Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,6,Mod(206,207)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.206");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 207.c (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: | \(33.1994507013\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
206.1 | − | 6.22736i | 0 | −6.78003 | −105.915 | 0 | − | 86.6285i | − | 157.054i | 0 | 659.570i | |||||||||||||||
206.2 | 6.22736i | 0 | −6.78003 | −105.915 | 0 | 86.6285i | 157.054i | 0 | − | 659.570i | |||||||||||||||||
206.3 | − | 1.91117i | 0 | 28.3474 | 86.4475 | 0 | 91.5099i | − | 115.334i | 0 | − | 165.216i | |||||||||||||||
206.4 | 1.91117i | 0 | 28.3474 | 86.4475 | 0 | − | 91.5099i | 115.334i | 0 | 165.216i | |||||||||||||||||
206.5 | − | 10.4484i | 0 | −77.1695 | −72.3402 | 0 | − | 9.31885i | 471.949i | 0 | 755.841i | ||||||||||||||||
206.6 | 10.4484i | 0 | −77.1695 | −72.3402 | 0 | 9.31885i | − | 471.949i | 0 | − | 755.841i | ||||||||||||||||
206.7 | − | 6.00386i | 0 | −4.04628 | −54.7821 | 0 | 232.221i | − | 167.830i | 0 | 328.904i | ||||||||||||||||
206.8 | 6.00386i | 0 | −4.04628 | −54.7821 | 0 | − | 232.221i | 167.830i | 0 | − | 328.904i | ||||||||||||||||
206.9 | − | 3.32542i | 0 | 20.9416 | −58.3890 | 0 | 22.0963i | − | 176.053i | 0 | 194.168i | ||||||||||||||||
206.10 | 3.32542i | 0 | 20.9416 | −58.3890 | 0 | − | 22.0963i | 176.053i | 0 | − | 194.168i | ||||||||||||||||
206.11 | − | 10.4590i | 0 | −77.3916 | 44.1133 | 0 | − | 173.403i | 474.753i | 0 | − | 461.383i | |||||||||||||||
206.12 | 10.4590i | 0 | −77.3916 | 44.1133 | 0 | 173.403i | − | 474.753i | 0 | 461.383i | |||||||||||||||||
206.13 | − | 7.94564i | 0 | −31.1332 | −36.1472 | 0 | − | 129.914i | − | 6.88755i | 0 | 287.213i | |||||||||||||||
206.14 | 7.94564i | 0 | −31.1332 | −36.1472 | 0 | 129.914i | 6.88755i | 0 | − | 287.213i | |||||||||||||||||
206.15 | − | 8.84990i | 0 | −46.3207 | −32.1196 | 0 | 32.0928i | 126.737i | 0 | 284.255i | |||||||||||||||||
206.16 | 8.84990i | 0 | −46.3207 | −32.1196 | 0 | − | 32.0928i | − | 126.737i | 0 | − | 284.255i | |||||||||||||||
206.17 | − | 1.13533i | 0 | 30.7110 | −11.6464 | 0 | − | 79.1972i | − | 71.1980i | 0 | 13.2226i | |||||||||||||||
206.18 | 1.13533i | 0 | 30.7110 | −11.6464 | 0 | 79.1972i | 71.1980i | 0 | − | 13.2226i | |||||||||||||||||
206.19 | − | 4.37708i | 0 | 12.8412 | −13.0959 | 0 | 213.282i | − | 196.273i | 0 | 57.3218i | ||||||||||||||||
206.20 | 4.37708i | 0 | 12.8412 | −13.0959 | 0 | − | 213.282i | 196.273i | 0 | − | 57.3218i | ||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
69.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 207.6.c.a | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 207.6.c.a | ✓ | 40 |
23.b | odd | 2 | 1 | inner | 207.6.c.a | ✓ | 40 |
69.c | even | 2 | 1 | inner | 207.6.c.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
207.6.c.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
207.6.c.a | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
207.6.c.a | ✓ | 40 | 23.b | odd | 2 | 1 | inner |
207.6.c.a | ✓ | 40 | 69.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(207, [\chi])\).