Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [672,4,Mod(31,672)]
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, 0, 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("672.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 672 = 2^{5} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 672.bl (of order \(6\), degree \(2\), 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: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | 1.50000 | − | 2.59808i | 0 | −18.1004 | + | 10.4503i | 0 | 1.74956 | + | 18.4374i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.2 | 0 | 1.50000 | − | 2.59808i | 0 | −16.1208 | + | 9.30737i | 0 | −4.52171 | − | 17.9598i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.3 | 0 | 1.50000 | − | 2.59808i | 0 | −14.0388 | + | 8.10531i | 0 | −18.0605 | + | 4.10082i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.4 | 0 | 1.50000 | − | 2.59808i | 0 | −13.6079 | + | 7.85650i | 0 | 12.9788 | + | 13.2118i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.5 | 0 | 1.50000 | − | 2.59808i | 0 | −12.1968 | + | 7.04181i | 0 | 18.4850 | − | 1.14200i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.6 | 0 | 1.50000 | − | 2.59808i | 0 | −11.2973 | + | 6.52253i | 0 | −14.5303 | − | 11.4834i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.7 | 0 | 1.50000 | − | 2.59808i | 0 | −8.26531 | + | 4.77198i | 0 | −16.7811 | + | 7.83539i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.8 | 0 | 1.50000 | − | 2.59808i | 0 | −7.66468 | + | 4.42521i | 0 | 11.0120 | − | 14.8908i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.9 | 0 | 1.50000 | − | 2.59808i | 0 | −4.30657 | + | 2.48640i | 0 | −9.80887 | − | 15.7094i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.10 | 0 | 1.50000 | − | 2.59808i | 0 | −1.65231 | + | 0.953964i | 0 | 17.8528 | − | 4.92720i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.11 | 0 | 1.50000 | − | 2.59808i | 0 | −1.40368 | + | 0.810413i | 0 | 8.73076 | + | 16.3332i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.12 | 0 | 1.50000 | − | 2.59808i | 0 | 0.507989 | − | 0.293287i | 0 | −9.34878 | + | 15.9875i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.13 | 0 | 1.50000 | − | 2.59808i | 0 | 0.775884 | − | 0.447957i | 0 | −6.80305 | + | 17.2255i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.14 | 0 | 1.50000 | − | 2.59808i | 0 | 0.935302 | − | 0.539997i | 0 | 15.2722 | − | 10.4766i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.15 | 0 | 1.50000 | − | 2.59808i | 0 | 3.31399 | − | 1.91333i | 0 | 12.8548 | + | 13.3324i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.16 | 0 | 1.50000 | − | 2.59808i | 0 | 6.83580 | − | 3.94665i | 0 | −4.47632 | − | 17.9712i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.17 | 0 | 1.50000 | − | 2.59808i | 0 | 7.11284 | − | 4.10660i | 0 | −18.3102 | − | 2.78176i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.18 | 0 | 1.50000 | − | 2.59808i | 0 | 7.80105 | − | 4.50394i | 0 | −15.6161 | − | 9.95675i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.19 | 0 | 1.50000 | − | 2.59808i | 0 | 7.93810 | − | 4.58307i | 0 | 13.2705 | − | 12.9187i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
31.20 | 0 | 1.50000 | − | 2.59808i | 0 | 8.75646 | − | 5.05555i | 0 | −3.07975 | + | 18.2624i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
28.f | 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.bl.b | yes | 48 |
4.b | odd | 2 | 1 | 672.4.bl.a | ✓ | 48 | |
7.d | odd | 6 | 1 | 672.4.bl.a | ✓ | 48 | |
28.f | even | 6 | 1 | inner | 672.4.bl.b | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
672.4.bl.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
672.4.bl.a | ✓ | 48 | 7.d | odd | 6 | 1 | |
672.4.bl.b | yes | 48 | 1.a | even | 1 | 1 | trivial |
672.4.bl.b | yes | 48 | 28.f | even | 6 | 1 | inner |