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.
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 |
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 \(S_{2}^{\mathrm{new}}(672, [\chi])\).