Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,4,Mod(863,864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("864.863");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 864.c (of order \(2\), degree \(1\), 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: | \(50.9776502450\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
863.1 | 0 | 0 | 0 | − | 18.7846i | 0 | − | 30.8989i | 0 | 0 | 0 | ||||||||||||||||
863.2 | 0 | 0 | 0 | − | 18.7846i | 0 | 30.8989i | 0 | 0 | 0 | |||||||||||||||||
863.3 | 0 | 0 | 0 | − | 18.0256i | 0 | − | 1.63267i | 0 | 0 | 0 | ||||||||||||||||
863.4 | 0 | 0 | 0 | − | 18.0256i | 0 | 1.63267i | 0 | 0 | 0 | |||||||||||||||||
863.5 | 0 | 0 | 0 | − | 11.4141i | 0 | − | 5.04782i | 0 | 0 | 0 | ||||||||||||||||
863.6 | 0 | 0 | 0 | − | 11.4141i | 0 | 5.04782i | 0 | 0 | 0 | |||||||||||||||||
863.7 | 0 | 0 | 0 | − | 8.99651i | 0 | − | 35.3962i | 0 | 0 | 0 | ||||||||||||||||
863.8 | 0 | 0 | 0 | − | 8.99651i | 0 | 35.3962i | 0 | 0 | 0 | |||||||||||||||||
863.9 | 0 | 0 | 0 | − | 5.95128i | 0 | − | 12.0727i | 0 | 0 | 0 | ||||||||||||||||
863.10 | 0 | 0 | 0 | − | 5.95128i | 0 | 12.0727i | 0 | 0 | 0 | |||||||||||||||||
863.11 | 0 | 0 | 0 | − | 4.19264i | 0 | − | 15.8895i | 0 | 0 | 0 | ||||||||||||||||
863.12 | 0 | 0 | 0 | − | 4.19264i | 0 | 15.8895i | 0 | 0 | 0 | |||||||||||||||||
863.13 | 0 | 0 | 0 | 4.19264i | 0 | − | 15.8895i | 0 | 0 | 0 | |||||||||||||||||
863.14 | 0 | 0 | 0 | 4.19264i | 0 | 15.8895i | 0 | 0 | 0 | ||||||||||||||||||
863.15 | 0 | 0 | 0 | 5.95128i | 0 | − | 12.0727i | 0 | 0 | 0 | |||||||||||||||||
863.16 | 0 | 0 | 0 | 5.95128i | 0 | 12.0727i | 0 | 0 | 0 | ||||||||||||||||||
863.17 | 0 | 0 | 0 | 8.99651i | 0 | − | 35.3962i | 0 | 0 | 0 | |||||||||||||||||
863.18 | 0 | 0 | 0 | 8.99651i | 0 | 35.3962i | 0 | 0 | 0 | ||||||||||||||||||
863.19 | 0 | 0 | 0 | 11.4141i | 0 | − | 5.04782i | 0 | 0 | 0 | |||||||||||||||||
863.20 | 0 | 0 | 0 | 11.4141i | 0 | 5.04782i | 0 | 0 | 0 | ||||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 864.4.c.b | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 864.4.c.b | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 864.4.c.b | ✓ | 24 |
8.b | even | 2 | 1 | 1728.4.c.k | 24 | ||
8.d | odd | 2 | 1 | 1728.4.c.k | 24 | ||
12.b | even | 2 | 1 | inner | 864.4.c.b | ✓ | 24 |
24.f | even | 2 | 1 | 1728.4.c.k | 24 | ||
24.h | odd | 2 | 1 | 1728.4.c.k | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
864.4.c.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
864.4.c.b | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
864.4.c.b | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
864.4.c.b | ✓ | 24 | 12.b | even | 2 | 1 | inner |
1728.4.c.k | 24 | 8.b | even | 2 | 1 | ||
1728.4.c.k | 24 | 8.d | odd | 2 | 1 | ||
1728.4.c.k | 24 | 24.f | even | 2 | 1 | ||
1728.4.c.k | 24 | 24.h | odd | 2 | 1 |