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Properties

Label 672.4.p.a
Level $672$
Weight $4$
Character orbit 672.p
Analytic conductor $39.649$
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] = [672,4,Mod(559,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.559");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.p (of order \(2\), degree \(1\), 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: \(39.6492835239\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 432 q^{9} + 40 q^{11} + 1200 q^{25} + 456 q^{35} + 808 q^{43} + 360 q^{49} + 336 q^{57} + 4104 q^{67} + 3888 q^{81} - 104 q^{91} - 360 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1 0 − 3.00000i 0 9.00358 0 15.3096 − 10.4219i 0 −9.00000 0
559.2 0 3.00000i 0 9.00358 0 15.3096 + 10.4219i 0 −9.00000 0
559.3 0 − 3.00000i 0 −18.1258 0 −17.7395 − 5.32058i 0 −9.00000 0
559.4 0 3.00000i 0 −18.1258 0 −17.7395 + 5.32058i 0 −9.00000 0
559.5 0 − 3.00000i 0 −7.23730 0 −18.0963 − 3.94024i 0 −9.00000 0
559.6 0 3.00000i 0 −7.23730 0 −18.0963 + 3.94024i 0 −9.00000 0
559.7 0 − 3.00000i 0 1.47916 0 −7.43403 − 16.9628i 0 −9.00000 0
559.8 0 3.00000i 0 1.47916 0 −7.43403 + 16.9628i 0 −9.00000 0
559.9 0 − 3.00000i 0 −1.47916 0 7.43403 + 16.9628i 0 −9.00000 0
559.10 0 3.00000i 0 −1.47916 0 7.43403 − 16.9628i 0 −9.00000 0
559.11 0 − 3.00000i 0 14.5659 0 11.7275 − 14.3341i 0 −9.00000 0
559.12 0 3.00000i 0 14.5659 0 11.7275 + 14.3341i 0 −9.00000 0
559.13 0 − 3.00000i 0 4.56033 0 17.6399 + 5.64228i 0 −9.00000 0
559.14 0 3.00000i 0 4.56033 0 17.6399 − 5.64228i 0 −9.00000 0
559.15 0 − 3.00000i 0 −7.41253 0 2.46535 − 18.3554i 0 −9.00000 0
559.16 0 3.00000i 0 −7.41253 0 2.46535 + 18.3554i 0 −9.00000 0
559.17 0 − 3.00000i 0 7.41253 0 −2.46535 + 18.3554i 0 −9.00000 0
559.18 0 3.00000i 0 7.41253 0 −2.46535 − 18.3554i 0 −9.00000 0
559.19 0 − 3.00000i 0 −20.8990 0 18.5132 − 0.511483i 0 −9.00000 0
559.20 0 3.00000i 0 −20.8990 0 18.5132 + 0.511483i 0 −9.00000 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.p.a 48
4.b odd 2 1 168.4.p.a ✓ 48
7.b odd 2 1 inner 672.4.p.a 48
8.b even 2 1 168.4.p.a ✓ 48
8.d odd 2 1 inner 672.4.p.a 48
28.d even 2 1 168.4.p.a ✓ 48
56.e even 2 1 inner 672.4.p.a 48
56.h odd 2 1 168.4.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.p.a ✓ 48 4.b odd 2 1
168.4.p.a ✓ 48 8.b even 2 1
168.4.p.a ✓ 48 28.d even 2 1
168.4.p.a ✓ 48 56.h odd 2 1
672.4.p.a 48 1.a even 1 1 trivial
672.4.p.a 48 7.b odd 2 1 inner
672.4.p.a 48 8.d odd 2 1 inner
672.4.p.a 48 56.e even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(672, [\chi])\).