Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [867,2,Mod(65,867)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([8, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("867.65");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 867 = 3 \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 867.i (of order \(16\), degree \(8\), 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: | \(6.92302985525\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | −0.855706 | − | 2.06586i | −1.00198 | + | 1.41281i | −2.12132 | + | 2.12132i | 1.57139 | + | 2.35175i | 3.77606 | + | 0.861009i | −2.62934 | − | 1.75687i | 2.06586 | + | 0.855706i | −0.992053 | − | 2.83122i | 3.51373 | − | 5.25868i |
65.2 | −0.855706 | − | 2.06586i | 1.00198 | − | 1.41281i | −2.12132 | + | 2.12132i | −1.57139 | − | 2.35175i | −3.77606 | − | 0.861009i | 2.62934 | + | 1.75687i | 2.06586 | + | 0.855706i | −0.992053 | − | 2.83122i | −3.51373 | + | 5.25868i |
65.3 | 0.855706 | + | 2.06586i | −0.385055 | − | 1.68871i | −2.12132 | + | 2.12132i | 1.57139 | + | 2.35175i | 3.15913 | − | 2.24051i | 2.62934 | + | 1.75687i | −2.06586 | − | 0.855706i | −2.70347 | + | 1.30049i | −3.51373 | + | 5.25868i |
65.4 | 0.855706 | + | 2.06586i | 0.385055 | + | 1.68871i | −2.12132 | + | 2.12132i | −1.57139 | − | 2.35175i | −3.15913 | + | 2.24051i | −2.62934 | − | 1.75687i | −2.06586 | − | 0.855706i | −2.70347 | + | 1.30049i | 3.51373 | − | 5.25868i |
131.1 | −2.06586 | + | 0.855706i | −1.70752 | + | 0.290496i | 2.12132 | − | 2.12132i | 0.551799 | − | 2.77408i | 3.27891 | − | 2.06126i | 3.10152 | − | 0.616930i | −0.855706 | + | 2.06586i | 2.83122 | − | 0.992053i | 1.23386 | + | 6.20303i |
131.2 | −2.06586 | + | 0.855706i | 1.70752 | − | 0.290496i | 2.12132 | − | 2.12132i | −0.551799 | + | 2.77408i | −3.27891 | + | 2.06126i | −3.10152 | + | 0.616930i | −0.855706 | + | 2.06586i | 2.83122 | − | 0.992053i | −1.23386 | − | 6.20303i |
131.3 | 2.06586 | − | 0.855706i | −0.921821 | + | 1.46637i | 2.12132 | − | 2.12132i | −0.551799 | + | 2.77408i | −0.649569 | + | 3.81812i | 3.10152 | − | 0.616930i | 0.855706 | − | 2.06586i | −1.30049 | − | 2.70347i | 1.23386 | + | 6.20303i |
131.4 | 2.06586 | − | 0.855706i | 0.921821 | − | 1.46637i | 2.12132 | − | 2.12132i | 0.551799 | − | 2.77408i | 0.649569 | − | 3.81812i | −3.10152 | + | 0.616930i | 0.855706 | − | 2.06586i | −1.30049 | − | 2.70347i | −1.23386 | − | 6.20303i |
158.1 | −2.06586 | + | 0.855706i | −1.46637 | − | 0.921821i | 2.12132 | − | 2.12132i | −2.77408 | − | 0.551799i | 3.81812 | + | 0.649569i | −0.616930 | − | 3.10152i | −0.855706 | + | 2.06586i | 1.30049 | + | 2.70347i | 6.20303 | − | 1.23386i |
158.2 | −2.06586 | + | 0.855706i | 1.46637 | + | 0.921821i | 2.12132 | − | 2.12132i | 2.77408 | + | 0.551799i | −3.81812 | − | 0.649569i | 0.616930 | + | 3.10152i | −0.855706 | + | 2.06586i | 1.30049 | + | 2.70347i | −6.20303 | + | 1.23386i |
158.3 | 2.06586 | − | 0.855706i | −0.290496 | − | 1.70752i | 2.12132 | − | 2.12132i | 2.77408 | + | 0.551799i | −2.06126 | − | 3.27891i | −0.616930 | − | 3.10152i | 0.855706 | − | 2.06586i | −2.83122 | + | 0.992053i | 6.20303 | − | 1.23386i |
158.4 | 2.06586 | − | 0.855706i | 0.290496 | + | 1.70752i | 2.12132 | − | 2.12132i | −2.77408 | − | 0.551799i | 2.06126 | + | 3.27891i | 0.616930 | + | 3.10152i | 0.855706 | − | 2.06586i | −2.83122 | + | 0.992053i | −6.20303 | + | 1.23386i |
