Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1020,3,Mod(217,1020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1020, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1020.217");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1020.bm (of order \(4\), 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: | \(27.7929869648\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(36\) over \(\Q(i)\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
217.1 | 0 | −1.73205 | 0 | 1.12936 | + | 4.87079i | 0 | −12.6614 | 0 | 3.00000 | 0 | ||||||||||||||||
217.2 | 0 | −1.73205 | 0 | 0.197503 | − | 4.99610i | 0 | 11.2230 | 0 | 3.00000 | 0 | ||||||||||||||||
217.3 | 0 | −1.73205 | 0 | 3.95434 | + | 3.05994i | 0 | 9.26802 | 0 | 3.00000 | 0 | ||||||||||||||||
217.4 | 0 | −1.73205 | 0 | −1.91114 | + | 4.62034i | 0 | 8.52539 | 0 | 3.00000 | 0 | ||||||||||||||||
217.5 | 0 | −1.73205 | 0 | 3.36666 | − | 3.69670i | 0 | −7.60049 | 0 | 3.00000 | 0 | ||||||||||||||||
217.6 | 0 | −1.73205 | 0 | 0.207571 | − | 4.99569i | 0 | −7.23740 | 0 | 3.00000 | 0 | ||||||||||||||||
217.7 | 0 | −1.73205 | 0 | 4.98759 | + | 0.351991i | 0 | 6.16656 | 0 | 3.00000 | 0 | ||||||||||||||||
217.8 | 0 | −1.73205 | 0 | −1.48152 | + | 4.77547i | 0 | −6.51708 | 0 | 3.00000 | 0 | ||||||||||||||||
217.9 | 0 | −1.73205 | 0 | 2.10409 | − | 4.53573i | 0 | 6.12467 | 0 | 3.00000 | 0 | ||||||||||||||||
217.10 | 0 | −1.73205 | 0 | 1.79402 | + | 4.66706i | 0 | 3.77059 | 0 | 3.00000 | 0 | ||||||||||||||||
217.11 | 0 | −1.73205 | 0 | −2.43367 | − | 4.36775i | 0 | 1.56608 | 0 | 3.00000 | 0 | ||||||||||||||||
217.12 | 0 | −1.73205 | 0 | −4.74243 | + | 1.58410i | 0 | 0.985444 | 0 | 3.00000 | 0 | ||||||||||||||||
217.13 | 0 | −1.73205 | 0 | −4.23232 | − | 2.66222i | 0 | 1.92496 | 0 | 3.00000 | 0 | ||||||||||||||||
217.14 | 0 | −1.73205 | 0 | −4.30375 | − | 2.54514i | 0 | −4.70111 | 0 | 3.00000 | 0 | ||||||||||||||||
217.15 | 0 | −1.73205 | 0 | 4.84913 | − | 1.21901i | 0 | −4.97931 | 0 | 3.00000 | 0 | ||||||||||||||||
217.16 | 0 | −1.73205 | 0 | 4.95782 | + | 0.648100i | 0 | −6.22839 | 0 | 3.00000 | 0 | ||||||||||||||||
217.17 | 0 | −1.73205 | 0 | −3.87944 | + | 3.15435i | 0 | 4.49402 | 0 | 3.00000 | 0 | ||||||||||||||||
217.18 | 0 | −1.73205 | 0 | −4.83174 | + | 1.28619i | 0 | −10.1235 | 0 | 3.00000 | 0 | ||||||||||||||||
217.19 | 0 | 1.73205 | 0 | −4.13313 | − | 2.81375i | 0 | 12.4732 | 0 | 3.00000 | 0 | ||||||||||||||||
217.20 | 0 | 1.73205 | 0 | −3.15569 | + | 3.87835i | 0 | 11.5811 | 0 | 3.00000 | 0 | ||||||||||||||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
85.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1020.3.bm.a | yes | 72 |
5.c | odd | 4 | 1 | 1020.3.t.a | ✓ | 72 | |
17.c | even | 4 | 1 | 1020.3.t.a | ✓ | 72 | |
85.i | odd | 4 | 1 | inner | 1020.3.bm.a | yes | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1020.3.t.a | ✓ | 72 | 5.c | odd | 4 | 1 | |
1020.3.t.a | ✓ | 72 | 17.c | even | 4 | 1 | |
1020.3.bm.a | yes | 72 | 1.a | even | 1 | 1 | trivial |
1020.3.bm.a | yes | 72 | 85.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1020, [\chi])\).