Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,2,Mod(19,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 810.p (of order \(18\), degree \(6\), 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: | \(6.46788256372\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{18})\) |
Twist minimal: | no (minimal twist has level 270) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | −2.12627 | − | 0.692069i | 0 | −1.94767 | + | 2.32114i | 0.866025 | + | 0.500000i | 0 | 0.0768969 | + | 2.23475i | ||||||
19.2 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | −2.00523 | + | 0.989469i | 0 | 0.650202 | − | 0.774881i | 0.866025 | + | 0.500000i | 0 | 1.61563 | + | 1.54588i | ||||||
19.3 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | −1.27128 | − | 1.83952i | 0 | 2.50497 | − | 2.98531i | 0.866025 | + | 0.500000i | 0 | −1.29378 | + | 1.82377i | ||||||
19.4 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | −0.326956 | + | 2.21204i | 0 | 2.21166 | − | 2.63576i | 0.866025 | + | 0.500000i | 0 | 2.19046 | − | 0.449323i | ||||||
19.5 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | −0.298468 | + | 2.21606i | 0 | −2.57922 | + | 3.07380i | 0.866025 | + | 0.500000i | 0 | 2.18450 | − | 0.477468i | ||||||
19.6 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | 1.09596 | + | 1.94907i | 0 | 0.582960 | − | 0.694744i | 0.866025 | + | 0.500000i | 0 | 1.45668 | − | 1.69649i | ||||||
19.7 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | 1.17383 | − | 1.90318i | 0 | −0.974601 | + | 1.16148i | 0.866025 | + | 0.500000i | 0 | −2.18988 | − | 0.452116i | ||||||
19.8 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | 1.78992 | − | 1.34021i | 0 | −1.09389 | + | 1.30365i | 0.866025 | + | 0.500000i | 0 | −1.87158 | − | 1.22360i | ||||||
19.9 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | 2.23453 | − | 0.0828267i | 0 | −1.70233 | + | 2.02876i | 0.866025 | + | 0.500000i | 0 | −0.842087 | − | 2.07145i | ||||||
19.10 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | −2.14036 | − | 0.647200i | 0 | 1.09389 | − | 1.30365i | −0.866025 | − | 0.500000i | 0 | −0.123877 | − | 2.23263i | ||||||
19.11 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | −2.12810 | + | 0.686424i | 0 | 1.70233 | − | 2.02876i | −0.866025 | − | 0.500000i | 0 | −1.37288 | − | 1.76499i | ||||||
19.12 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | −1.75397 | − | 1.38693i | 0 | 0.974601 | − | 1.16148i | −0.866025 | − | 0.500000i | 0 | 0.703398 | − | 2.12255i | ||||||
19.13 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | −0.363249 | + | 2.20637i | 0 | −0.582960 | + | 0.694744i | −0.866025 | − | 0.500000i | 0 | −2.19754 | + | 0.413280i | ||||||
19.14 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | 0.565460 | − | 2.16339i | 0 | −2.50497 | + | 2.98531i | −0.866025 | − | 0.500000i | 0 | 2.22632 | − | 0.208565i | ||||||
19.15 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | 1.03841 | + | 1.98033i | 0 | 2.57922 | − | 3.07380i | −0.866025 | − | 0.500000i | 0 | −1.50575 | + | 1.65310i | ||||||
19.16 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | 1.06380 | + | 1.96681i | 0 | −2.21166 | + | 2.63576i | −0.866025 | − | 0.500000i | 0 | −1.48435 | + | 1.67233i | ||||||
19.17 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | 1.76134 | − | 1.37756i | 0 | 1.94767 | − | 2.32114i | −0.866025 | − | 0.500000i | 0 | 1.89690 | + | 1.18397i | ||||||
19.18 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | 2.22272 | + | 0.243967i | 0 | −0.650202 | + | 0.774881i | −0.866025 | − | 0.500000i | 0 | 0.530960 | + | 2.17211i | ||||||
199.1 | −0.642788 | − | 0.766044i | 0 | −0.173648 | + | 0.984808i | −2.21219 | + | 0.325886i | 0 | −0.531122 | + | 0.0936511i | 0.866025 | − | 0.500000i | 0 | 1.67161 | + | 1.48516i | ||||||
199.2 | −0.642788 | − | 0.766044i | 0 | −0.173648 | + | 0.984808i | −1.96492 | − | 1.06728i | 0 | 4.83804 | − | 0.853077i | 0.866025 | − | 0.500000i | 0 | 0.445446 | + | 2.19125i | ||||||
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
27.e | even | 9 | 1 | inner |
135.p | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 810.2.p.a | 108 | |
3.b | odd | 2 | 1 | 270.2.p.a | ✓ | 108 | |
5.b | even | 2 | 1 | inner | 810.2.p.a | 108 | |
15.d | odd | 2 | 1 | 270.2.p.a | ✓ | 108 | |
27.e | even | 9 | 1 | inner | 810.2.p.a | 108 | |
27.f | odd | 18 | 1 | 270.2.p.a | ✓ | 108 | |
135.n | odd | 18 | 1 | 270.2.p.a | ✓ | 108 | |
135.p | even | 18 | 1 | inner | 810.2.p.a | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
270.2.p.a | ✓ | 108 | 3.b | odd | 2 | 1 | |
270.2.p.a | ✓ | 108 | 15.d | odd | 2 | 1 | |
270.2.p.a | ✓ | 108 | 27.f | odd | 18 | 1 | |
270.2.p.a | ✓ | 108 | 135.n | odd | 18 | 1 | |
810.2.p.a | 108 | 1.a | even | 1 | 1 | trivial | |
810.2.p.a | 108 | 5.b | even | 2 | 1 | inner | |
810.2.p.a | 108 | 27.e | even | 9 | 1 | inner | |
810.2.p.a | 108 | 135.p | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(810, [\chi])\).