Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,3,Mod(269,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.269");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 810.j (of order \(6\), 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: | \(22.0709014132\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
269.1 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −0.222532 | + | 4.99505i | 0 | −6.62225 | − | 3.82336i | 2.82843 | 0 | −5.96030 | − | 3.80458i | ||||||||
269.2 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | 2.48719 | + | 4.33750i | 0 | −5.12645 | − | 2.95976i | 2.82843 | 0 | −7.07104 | − | 0.0209028i | ||||||||
269.3 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −3.56960 | − | 3.50113i | 0 | −0.315907 | − | 0.182389i | 2.82843 | 0 | 6.81209 | − | 1.89618i | ||||||||
269.4 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | 4.10456 | − | 2.85528i | 0 | 1.70200 | + | 0.982651i | 2.82843 | 0 | 0.594626 | + | 7.04602i | ||||||||
269.5 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −4.91871 | + | 0.897918i | 0 | 8.15650 | + | 4.70916i | 2.82843 | 0 | 2.37833 | − | 6.65909i | ||||||||
269.6 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | 4.24042 | − | 2.64931i | 0 | 8.20611 | + | 4.73780i | 2.82843 | 0 | 0.246293 | + | 7.06678i | ||||||||
269.7 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 0.222532 | − | 4.99505i | 0 | −6.62225 | − | 3.82336i | −2.82843 | 0 | −5.96030 | − | 3.80458i | ||||||||
269.8 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | −4.10456 | + | 2.85528i | 0 | 1.70200 | + | 0.982651i | −2.82843 | 0 | 0.594626 | + | 7.04602i | ||||||||
269.9 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | −2.48719 | − | 4.33750i | 0 | −5.12645 | − | 2.95976i | −2.82843 | 0 | −7.07104 | − | 0.0209028i | ||||||||
269.10 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 4.91871 | − | 0.897918i | 0 | 8.15650 | + | 4.70916i | −2.82843 | 0 | 2.37833 | − | 6.65909i | ||||||||
269.11 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 3.56960 | + | 3.50113i | 0 | −0.315907 | − | 0.182389i | −2.82843 | 0 | 6.81209 | − | 1.89618i | ||||||||
269.12 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | −4.24042 | + | 2.64931i | 0 | 8.20611 | + | 4.73780i | −2.82843 | 0 | 0.246293 | + | 7.06678i | ||||||||
539.1 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −0.222532 | − | 4.99505i | 0 | −6.62225 | + | 3.82336i | 2.82843 | 0 | −5.96030 | + | 3.80458i | ||||||||
539.2 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | 2.48719 | − | 4.33750i | 0 | −5.12645 | + | 2.95976i | 2.82843 | 0 | −7.07104 | + | 0.0209028i | ||||||||
539.3 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −3.56960 | + | 3.50113i | 0 | −0.315907 | + | 0.182389i | 2.82843 | 0 | 6.81209 | + | 1.89618i | ||||||||
539.4 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | 4.10456 | + | 2.85528i | 0 | 1.70200 | − | 0.982651i | 2.82843 | 0 | 0.594626 | − | 7.04602i | ||||||||
539.5 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −4.91871 | − | 0.897918i | 0 | 8.15650 | − | 4.70916i | 2.82843 | 0 | 2.37833 | + | 6.65909i | ||||||||
539.6 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | 4.24042 | + | 2.64931i | 0 | 8.20611 | − | 4.73780i | 2.82843 | 0 | 0.246293 | − | 7.06678i | ||||||||
539.7 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | 0.222532 | + | 4.99505i | 0 | −6.62225 | + | 3.82336i | −2.82843 | 0 | −5.96030 | + | 3.80458i | ||||||||
539.8 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | −2.48719 | + | 4.33750i | 0 | −5.12645 | + | 2.95976i | −2.82843 | 0 | −7.07104 | + | 0.0209028i | ||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
45.h | odd | 6 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 810.3.j.h | 24 | |
3.b | odd | 2 | 1 | inner | 810.3.j.h | 24 | |
5.b | even | 2 | 1 | 810.3.j.g | 24 | ||
9.c | even | 3 | 1 | 810.3.b.c | ✓ | 24 | |
9.c | even | 3 | 1 | 810.3.j.g | 24 | ||
9.d | odd | 6 | 1 | 810.3.b.c | ✓ | 24 | |
9.d | odd | 6 | 1 | 810.3.j.g | 24 | ||
15.d | odd | 2 | 1 | 810.3.j.g | 24 | ||
45.h | odd | 6 | 1 | 810.3.b.c | ✓ | 24 | |
45.h | odd | 6 | 1 | inner | 810.3.j.h | 24 | |
45.j | even | 6 | 1 | 810.3.b.c | ✓ | 24 | |
45.j | even | 6 | 1 | inner | 810.3.j.h | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
810.3.b.c | ✓ | 24 | 9.c | even | 3 | 1 | |
810.3.b.c | ✓ | 24 | 9.d | odd | 6 | 1 | |
810.3.b.c | ✓ | 24 | 45.h | odd | 6 | 1 | |
810.3.b.c | ✓ | 24 | 45.j | even | 6 | 1 | |
810.3.j.g | 24 | 5.b | even | 2 | 1 | ||
810.3.j.g | 24 | 9.c | even | 3 | 1 | ||
810.3.j.g | 24 | 9.d | odd | 6 | 1 | ||
810.3.j.g | 24 | 15.d | odd | 2 | 1 | ||
810.3.j.h | 24 | 1.a | even | 1 | 1 | trivial | |
810.3.j.h | 24 | 3.b | odd | 2 | 1 | inner | |
810.3.j.h | 24 | 45.h | odd | 6 | 1 | inner | |
810.3.j.h | 24 | 45.j | 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 \(S_{3}^{\mathrm{new}}(810, [\chi])\):
\( T_{7}^{12} - 12 T_{7}^{11} - 66 T_{7}^{10} + 1368 T_{7}^{9} + 5598 T_{7}^{8} - 97800 T_{7}^{7} + \cdots + 8386816 \) |
\( T_{17}^{12} - 1464 T_{17}^{10} + 793137 T_{17}^{8} - 199784716 T_{17}^{6} + 23433607236 T_{17}^{4} + \cdots + 1460109722500 \) |