Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,2,Mod(6,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.6");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.bc (of order \(12\), degree \(4\), 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: | \(0.726638658394\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | −0.607529 | − | 2.26733i | −2.39046 | − | 1.38013i | −3.03963 | + | 1.75493i | 1.12678 | + | 1.12678i | −1.67694 | + | 6.25841i | −0.437995 | − | 2.60925i | 2.50608 | + | 2.50608i | 2.30952 | + | 4.00020i | 1.87023 | − | 3.23933i |
6.2 | −0.607529 | − | 2.26733i | 2.39046 | + | 1.38013i | −3.03963 | + | 1.75493i | −1.12678 | − | 1.12678i | 1.67694 | − | 6.25841i | 1.68394 | − | 2.04067i | 2.50608 | + | 2.50608i | 2.30952 | + | 4.00020i | −1.87023 | + | 3.23933i |
6.3 | −0.357317 | − | 1.33353i | −0.928437 | − | 0.536034i | 0.0814361 | − | 0.0470171i | −2.02925 | − | 2.02925i | −0.383068 | + | 1.42963i | 1.39092 | + | 2.25063i | −2.04421 | − | 2.04421i | −0.925336 | − | 1.60273i | −1.98097 | + | 3.43114i |
6.4 | −0.357317 | − | 1.33353i | 0.928437 | + | 0.536034i | 0.0814361 | − | 0.0470171i | 2.02925 | + | 2.02925i | 0.383068 | − | 1.42963i | −2.32989 | + | 1.25364i | −2.04421 | − | 2.04421i | −0.925336 | − | 1.60273i | 1.98097 | − | 3.43114i |
6.5 | 0.218036 | + | 0.813723i | −1.24122 | − | 0.716618i | 1.11745 | − | 0.645157i | 1.01306 | + | 1.01306i | 0.312498 | − | 1.16626i | 2.43848 | + | 1.02656i | 1.96000 | + | 1.96000i | −0.472916 | − | 0.819114i | −0.603466 | + | 1.04523i |
6.6 | 0.218036 | + | 0.813723i | 1.24122 | + | 0.716618i | 1.11745 | − | 0.645157i | −1.01306 | − | 1.01306i | −0.312498 | + | 1.16626i | −2.62506 | − | 0.330214i | 1.96000 | + | 1.96000i | −0.472916 | − | 0.819114i | 0.603466 | − | 1.04523i |
6.7 | 0.612835 | + | 2.28713i | −1.04118 | − | 0.601128i | −3.12335 | + | 1.80327i | 2.25527 | + | 2.25527i | 0.736784 | − | 2.74971i | −2.60590 | − | 0.457468i | −2.68981 | − | 2.68981i | −0.777291 | − | 1.34631i | −3.77599 | + | 6.54020i |
6.8 | 0.612835 | + | 2.28713i | 1.04118 | + | 0.601128i | −3.12335 | + | 1.80327i | −2.25527 | − | 2.25527i | −0.736784 | + | 2.74971i | 2.48551 | + | 0.906771i | −2.68981 | − | 2.68981i | −0.777291 | − | 1.34631i | 3.77599 | − | 6.54020i |
20.1 | −2.47996 | + | 0.664504i | −2.33506 | − | 1.34815i | 3.97660 | − | 2.29589i | −0.281686 | + | 0.281686i | 6.68671 | + | 1.79170i | −0.201802 | + | 2.63804i | −4.70528 | + | 4.70528i | 2.13500 | + | 3.69793i | 0.511390 | − | 0.885754i |
20.2 | −2.47996 | + | 0.664504i | 2.33506 | + | 1.34815i | 3.97660 | − | 2.29589i | 0.281686 | − | 0.281686i | −6.68671 | − | 1.79170i | 1.14426 | − | 2.38551i | −4.70528 | + | 4.70528i | 2.13500 | + | 3.69793i | −0.511390 | + | 0.885754i |
20.3 | −0.892257 | + | 0.239080i | −0.988196 | − | 0.570535i | −0.993087 | + | 0.573359i | 2.80028 | − | 2.80028i | 1.01813 | + | 0.272807i | 2.62900 | − | 0.297264i | 2.05537 | − | 2.05537i | −0.848979 | − | 1.47047i | −1.82908 | + | 3.16806i |
20.4 | −0.892257 | + | 0.239080i | 0.988196 | + | 0.570535i | −0.993087 | + | 0.573359i | −2.80028 | + | 2.80028i | −1.01813 | − | 0.272807i | 2.12815 | + | 1.57194i | 2.05537 | − | 2.05537i | −0.848979 | − | 1.47047i | 1.82908 | − | 3.16806i |
20.5 | −0.112775 | + | 0.0302180i | −2.25055 | − | 1.29935i | −1.72025 | + | 0.993184i | −1.24841 | + | 1.24841i | 0.293070 | + | 0.0785278i | −2.18126 | − | 1.49736i | 0.329103 | − | 0.329103i | 1.87664 | + | 3.25044i | 0.103066 | − | 0.178515i |
