Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,2,Mod(17,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.j (of order \(22\), degree \(10\), 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: | \(3.30580664368\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −0.909632 | − | 0.415415i | 0 | 0.654861 | + | 0.755750i | −2.75649 | + | 1.77149i | 0 | −0.318197 | − | 0.0457498i | −0.281733 | − | 0.959493i | 0 | 3.24329 | − | 0.466315i | ||||||
17.2 | −0.909632 | − | 0.415415i | 0 | 0.654861 | + | 0.755750i | −2.01516 | + | 1.29506i | 0 | 0.691369 | + | 0.0994038i | −0.281733 | − | 0.959493i | 0 | 2.37104 | − | 0.340905i | ||||||
17.3 | −0.909632 | − | 0.415415i | 0 | 0.654861 | + | 0.755750i | 2.08613 | − | 1.34067i | 0 | 2.86484 | + | 0.411901i | −0.281733 | − | 0.959493i | 0 | −2.45454 | + | 0.352910i | ||||||
17.4 | −0.909632 | − | 0.415415i | 0 | 0.654861 | + | 0.755750i | 2.68552 | − | 1.72588i | 0 | −3.23801 | − | 0.465555i | −0.281733 | − | 0.959493i | 0 | −3.15979 | + | 0.454310i | ||||||
17.5 | 0.909632 | + | 0.415415i | 0 | 0.654861 | + | 0.755750i | −2.68552 | + | 1.72588i | 0 | −3.23801 | − | 0.465555i | 0.281733 | + | 0.959493i | 0 | −3.15979 | + | 0.454310i | ||||||
17.6 | 0.909632 | + | 0.415415i | 0 | 0.654861 | + | 0.755750i | −2.08613 | + | 1.34067i | 0 | 2.86484 | + | 0.411901i | 0.281733 | + | 0.959493i | 0 | −2.45454 | + | 0.352910i | ||||||
17.7 | 0.909632 | + | 0.415415i | 0 | 0.654861 | + | 0.755750i | 2.01516 | − | 1.29506i | 0 | 0.691369 | + | 0.0994038i | 0.281733 | + | 0.959493i | 0 | 2.37104 | − | 0.340905i | ||||||
17.8 | 0.909632 | + | 0.415415i | 0 | 0.654861 | + | 0.755750i | 2.75649 | − | 1.77149i | 0 | −0.318197 | − | 0.0457498i | 0.281733 | + | 0.959493i | 0 | 3.24329 | − | 0.466315i | ||||||
53.1 | −0.755750 | − | 0.654861i | 0 | 0.142315 | + | 0.989821i | −0.959224 | − | 2.10041i | 0 | 0.976881 | + | 3.32695i | 0.540641 | − | 0.841254i | 0 | −0.650541 | + | 2.21554i | ||||||
53.2 | −0.755750 | − | 0.654861i | 0 | 0.142315 | + | 0.989821i | −0.922959 | − | 2.02100i | 0 | 0.0130977 | + | 0.0446065i | 0.540641 | − | 0.841254i | 0 | −0.625947 | + | 2.13178i | ||||||
53.3 | −0.755750 | − | 0.654861i | 0 | 0.142315 | + | 0.989821i | 0.323986 | + | 0.709431i | 0 | −0.374655 | − | 1.27596i | 0.540641 | − | 0.841254i | 0 | 0.219726 | − | 0.748318i | ||||||
53.4 | −0.755750 | − | 0.654861i | 0 | 0.142315 | + | 0.989821i | 1.55820 | + | 3.41197i | 0 | −0.615324 | − | 2.09560i | 0.540641 | − | 0.841254i | 0 | 1.05676 | − | 3.59900i | ||||||
53.5 | 0.755750 | + | 0.654861i | 0 | 0.142315 | + | 0.989821i | −1.55820 | − | 3.41197i | 0 | −0.615324 | − | 2.09560i | −0.540641 | + | 0.841254i | 0 | 1.05676 | − | 3.59900i | ||||||
53.6 | 0.755750 | + | 0.654861i | 0 | 0.142315 | + | 0.989821i | −0.323986 | − | 0.709431i | 0 | −0.374655 | − | 1.27596i | −0.540641 | + | 0.841254i | 0 | 0.219726 | − | 0.748318i | ||||||
53.7 | 0.755750 | + | 0.654861i | 0 | 0.142315 | + | 0.989821i | 0.922959 | + | 2.02100i | 0 | 0.0130977 | + | 0.0446065i | −0.540641 | + | 0.841254i | 0 | −0.625947 | + | 2.13178i | ||||||
53.8 | 0.755750 | + | 0.654861i | 0 | 0.142315 | + | 0.989821i | 0.959224 | + | 2.10041i | 0 | 0.976881 | + | 3.32695i | −0.540641 | + | 0.841254i | 0 | −0.650541 | + | 2.21554i | ||||||
89.1 | −0.989821 | + | 0.142315i | 0 | 0.959493 | − | 0.281733i | −0.922790 | + | 1.06496i | 0 | 2.78807 | + | 4.33833i | −0.909632 | + | 0.415415i | 0 | 0.761838 | − | 1.18544i | ||||||
89.2 | −0.989821 | + | 0.142315i | 0 | 0.959493 | − | 0.281733i | −0.762054 | + | 0.879457i | 0 | −1.27448 | − | 1.98312i | −0.909632 | + | 0.415415i | 0 | 0.629138 | − | 0.978957i | ||||||
89.3 | −0.989821 | + | 0.142315i | 0 | 0.959493 | − | 0.281733i | −0.408438 | + | 0.471363i | 0 | −2.27479 | − | 3.53964i | −0.909632 | + | 0.415415i | 0 | 0.337199 | − | 0.524692i | ||||||
89.4 | −0.989821 | + | 0.142315i | 0 | 0.959493 | − | 0.281733i | 2.09328 | − | 2.41578i | 0 | 0.761188 | + | 1.18443i | −0.909632 | + | 0.415415i | 0 | −1.72817 | + | 2.68909i | ||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 414.2.j.a | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 414.2.j.a | ✓ | 80 |
23.d | odd | 22 | 1 | inner | 414.2.j.a | ✓ | 80 |
69.g | even | 22 | 1 | inner | 414.2.j.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.2.j.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
414.2.j.a | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
414.2.j.a | ✓ | 80 | 23.d | odd | 22 | 1 | inner |
414.2.j.a | ✓ | 80 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(414, [\chi])\).