Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,3,Mod(79,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.79");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.q (of order \(4\), 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: | \(26.1581053786\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 240) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
79.1 | 0 | −1.22474 | + | 1.22474i | 0 | −4.72148 | + | 1.64548i | 0 | 11.4402i | 0 | − | 3.00000i | 0 | |||||||||||||
79.2 | 0 | −1.22474 | + | 1.22474i | 0 | 4.59527 | + | 1.97065i | 0 | 11.1879i | 0 | − | 3.00000i | 0 | |||||||||||||
79.3 | 0 | −1.22474 | + | 1.22474i | 0 | −0.515045 | − | 4.97340i | 0 | 10.8255i | 0 | − | 3.00000i | 0 | |||||||||||||
79.4 | 0 | −1.22474 | + | 1.22474i | 0 | −2.87557 | + | 4.09037i | 0 | 10.3500i | 0 | − | 3.00000i | 0 | |||||||||||||
79.5 | 0 | −1.22474 | + | 1.22474i | 0 | −0.367849 | − | 4.98645i | 0 | 9.62585i | 0 | − | 3.00000i | 0 | |||||||||||||
79.6 | 0 | −1.22474 | + | 1.22474i | 0 | 4.99865 | − | 0.116331i | 0 | 5.97898i | 0 | − | 3.00000i | 0 | |||||||||||||
79.7 | 0 | −1.22474 | + | 1.22474i | 0 | 4.17527 | − | 2.75084i | 0 | 4.98546i | 0 | − | 3.00000i | 0 | |||||||||||||
79.8 | 0 | −1.22474 | + | 1.22474i | 0 | −2.26868 | + | 4.45568i | 0 | 4.16156i | 0 | − | 3.00000i | 0 | |||||||||||||
79.9 | 0 | −1.22474 | + | 1.22474i | 0 | −4.10833 | − | 2.84985i | 0 | 3.77669i | 0 | − | 3.00000i | 0 | |||||||||||||
79.10 | 0 | −1.22474 | + | 1.22474i | 0 | −4.99932 | + | 0.0824727i | 0 | − | 0.574521i | 0 | − | 3.00000i | 0 | ||||||||||||
79.11 | 0 | −1.22474 | + | 1.22474i | 0 | −0.437516 | + | 4.98082i | 0 | 0.666894i | 0 | − | 3.00000i | 0 | |||||||||||||
79.12 | 0 | −1.22474 | + | 1.22474i | 0 | 2.85432 | + | 4.10522i | 0 | 1.41651i | 0 | − | 3.00000i | 0 | |||||||||||||
79.13 | 0 | −1.22474 | + | 1.22474i | 0 | 3.84224 | − | 3.19957i | 0 | − | 1.61537i | 0 | − | 3.00000i | 0 | ||||||||||||
79.14 | 0 | −1.22474 | + | 1.22474i | 0 | −1.95649 | − | 4.60132i | 0 | − | 1.97864i | 0 | − | 3.00000i | 0 | ||||||||||||
79.15 | 0 | −1.22474 | + | 1.22474i | 0 | 4.31221 | + | 2.53078i | 0 | 2.25392i | 0 | − | 3.00000i | 0 | |||||||||||||
79.16 | 0 | −1.22474 | + | 1.22474i | 0 | −4.89181 | + | 1.03448i | 0 | − | 4.36400i | 0 | − | 3.00000i | 0 | ||||||||||||
79.17 | 0 | −1.22474 | + | 1.22474i | 0 | −3.86518 | − | 3.17182i | 0 | − | 4.51781i | 0 | − | 3.00000i | 0 | ||||||||||||
79.18 | 0 | −1.22474 | + | 1.22474i | 0 | 1.19637 | − | 4.85476i | 0 | − | 5.60935i | 0 | − | 3.00000i | 0 | ||||||||||||
79.19 | 0 | −1.22474 | + | 1.22474i | 0 | −3.11729 | + | 3.90928i | 0 | − | 6.76702i | 0 | − | 3.00000i | 0 | ||||||||||||
79.20 | 0 | −1.22474 | + | 1.22474i | 0 | −0.338767 | + | 4.98851i | 0 | − | 7.33616i | 0 | − | 3.00000i | 0 | ||||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
80.k | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.3.q.a | 96 | |
4.b | odd | 2 | 1 | 240.3.q.a | ✓ | 96 | |
5.b | even | 2 | 1 | inner | 960.3.q.a | 96 | |
16.e | even | 4 | 1 | 240.3.q.a | ✓ | 96 | |
16.f | odd | 4 | 1 | inner | 960.3.q.a | 96 | |
20.d | odd | 2 | 1 | 240.3.q.a | ✓ | 96 | |
80.k | odd | 4 | 1 | inner | 960.3.q.a | 96 | |
80.q | even | 4 | 1 | 240.3.q.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.3.q.a | ✓ | 96 | 4.b | odd | 2 | 1 | |
240.3.q.a | ✓ | 96 | 16.e | even | 4 | 1 | |
240.3.q.a | ✓ | 96 | 20.d | odd | 2 | 1 | |
240.3.q.a | ✓ | 96 | 80.q | even | 4 | 1 | |
960.3.q.a | 96 | 1.a | even | 1 | 1 | trivial | |
960.3.q.a | 96 | 5.b | even | 2 | 1 | inner | |
960.3.q.a | 96 | 16.f | odd | 4 | 1 | inner | |
960.3.q.a | 96 | 80.k | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(960, [\chi])\).