Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1344,4,Mod(223,1344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("1344.223");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1344.p (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: | \(79.2985670477\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
223.1 | 0 | − | 3.00000i | 0 | −12.5168 | 0 | 18.2936 | + | 2.88837i | 0 | −9.00000 | 0 | |||||||||||||||
223.2 | 0 | 3.00000i | 0 | −12.5168 | 0 | 18.2936 | − | 2.88837i | 0 | −9.00000 | 0 | ||||||||||||||||
223.3 | 0 | − | 3.00000i | 0 | −12.8225 | 0 | 6.32908 | − | 17.4053i | 0 | −9.00000 | 0 | |||||||||||||||
223.4 | 0 | 3.00000i | 0 | −12.8225 | 0 | 6.32908 | + | 17.4053i | 0 | −9.00000 | 0 | ||||||||||||||||
223.5 | 0 | − | 3.00000i | 0 | 20.7003 | 0 | −11.8166 | − | 14.2607i | 0 | −9.00000 | 0 | |||||||||||||||
223.6 | 0 | 3.00000i | 0 | 20.7003 | 0 | −11.8166 | + | 14.2607i | 0 | −9.00000 | 0 | ||||||||||||||||
223.7 | 0 | − | 3.00000i | 0 | −12.5168 | 0 | −18.2936 | + | 2.88837i | 0 | −9.00000 | 0 | |||||||||||||||
223.8 | 0 | 3.00000i | 0 | −12.5168 | 0 | −18.2936 | − | 2.88837i | 0 | −9.00000 | 0 | ||||||||||||||||
223.9 | 0 | − | 3.00000i | 0 | 20.7003 | 0 | 11.8166 | − | 14.2607i | 0 | −9.00000 | 0 | |||||||||||||||
223.10 | 0 | 3.00000i | 0 | 20.7003 | 0 | 11.8166 | + | 14.2607i | 0 | −9.00000 | 0 | ||||||||||||||||
223.11 | 0 | − | 3.00000i | 0 | −12.8225 | 0 | −6.32908 | − | 17.4053i | 0 | −9.00000 | 0 | |||||||||||||||
223.12 | 0 | 3.00000i | 0 | −12.8225 | 0 | −6.32908 | + | 17.4053i | 0 | −9.00000 | 0 | ||||||||||||||||
223.13 | 0 | − | 3.00000i | 0 | −15.4018 | 0 | −6.29285 | + | 17.4184i | 0 | −9.00000 | 0 | |||||||||||||||
223.14 | 0 | 3.00000i | 0 | −15.4018 | 0 | −6.29285 | − | 17.4184i | 0 | −9.00000 | 0 | ||||||||||||||||
223.15 | 0 | − | 3.00000i | 0 | 1.64166 | 0 | 15.2747 | + | 10.4729i | 0 | −9.00000 | 0 | |||||||||||||||
223.16 | 0 | 3.00000i | 0 | 1.64166 | 0 | 15.2747 | − | 10.4729i | 0 | −9.00000 | 0 | ||||||||||||||||
223.17 | 0 | − | 3.00000i | 0 | 5.86484 | 0 | 18.3297 | − | 2.64978i | 0 | −9.00000 | 0 | |||||||||||||||
223.18 | 0 | 3.00000i | 0 | 5.86484 | 0 | 18.3297 | + | 2.64978i | 0 | −9.00000 | 0 | ||||||||||||||||
223.19 | 0 | − | 3.00000i | 0 | −3.15485 | 0 | 6.11026 | − | 17.4833i | 0 | −9.00000 | 0 | |||||||||||||||
223.20 | 0 | 3.00000i | 0 | −3.15485 | 0 | 6.11026 | + | 17.4833i | 0 | −9.00000 | 0 | ||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
56.e | even | 2 | 1 | inner |
56.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1344.4.p.c | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 1344.4.p.c | ✓ | 32 |
7.b | odd | 2 | 1 | 1344.4.p.d | yes | 32 | |
8.b | even | 2 | 1 | 1344.4.p.d | yes | 32 | |
8.d | odd | 2 | 1 | 1344.4.p.d | yes | 32 | |
28.d | even | 2 | 1 | 1344.4.p.d | yes | 32 | |
56.e | even | 2 | 1 | inner | 1344.4.p.c | ✓ | 32 |
56.h | odd | 2 | 1 | inner | 1344.4.p.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1344.4.p.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
1344.4.p.c | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
1344.4.p.c | ✓ | 32 | 56.e | even | 2 | 1 | inner |
1344.4.p.c | ✓ | 32 | 56.h | odd | 2 | 1 | inner |
1344.4.p.d | yes | 32 | 7.b | odd | 2 | 1 | |
1344.4.p.d | yes | 32 | 8.b | even | 2 | 1 | |
1344.4.p.d | yes | 32 | 8.d | odd | 2 | 1 | |
1344.4.p.d | yes | 32 | 28.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 640T_{5}^{6} - 1728T_{5}^{5} + 116464T_{5}^{4} + 458400T_{5}^{3} - 4875024T_{5}^{2} - 8581632T_{5} + 24385536 \) acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\).