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Properties

Label 588.3.g.h
Level $588$
Weight $3$
Character orbit 588.g
Analytic conductor $16.022$
Analytic rank $0$
Dimension $24$
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] = [588,3,Mod(295,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.295");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.g (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: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 12 q^{4} - 20 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 12 q^{4} - 20 q^{8} - 72 q^{9} - 60 q^{16} - 12 q^{18} + 168 q^{22} + 120 q^{25} + 64 q^{29} - 236 q^{32} - 36 q^{36} - 192 q^{37} - 360 q^{44} - 72 q^{46} + 532 q^{50} + 432 q^{53} + 240 q^{58} + 72 q^{60} - 372 q^{64} - 560 q^{65} + 60 q^{72} + 96 q^{74} + 216 q^{78} + 216 q^{81} + 144 q^{85} + 816 q^{86} - 72 q^{88} - 504 q^{92} - 96 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1 −1.96525 − 0.371196i − 1.73205i 3.72443 + 1.45899i 4.44657 −0.642930 + 3.40392i 0 −6.77787 − 4.24976i −3.00000 −8.73863 − 1.65055i
295.2 −1.96525 − 0.371196i 1.73205i 3.72443 + 1.45899i −4.44657 0.642930 − 3.40392i 0 −6.77787 − 4.24976i −3.00000 8.73863 + 1.65055i
295.3 −1.96525 + 0.371196i − 1.73205i 3.72443 − 1.45899i −4.44657 0.642930 + 3.40392i 0 −6.77787 + 4.24976i −3.00000 8.73863 − 1.65055i
295.4 −1.96525 + 0.371196i 1.73205i 3.72443 − 1.45899i 4.44657 −0.642930 − 3.40392i 0 −6.77787 + 4.24976i −3.00000 −8.73863 + 1.65055i
295.5 −1.44448 − 1.38328i − 1.73205i 0.173067 + 3.99625i 1.50286 −2.39591 + 2.50192i 0 5.27795 − 6.01192i −3.00000 −2.17086 − 2.07888i
295.6 −1.44448 − 1.38328i 1.73205i 0.173067 + 3.99625i −1.50286 2.39591 − 2.50192i 0 5.27795 − 6.01192i −3.00000 2.17086 + 2.07888i
295.7 −1.44448 + 1.38328i − 1.73205i 0.173067 − 3.99625i −1.50286 2.39591 + 2.50192i 0 5.27795 + 6.01192i −3.00000 2.17086 − 2.07888i
295.8 −1.44448 + 1.38328i 1.73205i 0.173067 − 3.99625i 1.50286 −2.39591 − 2.50192i 0 5.27795 + 6.01192i −3.00000 −2.17086 + 2.07888i
295.9 −0.249300 − 1.98440i − 1.73205i −3.87570 + 0.989422i −4.12200 −3.43708 + 0.431800i 0 2.92962 + 7.44428i −3.00000 1.02761 + 8.17970i
295.10 −0.249300 − 1.98440i 1.73205i −3.87570 + 0.989422i 4.12200 3.43708 − 0.431800i 0 2.92962 + 7.44428i −3.00000 −1.02761 − 8.17970i
295.11 −0.249300 + 1.98440i − 1.73205i −3.87570 − 0.989422i 4.12200 3.43708 + 0.431800i 0 2.92962 − 7.44428i −3.00000 −1.02761 + 8.17970i
295.12 −0.249300 + 1.98440i 1.73205i −3.87570 − 0.989422i −4.12200 −3.43708 − 0.431800i 0 2.92962 − 7.44428i −3.00000 1.02761 − 8.17970i
295.13 1.14501 − 1.63980i − 1.73205i −1.37788 − 3.75519i 8.62198 −2.84022 − 1.98322i 0 −7.73545 − 2.04030i −3.00000 9.87230 − 14.1383i
295.14 1.14501 − 1.63980i 1.73205i −1.37788 − 3.75519i −8.62198 2.84022 + 1.98322i 0 −7.73545 − 2.04030i −3.00000 −9.87230 + 14.1383i
295.15 1.14501 + 1.63980i − 1.73205i −1.37788 + 3.75519i −8.62198 2.84022 − 1.98322i 0 −7.73545 + 2.04030i −3.00000 −9.87230 − 14.1383i
295.16 1.14501 + 1.63980i 1.73205i −1.37788 + 3.75519i 8.62198 −2.84022 + 1.98322i 0 −7.73545 + 2.04030i −3.00000 9.87230 + 14.1383i
295.17 1.71300 − 1.03229i − 1.73205i 1.86874 − 3.53664i −2.98338 −1.78799 − 2.96700i 0 −0.449712 − 7.98735i −3.00000 −5.11054 + 3.07973i
295.18 1.71300 − 1.03229i 1.73205i 1.86874 − 3.53664i 2.98338 1.78799 + 2.96700i 0 −0.449712 − 7.98735i −3.00000 5.11054 − 3.07973i
295.19 1.71300 + 1.03229i − 1.73205i 1.86874 + 3.53664i 2.98338 1.78799 − 2.96700i 0 −0.449712 + 7.98735i −3.00000 5.11054 + 3.07973i
295.20 1.71300 + 1.03229i 1.73205i 1.86874 + 3.53664i −2.98338 −1.78799 + 2.96700i 0 −0.449712 + 7.98735i −3.00000 −5.11054 − 3.07973i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.g.h ✓ 24
4.b odd 2 1 inner 588.3.g.h ✓ 24
7.b odd 2 1 inner 588.3.g.h ✓ 24
28.d even 2 1 inner 588.3.g.h ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.g.h ✓ 24 1.a even 1 1 trivial
588.3.g.h ✓ 24 4.b odd 2 1 inner
588.3.g.h ✓ 24 7.b odd 2 1 inner
588.3.g.h ✓ 24 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 180T_{5}^{10} + 11388T_{5}^{8} - 311392T_{5}^{6} + 3888624T_{5}^{4} - 20155200T_{5}^{2} + 28987456 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display