Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1008,4,Mod(431,1008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.431");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.cq (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: | \(59.4739252858\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
431.1 | 0 | 0 | 0 | −18.6889 | + | 10.7900i | 0 | 13.4076 | + | 12.7764i | 0 | 0 | 0 | ||||||||||||||
431.2 | 0 | 0 | 0 | −15.2715 | + | 8.81701i | 0 | 7.72385 | − | 16.8328i | 0 | 0 | 0 | ||||||||||||||
431.3 | 0 | 0 | 0 | −14.3834 | + | 8.30425i | 0 | −18.2641 | − | 3.06985i | 0 | 0 | 0 | ||||||||||||||
431.4 | 0 | 0 | 0 | −8.33504 | + | 4.81224i | 0 | −15.5688 | − | 10.0306i | 0 | 0 | 0 | ||||||||||||||
431.5 | 0 | 0 | 0 | −6.99528 | + | 4.03872i | 0 | 11.5633 | − | 14.4669i | 0 | 0 | 0 | ||||||||||||||
431.6 | 0 | 0 | 0 | −5.40790 | + | 3.12225i | 0 | 3.20776 | + | 18.2403i | 0 | 0 | 0 | ||||||||||||||
431.7 | 0 | 0 | 0 | −4.97239 | + | 2.87081i | 0 | 18.5085 | + | 0.660448i | 0 | 0 | 0 | ||||||||||||||
431.8 | 0 | 0 | 0 | −1.11698 | + | 0.644890i | 0 | −11.5781 | + | 14.4550i | 0 | 0 | 0 | ||||||||||||||
431.9 | 0 | 0 | 0 | 1.11698 | − | 0.644890i | 0 | −11.5781 | + | 14.4550i | 0 | 0 | 0 | ||||||||||||||
431.10 | 0 | 0 | 0 | 4.97239 | − | 2.87081i | 0 | 18.5085 | + | 0.660448i | 0 | 0 | 0 | ||||||||||||||
431.11 | 0 | 0 | 0 | 5.40790 | − | 3.12225i | 0 | 3.20776 | + | 18.2403i | 0 | 0 | 0 | ||||||||||||||
431.12 | 0 | 0 | 0 | 6.99528 | − | 4.03872i | 0 | 11.5633 | − | 14.4669i | 0 | 0 | 0 | ||||||||||||||
431.13 | 0 | 0 | 0 | 8.33504 | − | 4.81224i | 0 | −15.5688 | − | 10.0306i | 0 | 0 | 0 | ||||||||||||||
431.14 | 0 | 0 | 0 | 14.3834 | − | 8.30425i | 0 | −18.2641 | − | 3.06985i | 0 | 0 | 0 | ||||||||||||||
431.15 | 0 | 0 | 0 | 15.2715 | − | 8.81701i | 0 | 7.72385 | − | 16.8328i | 0 | 0 | 0 | ||||||||||||||
431.16 | 0 | 0 | 0 | 18.6889 | − | 10.7900i | 0 | 13.4076 | + | 12.7764i | 0 | 0 | 0 | ||||||||||||||
863.1 | 0 | 0 | 0 | −18.6889 | − | 10.7900i | 0 | 13.4076 | − | 12.7764i | 0 | 0 | 0 | ||||||||||||||
863.2 | 0 | 0 | 0 | −15.2715 | − | 8.81701i | 0 | 7.72385 | + | 16.8328i | 0 | 0 | 0 | ||||||||||||||
863.3 | 0 | 0 | 0 | −14.3834 | − | 8.30425i | 0 | −18.2641 | + | 3.06985i | 0 | 0 | 0 | ||||||||||||||
863.4 | 0 | 0 | 0 | −8.33504 | − | 4.81224i | 0 | −15.5688 | + | 10.0306i | 0 | 0 | 0 | ||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
28.g | odd | 6 | 1 | inner |
84.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1008.4.cq.c | yes | 32 |
3.b | odd | 2 | 1 | inner | 1008.4.cq.c | yes | 32 |
4.b | odd | 2 | 1 | 1008.4.cq.a | ✓ | 32 | |
7.c | even | 3 | 1 | 1008.4.cq.a | ✓ | 32 | |
12.b | even | 2 | 1 | 1008.4.cq.a | ✓ | 32 | |
21.h | odd | 6 | 1 | 1008.4.cq.a | ✓ | 32 | |
28.g | odd | 6 | 1 | inner | 1008.4.cq.c | yes | 32 |
84.n | even | 6 | 1 | inner | 1008.4.cq.c | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1008.4.cq.a | ✓ | 32 | 4.b | odd | 2 | 1 | |
1008.4.cq.a | ✓ | 32 | 7.c | even | 3 | 1 | |
1008.4.cq.a | ✓ | 32 | 12.b | even | 2 | 1 | |
1008.4.cq.a | ✓ | 32 | 21.h | odd | 6 | 1 | |
1008.4.cq.c | yes | 32 | 1.a | even | 1 | 1 | trivial |
1008.4.cq.c | yes | 32 | 3.b | odd | 2 | 1 | inner |
1008.4.cq.c | yes | 32 | 28.g | odd | 6 | 1 | inner |
1008.4.cq.c | yes | 32 | 84.n | even | 6 | 1 | inner |