Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,3,Mod(67,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.67");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 136.e (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: | \(3.70573159530\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.82569 | − | 0.816612i | − | 4.38815i | 2.66629 | + | 2.98176i | −2.66031 | −3.58341 | + | 8.01140i | −7.32916 | −2.43288 | − | 7.62110i | −10.2558 | 4.85691 | + | 2.17244i | |||||||
67.2 | −1.82569 | − | 0.816612i | 4.38815i | 2.66629 | + | 2.98176i | 2.66031 | 3.58341 | − | 8.01140i | 7.32916 | −2.43288 | − | 7.62110i | −10.2558 | −4.85691 | − | 2.17244i | ||||||||
67.3 | −1.82569 | + | 0.816612i | − | 4.38815i | 2.66629 | − | 2.98176i | 2.66031 | 3.58341 | + | 8.01140i | 7.32916 | −2.43288 | + | 7.62110i | −10.2558 | −4.85691 | + | 2.17244i | |||||||
67.4 | −1.82569 | + | 0.816612i | 4.38815i | 2.66629 | − | 2.98176i | −2.66031 | −3.58341 | − | 8.01140i | −7.32916 | −2.43288 | + | 7.62110i | −10.2558 | 4.85691 | − | 2.17244i | ||||||||
67.5 | −1.58917 | − | 1.21430i | − | 2.87954i | 1.05094 | + | 3.85947i | 4.15587 | −3.49664 | + | 4.57610i | 8.29662 | 3.01643 | − | 7.40953i | 0.708224 | −6.60439 | − | 5.04647i | |||||||
67.6 | −1.58917 | − | 1.21430i | 2.87954i | 1.05094 | + | 3.85947i | −4.15587 | 3.49664 | − | 4.57610i | −8.29662 | 3.01643 | − | 7.40953i | 0.708224 | 6.60439 | + | 5.04647i | ||||||||
67.7 | −1.58917 | + | 1.21430i | − | 2.87954i | 1.05094 | − | 3.85947i | −4.15587 | 3.49664 | + | 4.57610i | −8.29662 | 3.01643 | + | 7.40953i | 0.708224 | 6.60439 | − | 5.04647i | |||||||
67.8 | −1.58917 | + | 1.21430i | 2.87954i | 1.05094 | − | 3.85947i | 4.15587 | −3.49664 | − | 4.57610i | 8.29662 | 3.01643 | + | 7.40953i | 0.708224 | −6.60439 | + | 5.04647i | ||||||||
67.9 | −0.842089 | − | 1.81408i | − | 0.434450i | −2.58177 | + | 3.05523i | 5.41022 | −0.788127 | + | 0.365846i | −0.0314051 | 7.71652 | + | 2.11076i | 8.81125 | −4.55589 | − | 9.81456i | |||||||
67.10 | −0.842089 | − | 1.81408i | 0.434450i | −2.58177 | + | 3.05523i | −5.41022 | 0.788127 | − | 0.365846i | 0.0314051 | 7.71652 | + | 2.11076i | 8.81125 | 4.55589 | + | 9.81456i | ||||||||
67.11 | −0.842089 | + | 1.81408i | − | 0.434450i | −2.58177 | − | 3.05523i | −5.41022 | 0.788127 | + | 0.365846i | 0.0314051 | 7.71652 | − | 2.11076i | 8.81125 | 4.55589 | − | 9.81456i | |||||||
67.12 | −0.842089 | + | 1.81408i | 0.434450i | −2.58177 | − | 3.05523i | 5.41022 | −0.788127 | − | 0.365846i | −0.0314051 | 7.71652 | − | 2.11076i | 8.81125 | −4.55589 | + | 9.81456i | ||||||||
