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Properties

Label 531.4.d.a
Level $531$
Weight $4$
Character orbit 531.d
Analytic conductor $31.330$
Analytic rank $0$
Dimension $60$
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] = [531,4,Mod(530,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.530");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.d (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: \(31.3300142130\)
Analytic rank: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 240 q^{4} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 240 q^{4} + 24 q^{7} + 816 q^{16} - 96 q^{22} - 2004 q^{25} - 576 q^{28} + 1248 q^{46} + 5172 q^{49} + 816 q^{64} + 2784 q^{76} + 1992 q^{79} + 2688 q^{85} - 9576 q^{88} + 672 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
530.1 −5.38598 0 21.0088 − 8.70296i 0 19.2691 −70.0651 0 46.8740i
530.2 −5.38598 0 21.0088 8.70296i 0 19.2691 −70.0651 0 − 46.8740i
530.3 −5.10709 0 18.0823 − 11.0008i 0 −35.1570 −51.4913 0 56.1819i
530.4 −5.10709 0 18.0823 11.0008i 0 −35.1570 −51.4913 0 − 56.1819i
530.5 −4.87972 0 15.8116 − 2.08030i 0 −9.27371 −38.1185 0 10.1513i
530.6 −4.87972 0 15.8116 2.08030i 0 −9.27371 −38.1185 0 − 10.1513i
530.7 −4.82853 0 15.3147 − 20.4999i 0 27.0026 −35.3195 0 98.9847i
530.8 −4.82853 0 15.3147 20.4999i 0 27.0026 −35.3195 0 − 98.9847i
530.9 −4.19390 0 9.58884 − 9.04091i 0 −1.21513 −6.66343 0 37.9167i
530.10 −4.19390 0 9.58884 9.04091i 0 −1.21513 −6.66343 0 − 37.9167i
530.11 −3.94706 0 7.57932 21.2511i 0 −20.5183 1.66045 0 − 83.8794i
530.12 −3.94706 0 7.57932 − 21.2511i 0 −20.5183 1.66045 0 83.8794i
530.13 −3.35507 0 3.25649 5.32423i 0 4.49346 15.9148 0 − 17.8632i
530.14 −3.35507 0 3.25649 − 5.32423i 0 4.49346 15.9148 0 17.8632i
530.15 −3.26742 0 2.67606 14.5237i 0 −9.44552 17.3956 0 − 47.4550i
530.16 −3.26742 0 2.67606 − 14.5237i 0 −9.44552 17.3956 0 47.4550i
530.17 −3.16009 0 1.98619 − 1.99616i 0 22.5873 19.0042 0 6.30804i
530.18 −3.16009 0 1.98619 1.99616i 0 22.5873 19.0042 0 − 6.30804i
530.19 −2.05755 0 −3.76647 20.7416i 0 26.5362 24.2101 0 − 42.6769i
530.20 −2.05755 0 −3.76647 − 20.7416i 0 26.5362 24.2101 0 42.6769i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 530.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
59.b odd 2 1 inner
177.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.4.d.a ✓ 60
3.b odd 2 1 inner 531.4.d.a ✓ 60
59.b odd 2 1 inner 531.4.d.a ✓ 60
177.d even 2 1 inner 531.4.d.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.4.d.a ✓ 60 1.a even 1 1 trivial
531.4.d.a ✓ 60 3.b odd 2 1 inner
531.4.d.a ✓ 60 59.b odd 2 1 inner
531.4.d.a ✓ 60 177.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(531, [\chi])\).