224.1 | −0.855706 | − | 2.06586i | −1.68871 | + | 0.385055i | −2.12132 | + | 2.12132i | 2.35175 | − | 1.57139i | 2.24051 | + | 3.15913i | −1.75687 | + | 2.62934i | 2.06586 | + | 0.855706i | 2.70347 | − | 1.30049i | −5.25868 | − | 3.51373i |
224.2 | −0.855706 | − | 2.06586i | 1.68871 | − | 0.385055i | −2.12132 | + | 2.12132i | −2.35175 | + | 1.57139i | −2.24051 | − | 3.15913i | 1.75687 | − | 2.62934i | 2.06586 | + | 0.855706i | 2.70347 | − | 1.30049i | 5.25868 | + | 3.51373i |
224.3 | 0.855706 | + | 2.06586i | −1.41281 | − | 1.00198i | −2.12132 | + | 2.12132i | −2.35175 | + | 1.57139i | 0.861009 | − | 3.77606i | −1.75687 | + | 2.62934i | −2.06586 | − | 0.855706i | 0.992053 | + | 2.83122i | −5.25868 | − | 3.51373i |
224.4 | 0.855706 | + | 2.06586i | 1.41281 | + | 1.00198i | −2.12132 | + | 2.12132i | 2.35175 | − | 1.57139i | −0.861009 | + | 3.77606i | 1.75687 | − | 2.62934i | −2.06586 | − | 0.855706i | 0.992053 | + | 2.83122i | 5.25868 | + | 3.51373i |
329.1 | −0.855706 | + | 2.06586i | −1.68871 | − | 0.385055i | −2.12132 | − | 2.12132i | 2.35175 | + | 1.57139i | 2.24051 | − | 3.15913i | −1.75687 | − | 2.62934i | 2.06586 | − | 0.855706i | 2.70347 | + | 1.30049i | −5.25868 | + | 3.51373i |
329.2 | −0.855706 | + | 2.06586i | 1.68871 | + | 0.385055i | −2.12132 | − | 2.12132i | −2.35175 | − | 1.57139i | −2.24051 | + | 3.15913i | 1.75687 | + | 2.62934i | 2.06586 | − | 0.855706i | 2.70347 | + | 1.30049i | 5.25868 | − | 3.51373i |
329.3 | 0.855706 | − | 2.06586i | −1.41281 | + | 1.00198i | −2.12132 | − | 2.12132i | −2.35175 | − | 1.57139i | 0.861009 | + | 3.77606i | −1.75687 | − | 2.62934i | −2.06586 | + | 0.855706i | 0.992053 | − | 2.83122i | −5.25868 | + | 3.51373i |
329.4 | 0.855706 | − | 2.06586i | 1.41281 | − | 1.00198i | −2.12132 | − | 2.12132i | 2.35175 | + | 1.57139i | −0.861009 | − | 3.77606i | 1.75687 | + | 2.62934i | −2.06586 | + | 0.855706i | 0.992053 | − | 2.83122i | 5.25868 | − | 3.51373i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
17.b | even | 2 | 1 | inner |
17.c | even | 4 | 2 | inner |
17.d | even | 8 | 4 | inner |
17.e | odd | 16 | 8 | inner |
51.c | odd | 2 | 1 | inner |
51.f | odd | 4 | 2 | inner |
51.g | odd | 8 | 4 | inner |
51.i | even | 16 | 8 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 867.2.i.e | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 867.2.i.e | ✓ | 32 |
17.b | even | 2 | 1 | inner | 867.2.i.e | ✓ | 32 |
17.c | even | 4 | 2 | inner | 867.2.i.e | ✓ | 32 |
17.d | even | 8 | 4 | inner | 867.2.i.e | ✓ | 32 |
17.e | odd | 16 | 8 | inner | 867.2.i.e | ✓ | 32 |
51.c | odd | 2 | 1 | inner | 867.2.i.e | ✓ | 32 |
51.f | odd | 4 | 2 | inner | 867.2.i.e | ✓ | 32 |
51.g | odd | 8 | 4 | inner | 867.2.i.e | ✓ | 32 |
51.i | even | 16 | 8 | inner | 867.2.i.e | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
867.2.i.e | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
867.2.i.e | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
867.2.i.e | ✓ | 32 | 17.b | even | 2 | 1 | inner |
867.2.i.e | ✓ | 32 | 17.c | even | 4 | 2 | inner |
867.2.i.e | ✓ | 32 | 17.d | even | 8 | 4 | inner |
867.2.i.e | ✓ | 32 | 17.e | odd | 16 | 8 | inner |
867.2.i.e | ✓ | 32 | 51.c | odd | 2 | 1 | inner |
867.2.i.e | ✓ | 32 | 51.f | odd | 4 | 2 | inner |
867.2.i.e | ✓ | 32 | 51.g | odd | 8 | 4 | inner |
867.2.i.e | ✓ | 32 | 51.i | even | 16 | 8 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):
\( T_{2}^{8} + 625 \) |
\( T_{5}^{16} + 16777216 \) |
\( T_{7}^{16} + 100000000 \) |