Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,4,Mod(49,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 5, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.49");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 384.n (of order \(8\), degree \(4\), 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: | \(22.6567334422\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{8})\) |
Twist minimal: | no (minimal twist has level 96) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | 0 | −2.77164 | + | 1.14805i | 0 | −3.82395 | + | 9.23183i | 0 | 0.902320 | + | 0.902320i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.2 | 0 | −2.77164 | + | 1.14805i | 0 | −3.59912 | + | 8.68905i | 0 | −4.07596 | − | 4.07596i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.3 | 0 | −2.77164 | + | 1.14805i | 0 | 3.92474 | − | 9.47517i | 0 | 9.05205 | + | 9.05205i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.4 | 0 | −2.77164 | + | 1.14805i | 0 | −2.47683 | + | 5.97959i | 0 | −17.4348 | − | 17.4348i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.5 | 0 | −2.77164 | + | 1.14805i | 0 | 1.39870 | − | 3.37677i | 0 | −19.0279 | − | 19.0279i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.6 | 0 | −2.77164 | + | 1.14805i | 0 | 0.245889 | − | 0.593629i | 0 | 22.6211 | + | 22.6211i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.7 | 0 | −2.77164 | + | 1.14805i | 0 | 0.637895 | − | 1.54002i | 0 | 21.6359 | + | 21.6359i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.8 | 0 | −2.77164 | + | 1.14805i | 0 | 4.92948 | − | 11.9008i | 0 | −3.52308 | − | 3.52308i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.9 | 0 | −2.77164 | + | 1.14805i | 0 | 5.64591 | − | 13.6304i | 0 | −16.7100 | − | 16.7100i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.10 | 0 | −2.77164 | + | 1.14805i | 0 | −5.33520 | + | 12.8803i | 0 | −5.00776 | − | 5.00776i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.11 | 0 | −2.77164 | + | 1.14805i | 0 | −5.92903 | + | 14.3139i | 0 | 16.3325 | + | 16.3325i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.12 | 0 | −2.77164 | + | 1.14805i | 0 | 8.20833 | − | 19.8167i | 0 | 5.13502 | + | 5.13502i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.13 | 0 | 2.77164 | − | 1.14805i | 0 | 7.74531 | − | 18.6988i | 0 | −17.6550 | − | 17.6550i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.14 | 0 | 2.77164 | − | 1.14805i | 0 | 5.41058 | − | 13.0623i | 0 | 20.8641 | + | 20.8641i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.15 | 0 | 2.77164 | − | 1.14805i | 0 | 3.34846 | − | 8.08390i | 0 | −8.76230 | − | 8.76230i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.16 | 0 | 2.77164 | − | 1.14805i | 0 | −2.19526 | + | 5.29982i | 0 | 10.7161 | + | 10.7161i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.17 | 0 | 2.77164 | − | 1.14805i | 0 | −1.90077 | + | 4.58887i | 0 | −10.6034 | − | 10.6034i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.18 | 0 | 2.77164 | − | 1.14805i | 0 | 1.12181 | − | 2.70830i | 0 | 2.57989 | + | 2.57989i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.19 | 0 | 2.77164 | − | 1.14805i | 0 | 0.866110 | − | 2.09098i | 0 | 11.6959 | + | 11.6959i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
49.20 | 0 | 2.77164 | − | 1.14805i | 0 | 5.49171 | − | 13.2582i | 0 | 1.23403 | + | 1.23403i | 0 | 6.36396 | − | 6.36396i | 0 | ||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 384.4.n.a | 96 | |
4.b | odd | 2 | 1 | 96.4.n.a | ✓ | 96 | |
32.g | even | 8 | 1 | inner | 384.4.n.a | 96 | |
32.h | odd | 8 | 1 | 96.4.n.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.4.n.a | ✓ | 96 | 4.b | odd | 2 | 1 | |
96.4.n.a | ✓ | 96 | 32.h | odd | 8 | 1 | |
384.4.n.a | 96 | 1.a | even | 1 | 1 | trivial | |
384.4.n.a | 96 | 32.g | even | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(384, [\chi])\).