Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(65,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.65");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 462.n (of order \(6\), 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: | \(3.68908857338\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 | 0.500000 | − | 0.866025i | −1.72951 | − | 0.0938734i | −0.500000 | − | 0.866025i | −3.23623 | − | 1.86844i | −0.946049 | + | 1.45086i | 0.830341 | − | 2.51208i | −1.00000 | 2.98238 | + | 0.324709i | −3.23623 | + | 1.86844i | ||
65.2 | 0.500000 | − | 0.866025i | −1.64448 | − | 0.543759i | −0.500000 | − | 0.866025i | 1.63087 | + | 0.941585i | −1.29315 | + | 1.15228i | 1.76745 | + | 1.96879i | −1.00000 | 2.40865 | + | 1.78841i | 1.63087 | − | 0.941585i | ||
65.3 | 0.500000 | − | 0.866025i | −1.20836 | − | 1.24091i | −0.500000 | − | 0.866025i | −1.00577 | − | 0.580683i | −1.67884 | + | 0.426020i | −2.19957 | + | 1.47034i | −1.00000 | −0.0797091 | + | 2.99894i | −1.00577 | + | 0.580683i | ||
65.4 | 0.500000 | − | 0.866025i | −0.993776 | + | 1.41859i | −0.500000 | − | 0.866025i | 0.246102 | + | 0.142087i | 0.731651 | + | 1.56993i | −2.09441 | − | 1.61662i | −1.00000 | −1.02482 | − | 2.81953i | 0.246102 | − | 0.142087i | ||
65.5 | 0.500000 | − | 0.866025i | −0.470476 | − | 1.66693i | −0.500000 | − | 0.866025i | 1.00577 | + | 0.580683i | −1.67884 | − | 0.426020i | 2.19957 | − | 1.47034i | −1.00000 | −2.55730 | + | 1.56850i | 1.00577 | − | 0.580683i | ||
65.6 | 0.500000 | − | 0.866025i | 0.125098 | + | 1.72753i | −0.500000 | − | 0.866025i | 3.72427 | + | 2.15021i | 1.55863 | + | 0.755426i | 2.62857 | + | 0.301041i | −1.00000 | −2.96870 | + | 0.432220i | 3.72427 | − | 2.15021i | ||
65.7 | 0.500000 | − | 0.866025i | 0.301239 | + | 1.70565i | −0.500000 | − | 0.866025i | −1.38736 | − | 0.800992i | 1.62776 | + | 0.591946i | −0.229478 | + | 2.63578i | −1.00000 | −2.81851 | + | 1.02762i | −1.38736 | + | 0.800992i | ||
65.8 | 0.500000 | − | 0.866025i | 0.351333 | − | 1.69604i | −0.500000 | − | 0.866025i | −1.63087 | − | 0.941585i | −1.29315 | − | 1.15228i | −1.76745 | − | 1.96879i | −1.00000 | −2.75313 | − | 1.19175i | −1.63087 | + | 0.941585i | ||
65.9 | 0.500000 | − | 0.866025i | 0.783456 | − | 1.54473i | −0.500000 | − | 0.866025i | 3.23623 | + | 1.86844i | −0.946049 | − | 1.45086i | −0.830341 | + | 2.51208i | −1.00000 | −1.77239 | − | 2.42046i | 3.23623 | − | 1.86844i | ||
65.10 | 0.500000 | − | 0.866025i | 1.32652 | + | 1.11371i | −0.500000 | − | 0.866025i | 1.38736 | + | 0.800992i | 1.62776 | − | 0.591946i | 0.229478 | − | 2.63578i | −1.00000 | 0.519310 | + | 2.95471i | 1.38736 | − | 0.800992i | ||
65.11 | 0.500000 | − | 0.866025i | 1.43353 | + | 0.972102i | −0.500000 | − | 0.866025i | −3.72427 | − | 2.15021i | 1.55863 | − | 0.755426i | −2.62857 | − | 0.301041i | −1.00000 | 1.11004 | + | 2.78708i | −3.72427 | + | 2.15021i | ||
65.12 | 0.500000 | − | 0.866025i | 1.72543 | − | 0.151338i | −0.500000 | − | 0.866025i | −0.246102 | − | 0.142087i | 0.731651 | − | 1.56993i | 2.09441 | + | 1.61662i | −1.00000 | 2.95419 | − | 0.522246i | −0.246102 | + | 0.142087i | ||
