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Properties

Label 672.2.bd.a
Level $672$
Weight $2$
Character orbit 672.bd
Analytic conductor $5.366$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(431,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bd (of order \(6\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{3} - 2 q^{9} + 4 q^{19} - 16 q^{25} + 8 q^{27} - 14 q^{33} + 16 q^{43} - 16 q^{49} + 34 q^{51} + 4 q^{57} + 36 q^{67} + 4 q^{73} - 10 q^{81} - 72 q^{91} - 32 q^{97} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1 0 −1.72608 − 0.143748i 0 −1.77783 − 3.07929i 0 −0.793456 − 2.52397i 0 2.95867 + 0.496241i 0
431.2 0 −1.72608 − 0.143748i 0 1.77783 + 3.07929i 0 0.793456 + 2.52397i 0 2.95867 + 0.496241i 0
431.3 0 −1.66017 + 0.493786i 0 −1.09212 − 1.89161i 0 −0.451847 + 2.60688i 0 2.51235 − 1.63954i 0
431.4 0 −1.66017 + 0.493786i 0 1.09212 + 1.89161i 0 0.451847 − 2.60688i 0 2.51235 − 1.63954i 0
431.5 0 −1.28463 − 1.16178i 0 −0.646402 − 1.11960i 0 −2.42742 + 1.05244i 0 0.300539 + 2.98491i 0
431.6 0 −1.28463 − 1.16178i 0 0.646402 + 1.11960i 0 2.42742 − 1.05244i 0 0.300539 + 2.98491i 0
431.7 0 −1.03943 − 1.38549i 0 −0.692094 − 1.19874i 0 2.08863 + 1.62408i 0 −0.839162 + 2.88024i 0
431.8 0 −1.03943 − 1.38549i 0 0.692094 + 1.19874i 0 −2.08863 − 1.62408i 0 −0.839162 + 2.88024i 0
431.9 0 −0.996948 + 1.41637i 0 −1.00081 − 1.73346i 0 −2.50986 − 0.837013i 0 −1.01219 − 2.82409i 0
431.10 0 −0.996948 + 1.41637i 0 1.00081 + 1.73346i 0 2.50986 + 0.837013i 0 −1.01219 − 2.82409i 0
431.11 0 −0.728136 + 1.57157i 0 −1.00081 − 1.73346i 0 2.50986 + 0.837013i 0 −1.93964 − 2.28863i 0
431.12 0 −0.728136 + 1.57157i 0 1.00081 + 1.73346i 0 −2.50986 − 0.837013i 0 −1.93964 − 2.28863i 0
431.13 0 0.0929747 − 1.72955i 0 −0.187787 − 0.325257i 0 0.198565 − 2.63829i 0 −2.98271 − 0.321609i 0
431.14 0 0.0929747 − 1.72955i 0 0.187787 + 0.325257i 0 −0.198565 + 2.63829i 0 −2.98271 − 0.321609i 0
431.15 0 0.316681 − 1.70285i 0 −1.86088 − 3.22314i 0 2.46429 − 0.962962i 0 −2.79943 − 1.07852i 0
431.16 0 0.316681 − 1.70285i 0 1.86088 + 3.22314i 0 −2.46429 + 0.962962i 0 −2.79943 − 1.07852i 0
431.17 0 0.402456 + 1.68465i 0 −1.09212 − 1.89161i 0 0.451847 − 2.60688i 0 −2.67606 + 1.35599i 0
431.18 0 0.402456 + 1.68465i 0 1.09212 + 1.89161i 0 −0.451847 + 2.60688i 0 −2.67606 + 1.35599i 0
431.19 0 0.987527 + 1.42295i 0 −1.77783 − 3.07929i 0 0.793456 + 2.52397i 0 −1.04958 + 2.81041i 0
431.20 0 0.987527 + 1.42295i 0 1.77783 + 3.07929i 0 −0.793456 − 2.52397i 0 −1.04958 + 2.81041i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
8.d odd 2 1 inner
21.h odd 6 1 inner
24.f even 2 1 inner
56.k odd 6 1 inner
168.v even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bd.a 56
3.b odd 2 1 inner 672.2.bd.a 56
4.b odd 2 1 168.2.v.a ✓ 56
7.c even 3 1 inner 672.2.bd.a 56
8.b even 2 1 168.2.v.a ✓ 56
8.d odd 2 1 inner 672.2.bd.a 56
12.b even 2 1 168.2.v.a ✓ 56
21.h odd 6 1 inner 672.2.bd.a 56
24.f even 2 1 inner 672.2.bd.a 56
24.h odd 2 1 168.2.v.a ✓ 56
28.g odd 6 1 168.2.v.a ✓ 56
56.k odd 6 1 inner 672.2.bd.a 56
56.p even 6 1 168.2.v.a ✓ 56
84.n even 6 1 168.2.v.a ✓ 56
168.s odd 6 1 168.2.v.a ✓ 56
168.v even 6 1 inner 672.2.bd.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.v.a ✓ 56 4.b odd 2 1
168.2.v.a ✓ 56 8.b even 2 1
168.2.v.a ✓ 56 12.b even 2 1
168.2.v.a ✓ 56 24.h odd 2 1
168.2.v.a ✓ 56 28.g odd 6 1
168.2.v.a ✓ 56 56.p even 6 1
168.2.v.a ✓ 56 84.n even 6 1
168.2.v.a ✓ 56 168.s odd 6 1
672.2.bd.a 56 1.a even 1 1 trivial
672.2.bd.a 56 3.b odd 2 1 inner
672.2.bd.a 56 7.c even 3 1 inner
672.2.bd.a 56 8.d odd 2 1 inner
672.2.bd.a 56 21.h odd 6 1 inner
672.2.bd.a 56 24.f even 2 1 inner
672.2.bd.a 56 56.k odd 6 1 inner
672.2.bd.a 56 168.v even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).