 3496 q^{91} - 7728 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 0 0 0 −14.6111 + 14.6111i 0 − 26.8889i 0 0 0
145.2 0 0 0 −11.7911 + 11.7911i 0 − 12.5754i 0 0 0
145.3 0 0 0 −10.2951 + 10.2951i 0 32.8369i 0 0 0
145.4 0 0 0 −3.22588 + 3.22588i 0 24.6080i 0 0 0
145.5 0 0 0 −2.24191 + 2.24191i 0 9.00196i 0 0 0
145.6 0 0 0 −0.706564 + 0.706564i 0 − 4.44122i 0 0 0
145.7 0 0 0 −0.644922 + 0.644922i 0 7.13926i 0 0 0
145.8 0 0 0 3.72414 − 3.72414i 0 − 20.2675i 0 0 0
145.9 0 0 0 7.29121 − 7.29121i 0 − 22.1610i 0 0 0
145.10 0 0 0 8.83384 − 8.83384i 0 − 29.4760i 0 0 0
145.11 0 0 0 11.7719 − 11.7719i 0 14.7089i 0 0 0
145.12 0 0 0 11.8955 − 11.8955i 0 − 0.485059i 0 0 0
433.1 0 0 0 −14.6111 − 14.6111i 0 26.8889i 0 0 0
433.2 0 0 0 −11.7911 − 11.7911i 0 12.5754i 0 0 0
433.3 0 0 0 −10.2951 − 10.2951i 0 − 32.8369i 0 0 0
433.4 0 0 0 −3.22588 − 3.22588i 0 − 24.6080i 0 0 0
433.5 0 0 0 −2.24191 − 2.24191i 0 − 9.00196i 0 0 0
433.6 0 0 0 −0.706564 − 0.706564i 0 4.44122i 0 0 0
433.7 0 0 0 −0.644922 − 0.644922i 0 − 7.13926i 0 0 0
433.8 0 0 0 3.72414 + 3.72414i 0 20.2675i 0 0 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e 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.k.b 24
3.b odd 2 1 192.4.j.a 24
4.b odd 2 1 144.4.k.b 24
12.b even 2 1 48.4.j.a ✓ 24
16.e even 4 1 inner 576.4.k.b 24
16.f odd 4 1 144.4.k.b 24
24.f even 2 1 384.4.j.b 24
24.h odd 2 1 384.4.j.a 24
48.i odd 4 1 192.4.j.a 24
48.i odd 4 1 384.4.j.a 24
48.k even 4 1 48.4.j.a ✓ 24
48.k even 4 1 384.4.j.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.j.a ✓ 24 12.b even 2 1
48.4.j.a ✓ 24 48.k even 4 1
144.4.k.b 24 4.b odd 2 1
144.4.k.b 24 16.f odd 4 1
192.4.j.a 24 3.b odd 2 1
192.4.j.a 24 48.i odd 4 1
384.4.j.a 24 24.h odd 2 1
384.4.j.a 24 48.i odd 4 1
384.4.j.b 24 24.f even 2 1
384.4.j.b 24 48.k even 4 1
576.4.k.b 24 1.a even 1 1 trivial
576.4.k.b 24 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 1936 T_{5}^{21} + 249216 T_{5}^{20} - 752832 T_{5}^{19} + 1874048 T_{5}^{18} + \cdots + 15\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display