Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,2,Mod(47,384)]
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([4, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 384.o (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: | \(3.06625543762\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(14\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | 0 | −1.72928 | − | 0.0979076i | 0 | −0.180206 | + | 0.0746437i | 0 | 0.289055 | + | 0.289055i | 0 | 2.98083 | + | 0.338620i | 0 | ||||||||||
47.2 | 0 | −1.47792 | − | 0.903198i | 0 | −2.81491 | + | 1.16597i | 0 | 0.543879 | + | 0.543879i | 0 | 1.36847 | + | 2.66970i | 0 | ||||||||||
47.3 | 0 | −1.25135 | + | 1.19755i | 0 | −2.18808 | + | 0.906333i | 0 | 1.93241 | + | 1.93241i | 0 | 0.131733 | − | 2.99711i | 0 | ||||||||||
47.4 | 0 | −1.18890 | + | 1.25957i | 0 | 1.06973 | − | 0.443098i | 0 | −2.37247 | − | 2.37247i | 0 | −0.173046 | − | 2.99501i | 0 | ||||||||||
47.5 | 0 | −0.988287 | − | 1.42242i | 0 | 2.70066 | − | 1.11865i | 0 | 3.28204 | + | 3.28204i | 0 | −1.04658 | + | 2.81152i | 0 | ||||||||||
47.6 | 0 | −0.602068 | − | 1.62404i | 0 | 0.378520 | − | 0.156788i | 0 | −2.01144 | − | 2.01144i | 0 | −2.27503 | + | 1.95557i | 0 | ||||||||||
47.7 | 0 | −0.0499748 | + | 1.73133i | 0 | −1.06973 | + | 0.443098i | 0 | −2.37247 | − | 2.37247i | 0 | −2.99501 | − | 0.173046i | 0 | ||||||||||
47.8 | 0 | 0.0380372 | + | 1.73163i | 0 | 2.18808 | − | 0.906333i | 0 | 1.93241 | + | 1.93241i | 0 | −2.99711 | + | 0.131733i | 0 | ||||||||||
47.9 | 0 | 0.266294 | − | 1.71146i | 0 | −3.14689 | + | 1.30348i | 0 | −0.663471 | − | 0.663471i | 0 | −2.85817 | − | 0.911503i | 0 | ||||||||||
47.10 | 0 | 1.02188 | − | 1.39848i | 0 | 3.14689 | − | 1.30348i | 0 | −0.663471 | − | 0.663471i | 0 | −0.911503 | − | 2.85817i | 0 | ||||||||||
47.11 | 0 | 1.29202 | + | 1.15356i | 0 | 0.180206 | − | 0.0746437i | 0 | 0.289055 | + | 0.289055i | 0 | 0.338620 | + | 2.98083i | 0 | ||||||||||
47.12 | 0 | 1.57410 | − | 0.722645i | 0 | −0.378520 | + | 0.156788i | 0 | −2.01144 | − | 2.01144i | 0 | 1.95557 | − | 2.27503i | 0 | ||||||||||
47.13 | 0 | 1.68370 | + | 0.406387i | 0 | 2.81491 | − | 1.16597i | 0 | 0.543879 | + | 0.543879i | 0 | 2.66970 | + | 1.36847i | 0 | ||||||||||
47.14 | 0 | 1.70463 | − | 0.306981i | 0 | −2.70066 | + | 1.11865i | 0 | 3.28204 | + | 3.28204i | 0 | 2.81152 | − | 1.04658i | 0 | ||||||||||
143.1 | 0 | −1.69392 | + | 0.361451i | 0 | −0.348970 | + | 0.842489i | 0 | 0.471834 | − | 0.471834i | 0 | 2.73871 | − | 1.22453i | 0 | ||||||||||
143.2 | 0 | −1.45336 | + | 0.942196i | 0 | 0.348970 | − | 0.842489i | 0 | 0.471834 | − | 0.471834i | 0 | 1.22453 | − | 2.73871i | 0 | ||||||||||
143.3 | 0 | −1.44423 | − | 0.956131i | 0 | 1.56013 | − | 3.76650i | 0 | 0.838552 | − | 0.838552i | 0 | 1.17163 | + | 2.76175i | 0 | ||||||||||
143.4 | 0 | −1.14895 | − | 1.29612i | 0 | −0.970873 | + | 2.34389i | 0 | 1.60572 | − | 1.60572i | 0 | −0.359830 | + | 2.97834i | 0 | ||||||||||
143.5 | 0 | −0.684042 | − | 1.59125i | 0 | −0.296199 | + | 0.715088i | 0 | −2.77714 | + | 2.77714i | 0 | −2.06417 | + | 2.17697i | 0 | ||||||||||
143.6 | 0 | −0.345141 | + | 1.69731i | 0 | −1.56013 | + | 3.76650i | 0 | 0.838552 | − | 0.838552i | 0 | −2.76175 | − | 1.17163i | 0 | ||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
32.h | odd | 8 | 1 | inner |
96.o | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 384.2.o.a | 56 | |
3.b | odd | 2 | 1 | inner | 384.2.o.a | 56 | |
4.b | odd | 2 | 1 | 96.2.o.a | ✓ | 56 | |
8.b | even | 2 | 1 | 768.2.o.a | 56 | ||
8.d | odd | 2 | 1 | 768.2.o.b | 56 | ||
12.b | even | 2 | 1 | 96.2.o.a | ✓ | 56 | |
24.f | even | 2 | 1 | 768.2.o.b | 56 | ||
24.h | odd | 2 | 1 | 768.2.o.a | 56 | ||
32.g | even | 8 | 1 | 96.2.o.a | ✓ | 56 | |
32.g | even | 8 | 1 | 768.2.o.b | 56 | ||
32.h | odd | 8 | 1 | inner | 384.2.o.a | 56 | |
32.h | odd | 8 | 1 | 768.2.o.a | 56 | ||
96.o | even | 8 | 1 | inner | 384.2.o.a | 56 | |
96.o | even | 8 | 1 | 768.2.o.a | 56 | ||
96.p | odd | 8 | 1 | 96.2.o.a | ✓ | 56 | |
96.p | odd | 8 | 1 | 768.2.o.b | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.2.o.a | ✓ | 56 | 4.b | odd | 2 | 1 | |
96.2.o.a | ✓ | 56 | 12.b | even | 2 | 1 | |
96.2.o.a | ✓ | 56 | 32.g | even | 8 | 1 | |
96.2.o.a | ✓ | 56 | 96.p | odd | 8 | 1 | |
384.2.o.a | 56 | 1.a | even | 1 | 1 | trivial | |
384.2.o.a | 56 | 3.b | odd | 2 | 1 | inner | |
384.2.o.a | 56 | 32.h | odd | 8 | 1 | inner | |
384.2.o.a | 56 | 96.o | even | 8 | 1 | inner | |
768.2.o.a | 56 | 8.b | even | 2 | 1 | ||
768.2.o.a | 56 | 24.h | odd | 2 | 1 | ||
768.2.o.a | 56 | 32.h | odd | 8 | 1 | ||
768.2.o.a | 56 | 96.o | even | 8 | 1 | ||
768.2.o.b | 56 | 8.d | odd | 2 | 1 | ||
768.2.o.b | 56 | 24.f | even | 2 | 1 | ||
768.2.o.b | 56 | 32.g | even | 8 | 1 | ||
768.2.o.b | 56 | 96.p | odd | 8 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(384, [\chi])\).