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Properties

Label 672.4.bb.a
Level $672$
Weight $4$
Character orbit 672.bb
Analytic conductor $39.649$
Analytic rank $0$
Dimension $96$
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(271,672)]
 
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, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.271");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.bb (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: \(39.6492835239\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 432 q^{9} - 40 q^{11} - 1200 q^{25} - 456 q^{35} + 1616 q^{43} - 360 q^{49} - 336 q^{57} - 4128 q^{59} + 1440 q^{67} - 648 q^{73} - 3888 q^{81} + 104 q^{91} - 720 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1 0 −2.59808 + 1.50000i 0 −4.82508 + 8.35729i 0 14.2799 − 11.7934i 0 4.50000 − 7.79423i 0
271.2 0 −2.59808 + 1.50000i 0 4.82508 − 8.35729i 0 −14.2799 + 11.7934i 0 4.50000 − 7.79423i 0
271.3 0 −2.59808 + 1.50000i 0 −4.08960 + 7.08340i 0 −18.1007 − 3.91988i 0 4.50000 − 7.79423i 0
271.4 0 −2.59808 + 1.50000i 0 4.08960 − 7.08340i 0 18.1007 + 3.91988i 0 4.50000 − 7.79423i 0
271.5 0 −2.59808 + 1.50000i 0 −3.56336 + 6.17193i 0 −16.0311 + 9.27388i 0 4.50000 − 7.79423i 0
271.6 0 −2.59808 + 1.50000i 0 3.56336 − 6.17193i 0 16.0311 − 9.27388i 0 4.50000 − 7.79423i 0
271.7 0 −2.59808 + 1.50000i 0 −1.10134 + 1.90757i 0 −7.13695 + 17.0899i 0 4.50000 − 7.79423i 0
271.8 0 −2.59808 + 1.50000i 0 1.10134 − 1.90757i 0 7.13695 − 17.0899i 0 4.50000 − 7.79423i 0
271.9 0 −2.59808 + 1.50000i 0 −2.37912 + 4.12075i 0 −11.6311 − 14.4124i 0 4.50000 − 7.79423i 0
271.10 0 −2.59808 + 1.50000i 0 2.37912 − 4.12075i 0 11.6311 + 14.4124i 0 4.50000 − 7.79423i 0
271.11 0 −2.59808 + 1.50000i 0 −4.53700 + 7.85831i 0 18.3201 + 2.71568i 0 4.50000 − 7.79423i 0
271.12 0 −2.59808 + 1.50000i 0 4.53700 − 7.85831i 0 −18.3201 − 2.71568i 0 4.50000 − 7.79423i 0
271.13 0 −2.59808 + 1.50000i 0 −6.00810 + 10.4063i 0 −8.17074 + 16.6204i 0 4.50000 − 7.79423i 0
271.14 0 −2.59808 + 1.50000i 0 6.00810 − 10.4063i 0 8.17074 − 16.6204i 0 4.50000 − 7.79423i 0
271.15 0 −2.59808 + 1.50000i 0 −6.01586 + 10.4198i 0 −9.34266 + 15.9911i 0 4.50000 − 7.79423i 0
271.16 0 −2.59808 + 1.50000i 0 6.01586 − 10.4198i 0 9.34266 − 15.9911i 0 4.50000 − 7.79423i 0
271.17 0 −2.59808 + 1.50000i 0 −7.16825 + 12.4158i 0 12.0691 + 14.0477i 0 4.50000 − 7.79423i 0
271.18 0 −2.59808 + 1.50000i 0 7.16825 − 12.4158i 0 −12.0691 − 14.0477i 0 4.50000 − 7.79423i 0
271.19 0 −2.59808 + 1.50000i 0 −7.30307 + 12.6493i 0 0.782311 − 18.5037i 0 4.50000 − 7.79423i 0
271.20 0 −2.59808 + 1.50000i 0 7.30307 − 12.6493i 0 −0.782311 + 18.5037i 0 4.50000 − 7.79423i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.bb.a 96
4.b odd 2 1 168.4.t.a ✓ 96
7.d odd 6 1 inner 672.4.bb.a 96
8.b even 2 1 168.4.t.a ✓ 96
8.d odd 2 1 inner 672.4.bb.a 96
28.f even 6 1 168.4.t.a ✓ 96
56.j odd 6 1 168.4.t.a ✓ 96
56.m even 6 1 inner 672.4.bb.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.t.a ✓ 96 4.b odd 2 1
168.4.t.a ✓ 96 8.b even 2 1
168.4.t.a ✓ 96 28.f even 6 1
168.4.t.a ✓ 96 56.j odd 6 1
672.4.bb.a 96 1.a even 1 1 trivial
672.4.bb.a 96 7.d odd 6 1 inner
672.4.bb.a 96 8.d odd 2 1 inner
672.4.bb.a 96 56.m even 6 1 inner

Hecke kernels

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