20.6 | −0.112775 | + | 0.0302180i | 2.25055 | + | 1.29935i | −1.72025 | + | 0.993184i | 1.24841 | − | 1.24841i | −0.293070 | − | 0.0785278i | −2.63771 | + | 0.206123i | 0.329103 | − | 0.329103i | 1.87664 | + | 3.25044i | −0.103066 | + | 0.178515i |
20.7 | 1.61897 | − | 0.433802i | −0.552306 | − | 0.318874i | 0.700831 | − | 0.404625i | 1.42145 | − | 1.42145i | −1.03250 | − | 0.276656i | −1.11484 | + | 2.39940i | −1.41124 | + | 1.41124i | −1.29664 | − | 2.24584i | 1.68466 | − | 2.91792i |
20.8 | 1.61897 | − | 0.433802i | 0.552306 | + | 0.318874i | 0.700831 | − | 0.404625i | −1.42145 | + | 1.42145i | 1.03250 | + | 0.276656i | 0.234216 | − | 2.63536i | −1.41124 | + | 1.41124i | −1.29664 | − | 2.24584i | −1.68466 | + | 2.91792i |
41.1 | −2.47996 | − | 0.664504i | −2.33506 | + | 1.34815i | 3.97660 | + | 2.29589i | −0.281686 | − | 0.281686i | 6.68671 | − | 1.79170i | −0.201802 | − | 2.63804i | −4.70528 | − | 4.70528i | 2.13500 | − | 3.69793i | 0.511390 | + | 0.885754i |
41.2 | −2.47996 | − | 0.664504i | 2.33506 | − | 1.34815i | 3.97660 | + | 2.29589i | 0.281686 | + | 0.281686i | −6.68671 | + | 1.79170i | 1.14426 | + | 2.38551i | −4.70528 | − | 4.70528i | 2.13500 | − | 3.69793i | −0.511390 | − | 0.885754i |
41.3 | −0.892257 | − | 0.239080i | −0.988196 | + | 0.570535i | −0.993087 | − | 0.573359i | 2.80028 | + | 2.80028i | 1.01813 | − | 0.272807i | 2.62900 | + | 0.297264i | 2.05537 | + | 2.05537i | −0.848979 | + | 1.47047i | −1.82908 | − | 3.16806i |
41.4 | −0.892257 | − | 0.239080i | 0.988196 | − | 0.570535i | −0.993087 | − | 0.573359i | −2.80028 | − | 2.80028i | −1.01813 | + | 0.272807i | 2.12815 | − | 1.57194i | 2.05537 | + | 2.05537i | −0.848979 | + | 1.47047i | 1.82908 | + | 3.16806i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
91.bc | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.2.bc.a | ✓ | 32 |
3.b | odd | 2 | 1 | 819.2.fm.g | 32 | ||
7.b | odd | 2 | 1 | inner | 91.2.bc.a | ✓ | 32 |
7.c | even | 3 | 1 | 637.2.x.b | 32 | ||
7.c | even | 3 | 1 | 637.2.bb.b | 32 | ||
7.d | odd | 6 | 1 | 637.2.x.b | 32 | ||
7.d | odd | 6 | 1 | 637.2.bb.b | 32 | ||
13.f | odd | 12 | 1 | inner | 91.2.bc.a | ✓ | 32 |
21.c | even | 2 | 1 | 819.2.fm.g | 32 | ||
39.k | even | 12 | 1 | 819.2.fm.g | 32 | ||
91.w | even | 12 | 1 | 637.2.bb.b | 32 | ||
91.x | odd | 12 | 1 | 637.2.x.b | 32 | ||
91.ba | even | 12 | 1 | 637.2.x.b | 32 | ||
91.bc | even | 12 | 1 | inner | 91.2.bc.a | ✓ | 32 |
91.bd | odd | 12 | 1 | 637.2.bb.b | 32 | ||
273.ca | odd | 12 | 1 | 819.2.fm.g | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.bc.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
91.2.bc.a | ✓ | 32 | 7.b | odd | 2 | 1 | inner |
91.2.bc.a | ✓ | 32 | 13.f | odd | 12 | 1 | inner |
91.2.bc.a | ✓ | 32 | 91.bc | even | 12 | 1 | inner |
637.2.x.b | 32 | 7.c | even | 3 | 1 | ||
637.2.x.b | 32 | 7.d | odd | 6 | 1 | ||
637.2.x.b | 32 | 91.x | odd | 12 | 1 | ||
637.2.x.b | 32 | 91.ba | even | 12 | 1 | ||
637.2.bb.b | 32 | 7.c | even | 3 | 1 | ||
637.2.bb.b | 32 | 7.d | odd | 6 | 1 | ||
637.2.bb.b | 32 | 91.w | even | 12 | 1 | ||
637.2.bb.b | 32 | 91.bd | odd | 12 | 1 | ||
819.2.fm.g | 32 | 3.b | odd | 2 | 1 | ||
819.2.fm.g | 32 | 21.c | even | 2 | 1 | ||
819.2.fm.g | 32 | 39.k | even | 12 | 1 | ||
819.2.fm.g | 32 | 273.ca | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(91, [\chi])\).