263.1 | 0.500000 | + | 0.866025i | −1.72951 | + | 0.0938734i | −0.500000 | + | 0.866025i | −3.23623 | + | 1.86844i | −0.946049 | − | 1.45086i | 0.830341 | + | 2.51208i | −1.00000 | 2.98238 | − | 0.324709i | −3.23623 | − | 1.86844i | ||
263.2 | 0.500000 | + | 0.866025i | −1.64448 | + | 0.543759i | −0.500000 | + | 0.866025i | 1.63087 | − | 0.941585i | −1.29315 | − | 1.15228i | 1.76745 | − | 1.96879i | −1.00000 | 2.40865 | − | 1.78841i | 1.63087 | + | 0.941585i | ||
263.3 | 0.500000 | + | 0.866025i | −1.20836 | + | 1.24091i | −0.500000 | + | 0.866025i | −1.00577 | + | 0.580683i | −1.67884 | − | 0.426020i | −2.19957 | − | 1.47034i | −1.00000 | −0.0797091 | − | 2.99894i | −1.00577 | − | 0.580683i | ||
263.4 | 0.500000 | + | 0.866025i | −0.993776 | − | 1.41859i | −0.500000 | + | 0.866025i | 0.246102 | − | 0.142087i | 0.731651 | − | 1.56993i | −2.09441 | + | 1.61662i | −1.00000 | −1.02482 | + | 2.81953i | 0.246102 | + | 0.142087i | ||
263.5 | 0.500000 | + | 0.866025i | −0.470476 | + | 1.66693i | −0.500000 | + | 0.866025i | 1.00577 | − | 0.580683i | −1.67884 | + | 0.426020i | 2.19957 | + | 1.47034i | −1.00000 | −2.55730 | − | 1.56850i | 1.00577 | + | 0.580683i | ||
263.6 | 0.500000 | + | 0.866025i | 0.125098 | − | 1.72753i | −0.500000 | + | 0.866025i | 3.72427 | − | 2.15021i | 1.55863 | − | 0.755426i | 2.62857 | − | 0.301041i | −1.00000 | −2.96870 | − | 0.432220i | 3.72427 | + | 2.15021i | ||
263.7 | 0.500000 | + | 0.866025i | 0.301239 | − | 1.70565i | −0.500000 | + | 0.866025i | −1.38736 | + | 0.800992i | 1.62776 | − | 0.591946i | −0.229478 | − | 2.63578i | −1.00000 | −2.81851 | − | 1.02762i | −1.38736 | − | 0.800992i | ||
263.8 | 0.500000 | + | 0.866025i | 0.351333 | + | 1.69604i | −0.500000 | + | 0.866025i | −1.63087 | + | 0.941585i | −1.29315 | + | 1.15228i | −1.76745 | + | 1.96879i | −1.00000 | −2.75313 | + | 1.19175i | −1.63087 | − | 0.941585i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
33.d | even | 2 | 1 | inner |
231.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 462.2.n.f | yes | 24 |
3.b | odd | 2 | 1 | 462.2.n.e | ✓ | 24 | |
7.c | even | 3 | 1 | inner | 462.2.n.f | yes | 24 |
11.b | odd | 2 | 1 | 462.2.n.e | ✓ | 24 | |
21.h | odd | 6 | 1 | 462.2.n.e | ✓ | 24 | |
33.d | even | 2 | 1 | inner | 462.2.n.f | yes | 24 |
77.h | odd | 6 | 1 | 462.2.n.e | ✓ | 24 | |
231.l | even | 6 | 1 | inner | 462.2.n.f | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
462.2.n.e | ✓ | 24 | 3.b | odd | 2 | 1 | |
462.2.n.e | ✓ | 24 | 11.b | odd | 2 | 1 | |
462.2.n.e | ✓ | 24 | 21.h | odd | 6 | 1 | |
462.2.n.e | ✓ | 24 | 77.h | odd | 6 | 1 | |
462.2.n.f | yes | 24 | 1.a | even | 1 | 1 | trivial |
462.2.n.f | yes | 24 | 7.c | even | 3 | 1 | inner |
462.2.n.f | yes | 24 | 33.d | even | 2 | 1 | inner |
462.2.n.f | yes | 24 | 231.l | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):
\( T_{5}^{24} - 40 T_{5}^{22} + 1079 T_{5}^{20} - 15752 T_{5}^{18} + 164601 T_{5}^{16} - 922588 T_{5}^{14} + \cdots + 65536 \) |
\( T_{17}^{12} + 2 T_{17}^{11} + 47 T_{17}^{10} - 58 T_{17}^{9} + 1365 T_{17}^{8} - 1830 T_{17}^{7} + \cdots + 589824 \) |