Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1088,2,Mod(81,1088)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1088, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1088.81");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1088 = 2^{6} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1088.j (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: | \(8.68772373992\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
Relative dimension: | \(34\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 272) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
81.1 | 0 | − | 3.18433i | 0 | −3.46237 | 0 | −0.548963 | − | 0.548963i | 0 | −7.13997 | 0 | |||||||||||||||
81.2 | 0 | − | 3.03209i | 0 | 1.47873 | 0 | −3.44575 | − | 3.44575i | 0 | −6.19356 | 0 | |||||||||||||||
81.3 | 0 | − | 2.98090i | 0 | 0.726002 | 0 | 0.344548 | + | 0.344548i | 0 | −5.88576 | 0 | |||||||||||||||
81.4 | 0 | − | 2.55478i | 0 | 3.78051 | 0 | 1.48991 | + | 1.48991i | 0 | −3.52692 | 0 | |||||||||||||||
81.5 | 0 | − | 2.16143i | 0 | 1.75971 | 0 | −0.943570 | − | 0.943570i | 0 | −1.67179 | 0 | |||||||||||||||
81.6 | 0 | − | 2.05711i | 0 | −2.40124 | 0 | 2.53680 | + | 2.53680i | 0 | −1.23170 | 0 | |||||||||||||||
81.7 | 0 | − | 2.01369i | 0 | −2.90977 | 0 | −3.02165 | − | 3.02165i | 0 | −1.05494 | 0 | |||||||||||||||
81.8 | 0 | − | 1.97996i | 0 | −3.05104 | 0 | 2.13135 | + | 2.13135i | 0 | −0.920251 | 0 | |||||||||||||||
81.9 | 0 | − | 1.71343i | 0 | −1.98068 | 0 | 2.55435 | + | 2.55435i | 0 | 0.0641544 | 0 | |||||||||||||||
81.10 | 0 | − | 1.70026i | 0 | 2.90401 | 0 | 0.453147 | + | 0.453147i | 0 | 0.109126 | 0 | |||||||||||||||
81.11 | 0 | − | 1.68708i | 0 | 0.660632 | 0 | −0.710367 | − | 0.710367i | 0 | 0.153763 | 0 | |||||||||||||||
81.12 | 0 | − | 1.53523i | 0 | −1.14091 | 0 | −0.381270 | − | 0.381270i | 0 | 0.643071 | 0 | |||||||||||||||
81.13 | 0 | − | 0.572337i | 0 | −1.54731 | 0 | −2.11387 | − | 2.11387i | 0 | 2.67243 | 0 | |||||||||||||||
81.14 | 0 | − | 0.378959i | 0 | 3.16725 | 0 | 0.346423 | + | 0.346423i | 0 | 2.85639 | 0 | |||||||||||||||
81.15 | 0 | − | 0.342255i | 0 | 1.55667 | 0 | 3.34124 | + | 3.34124i | 0 | 2.88286 | 0 | |||||||||||||||
81.16 | 0 | − | 0.304704i | 0 | 0.222425 | 0 | 0.942993 | + | 0.942993i | 0 | 2.90716 | 0 | |||||||||||||||
81.17 | 0 | − | 0.00542376i | 0 | 2.38432 | 0 | −2.50734 | − | 2.50734i | 0 | 2.99997 | 0 | |||||||||||||||
81.18 | 0 | 0.410909i | 0 | −1.10087 | 0 | −1.57400 | − | 1.57400i | 0 | 2.83115 | 0 | ||||||||||||||||
81.19 | 0 | 0.583850i | 0 | −2.97401 | 0 | −1.20406 | − | 1.20406i | 0 | 2.65912 | 0 | ||||||||||||||||
81.20 | 0 | 0.696614i | 0 | 2.41672 | 0 | 3.18472 | + | 3.18472i | 0 | 2.51473 | 0 | ||||||||||||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
272.j | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1088.2.j.a | 68 | |
4.b | odd | 2 | 1 | 272.2.j.a | ✓ | 68 | |
16.e | even | 4 | 1 | 1088.2.s.a | 68 | ||
16.f | odd | 4 | 1 | 272.2.s.a | yes | 68 | |
17.c | even | 4 | 1 | 1088.2.s.a | 68 | ||
68.f | odd | 4 | 1 | 272.2.s.a | yes | 68 | |
272.j | even | 4 | 1 | inner | 1088.2.j.a | 68 | |
272.t | odd | 4 | 1 | 272.2.j.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
272.2.j.a | ✓ | 68 | 4.b | odd | 2 | 1 | |
272.2.j.a | ✓ | 68 | 272.t | odd | 4 | 1 | |
272.2.s.a | yes | 68 | 16.f | odd | 4 | 1 | |
272.2.s.a | yes | 68 | 68.f | odd | 4 | 1 | |
1088.2.j.a | 68 | 1.a | even | 1 | 1 | trivial | |
1088.2.j.a | 68 | 272.j | even | 4 | 1 | inner | |
1088.2.s.a | 68 | 16.e | even | 4 | 1 | ||
1088.2.s.a | 68 | 17.c | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1088, [\chi])\).