67.13 | −0.203558 | − | 1.98961i | − | 5.30587i | −3.91713 | + | 0.810002i | −6.46021 | −10.5566 | + | 1.08005i | 10.2774 | 2.40895 | + | 7.62869i | −19.1523 | 1.31502 | + | 12.8533i | |||||||
67.14 | −0.203558 | − | 1.98961i | 5.30587i | −3.91713 | + | 0.810002i | 6.46021 | 10.5566 | − | 1.08005i | −10.2774 | 2.40895 | + | 7.62869i | −19.1523 | −1.31502 | − | 12.8533i | ||||||||
67.15 | −0.203558 | + | 1.98961i | − | 5.30587i | −3.91713 | − | 0.810002i | 6.46021 | 10.5566 | + | 1.08005i | −10.2774 | 2.40895 | − | 7.62869i | −19.1523 | −1.31502 | + | 12.8533i | |||||||
67.16 | −0.203558 | + | 1.98961i | 5.30587i | −3.91713 | − | 0.810002i | −6.46021 | −10.5566 | − | 1.08005i | 10.2774 | 2.40895 | − | 7.62869i | −19.1523 | 1.31502 | − | 12.8533i | ||||||||
67.17 | 0.420841 | − | 1.95522i | − | 2.18661i | −3.64579 | − | 1.64567i | −1.66636 | −4.27530 | − | 0.920213i | −10.7711 | −4.75195 | + | 6.43575i | 4.21876 | −0.701272 | + | 3.25810i | |||||||
67.18 | 0.420841 | − | 1.95522i | 2.18661i | −3.64579 | − | 1.64567i | 1.66636 | 4.27530 | + | 0.920213i | 10.7711 | −4.75195 | + | 6.43575i | 4.21876 | 0.701272 | − | 3.25810i | ||||||||
67.19 | 0.420841 | + | 1.95522i | − | 2.18661i | −3.64579 | + | 1.64567i | 1.66636 | 4.27530 | − | 0.920213i | 10.7711 | −4.75195 | − | 6.43575i | 4.21876 | 0.701272 | + | 3.25810i | |||||||
67.20 | 0.420841 | + | 1.95522i | 2.18661i | −3.64579 | + | 1.64567i | −1.66636 | −4.27530 | + | 0.920213i | −10.7711 | −4.75195 | − | 6.43575i | 4.21876 | −0.701272 | − | 3.25810i | ||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
17.b | even | 2 | 1 | inner |
136.e | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 136.3.e.d | ✓ | 28 |
4.b | odd | 2 | 1 | 544.3.e.d | 28 | ||
8.b | even | 2 | 1 | 544.3.e.d | 28 | ||
8.d | odd | 2 | 1 | inner | 136.3.e.d | ✓ | 28 |
17.b | even | 2 | 1 | inner | 136.3.e.d | ✓ | 28 |
68.d | odd | 2 | 1 | 544.3.e.d | 28 | ||
136.e | odd | 2 | 1 | inner | 136.3.e.d | ✓ | 28 |
136.h | even | 2 | 1 | 544.3.e.d | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
136.3.e.d | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
136.3.e.d | ✓ | 28 | 8.d | odd | 2 | 1 | inner |
136.3.e.d | ✓ | 28 | 17.b | even | 2 | 1 | inner |
136.3.e.d | ✓ | 28 | 136.e | odd | 2 | 1 | inner |
544.3.e.d | 28 | 4.b | odd | 2 | 1 | ||
544.3.e.d | 28 | 8.b | even | 2 | 1 | ||
544.3.e.d | 28 | 68.d | odd | 2 | 1 | ||
544.3.e.d | 28 | 136.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(136, [\chi])\):
\( T_{3}^{14} + 84T_{3}^{12} + 2740T_{3}^{10} + 44264T_{3}^{8} + 373072T_{3}^{6} + 1571264T_{3}^{4} + 2682896T_{3}^{2} + 452864 \) |
\( T_{5}^{14} - 200 T_{5}^{12} + 14400 T_{5}^{10} - 491552 T_{5}^{8} + 8597184 T_{5}^{6} - 76550400 T_{5}^{4} + 315154688 T_{5}^{2} - 442003456 \) |