Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [464,2,Mod(33,464)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(464, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("464.33");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 464.y (of order \(14\), degree \(6\), 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: | \(3.70505865379\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{14})\) |
Twist minimal: | no (minimal twist has level 232) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
33.1 | 0 | −1.42415 | − | 2.95728i | 0 | −0.136467 | + | 0.597901i | 0 | −0.294470 | + | 0.141809i | 0 | −4.84685 | + | 6.07775i | 0 | ||||||||||
33.2 | 0 | −0.857903 | − | 1.78145i | 0 | −0.419644 | + | 1.83858i | 0 | 2.09267 | − | 1.00778i | 0 | −0.567114 | + | 0.711138i | 0 | ||||||||||
33.3 | 0 | −0.846741 | − | 1.75828i | 0 | 0.850744 | − | 3.72735i | 0 | −2.85493 | + | 1.37486i | 0 | −0.504093 | + | 0.632112i | 0 | ||||||||||
33.4 | 0 | −0.201238 | − | 0.417875i | 0 | −0.296646 | + | 1.29969i | 0 | −2.78319 | + | 1.34032i | 0 | 1.73635 | − | 2.17731i | 0 | ||||||||||
33.5 | 0 | 0.186194 | + | 0.386637i | 0 | 0.478397 | − | 2.09600i | 0 | 1.33207 | − | 0.641493i | 0 | 1.75565 | − | 2.20152i | 0 | ||||||||||
33.6 | 0 | 0.661075 | + | 1.37274i | 0 | −0.510076 | + | 2.23479i | 0 | −3.30380 | + | 1.59102i | 0 | 0.423086 | − | 0.530533i | 0 | ||||||||||
33.7 | 0 | 1.19841 | + | 2.48852i | 0 | −0.882074 | + | 3.86462i | 0 | 3.53526 | − | 1.70249i | 0 | −2.88609 | + | 3.61904i | 0 | ||||||||||
33.8 | 0 | 1.28436 | + | 2.66699i | 0 | 0.805849 | − | 3.53066i | 0 | 1.47444 | − | 0.710053i | 0 | −3.59282 | + | 4.50525i | 0 | ||||||||||
129.1 | 0 | −2.26073 | − | 1.80288i | 0 | −0.608430 | − | 0.293004i | 0 | 2.74476 | − | 3.44182i | 0 | 1.19300 | + | 5.22686i | 0 | ||||||||||
129.2 | 0 | −1.96464 | − | 1.56675i | 0 | 1.24505 | + | 0.599586i | 0 | −1.40365 | + | 1.76012i | 0 | 0.737555 | + | 3.23144i | 0 | ||||||||||
129.3 | 0 | −0.884679 | − | 0.705508i | 0 | 0.449269 | + | 0.216356i | 0 | −2.01029 | + | 2.52083i | 0 | −0.382647 | − | 1.67649i | 0 | ||||||||||
129.4 | 0 | −0.0131000 | − | 0.0104469i | 0 | 3.41718 | + | 1.64563i | 0 | 3.15424 | − | 3.95529i | 0 | −0.667500 | − | 2.92451i | 0 | ||||||||||
129.5 | 0 | 0.212201 | + | 0.169225i | 0 | −1.28207 | − | 0.617411i | 0 | −1.27595 | + | 1.59999i | 0 | −0.651170 | − | 2.85296i | 0 | ||||||||||
129.6 | 0 | 0.728009 | + | 0.580567i | 0 | −2.19459 | − | 1.05686i | 0 | 0.988136 | − | 1.23908i | 0 | −0.474625 | − | 2.07947i | 0 | ||||||||||
129.7 | 0 | 1.79043 | + | 1.42782i | 0 | 2.68076 | + | 1.29099i | 0 | −0.839902 | + | 1.05320i | 0 | 0.499400 | + | 2.18801i | 0 | ||||||||||
129.8 | 0 | 2.39252 | + | 1.90797i | 0 | −1.10330 | − | 0.531322i | 0 | 0.889636 | − | 1.11557i | 0 | 1.41624 | + | 6.20496i | 0 | ||||||||||
209.1 | 0 | −2.64359 | − | 0.603381i | 0 | −2.76955 | + | 3.47291i | 0 | −0.416896 | + | 1.82654i | 0 | 3.92157 | + | 1.88853i | 0 | ||||||||||
209.2 | 0 | −2.35308 | − | 0.537075i | 0 | 1.76863 | − | 2.21779i | 0 | 0.833292 | − | 3.65089i | 0 | 2.54562 | + | 1.22591i | 0 | ||||||||||
209.3 | 0 | −2.16441 | − | 0.494012i | 0 | 0.486884 | − | 0.610533i | 0 | −0.478099 | + | 2.09469i | 0 | 1.73771 | + | 0.836835i | 0 | ||||||||||
209.4 | 0 | −0.0690613 | − | 0.0157628i | 0 | −2.25017 | + | 2.82162i | 0 | 0.686126 | − | 3.00612i | 0 | −2.69839 | − | 1.29947i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.e | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 464.2.y.e | 48 | |
4.b | odd | 2 | 1 | 232.2.q.a | ✓ | 48 | |
29.e | even | 14 | 1 | inner | 464.2.y.e | 48 | |
116.h | odd | 14 | 1 | 232.2.q.a | ✓ | 48 | |
116.l | even | 28 | 1 | 6728.2.a.be | 24 | ||
116.l | even | 28 | 1 | 6728.2.a.bf | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
232.2.q.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
232.2.q.a | ✓ | 48 | 116.h | odd | 14 | 1 | |
464.2.y.e | 48 | 1.a | even | 1 | 1 | trivial | |
464.2.y.e | 48 | 29.e | even | 14 | 1 | inner | |
6728.2.a.be | 24 | 116.l | even | 28 | 1 | ||
6728.2.a.bf | 24 | 116.l | even | 28 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 15 T_{3}^{46} + 170 T_{3}^{44} - 1954 T_{3}^{42} + 22159 T_{3}^{40} - 1120 T_{3}^{39} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(464, [\chi])\).