Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,7,Mod(31,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.31");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 384.l (of order \(4\), degree \(2\), 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: | \(88.3407681100\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 48) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | −11.0227 | − | 11.0227i | 0 | 159.596 | + | 159.596i | 0 | −527.602 | 0 | 243.000i | 0 | ||||||||||||||
31.2 | 0 | −11.0227 | − | 11.0227i | 0 | 124.587 | + | 124.587i | 0 | 200.172 | 0 | 243.000i | 0 | ||||||||||||||
31.3 | 0 | −11.0227 | − | 11.0227i | 0 | 73.3870 | + | 73.3870i | 0 | 291.401 | 0 | 243.000i | 0 | ||||||||||||||
31.4 | 0 | −11.0227 | − | 11.0227i | 0 | 26.8357 | + | 26.8357i | 0 | −81.4485 | 0 | 243.000i | 0 | ||||||||||||||
31.5 | 0 | −11.0227 | − | 11.0227i | 0 | 26.6395 | + | 26.6395i | 0 | −403.840 | 0 | 243.000i | 0 | ||||||||||||||
31.6 | 0 | −11.0227 | − | 11.0227i | 0 | −38.1198 | − | 38.1198i | 0 | 34.3159 | 0 | 243.000i | 0 | ||||||||||||||
31.7 | 0 | −11.0227 | − | 11.0227i | 0 | 9.26689 | + | 9.26689i | 0 | −320.373 | 0 | 243.000i | 0 | ||||||||||||||
31.8 | 0 | −11.0227 | − | 11.0227i | 0 | −47.0845 | − | 47.0845i | 0 | 400.703 | 0 | 243.000i | 0 | ||||||||||||||
31.9 | 0 | −11.0227 | − | 11.0227i | 0 | 99.4317 | + | 99.4317i | 0 | 407.926 | 0 | 243.000i | 0 | ||||||||||||||
31.10 | 0 | −11.0227 | − | 11.0227i | 0 | −132.791 | − | 132.791i | 0 | −560.492 | 0 | 243.000i | 0 | ||||||||||||||
31.11 | 0 | −11.0227 | − | 11.0227i | 0 | −145.472 | − | 145.472i | 0 | 535.877 | 0 | 243.000i | 0 | ||||||||||||||
31.12 | 0 | −11.0227 | − | 11.0227i | 0 | −156.277 | − | 156.277i | 0 | 23.3623 | 0 | 243.000i | 0 | ||||||||||||||
31.13 | 0 | 11.0227 | + | 11.0227i | 0 | 160.484 | + | 160.484i | 0 | −53.5181 | 0 | 243.000i | 0 | ||||||||||||||
31.14 | 0 | 11.0227 | + | 11.0227i | 0 | −128.299 | − | 128.299i | 0 | 76.4178 | 0 | 243.000i | 0 | ||||||||||||||
31.15 | 0 | 11.0227 | + | 11.0227i | 0 | 98.7574 | + | 98.7574i | 0 | −218.381 | 0 | 243.000i | 0 | ||||||||||||||
31.16 | 0 | 11.0227 | + | 11.0227i | 0 | 95.7535 | + | 95.7535i | 0 | 338.697 | 0 | 243.000i | 0 | ||||||||||||||
31.17 | 0 | 11.0227 | + | 11.0227i | 0 | 59.6079 | + | 59.6079i | 0 | −661.000 | 0 | 243.000i | 0 | ||||||||||||||
31.18 | 0 | 11.0227 | + | 11.0227i | 0 | −45.7781 | − | 45.7781i | 0 | −565.544 | 0 | 243.000i | 0 | ||||||||||||||
31.19 | 0 | 11.0227 | + | 11.0227i | 0 | 55.7840 | + | 55.7840i | 0 | 496.753 | 0 | 243.000i | 0 | ||||||||||||||
31.20 | 0 | 11.0227 | + | 11.0227i | 0 | −29.4894 | − | 29.4894i | 0 | 461.206 | 0 | 243.000i | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.f | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 384.7.l.b | 48 | |
4.b | odd | 2 | 1 | 384.7.l.a | 48 | ||
8.b | even | 2 | 1 | 48.7.l.a | ✓ | 48 | |
8.d | odd | 2 | 1 | 192.7.l.a | 48 | ||
16.e | even | 4 | 1 | 192.7.l.a | 48 | ||
16.e | even | 4 | 1 | 384.7.l.a | 48 | ||
16.f | odd | 4 | 1 | 48.7.l.a | ✓ | 48 | |
16.f | odd | 4 | 1 | inner | 384.7.l.b | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.7.l.a | ✓ | 48 | 8.b | even | 2 | 1 | |
48.7.l.a | ✓ | 48 | 16.f | odd | 4 | 1 | |
192.7.l.a | 48 | 8.d | odd | 2 | 1 | ||
192.7.l.a | 48 | 16.e | even | 4 | 1 | ||
384.7.l.a | 48 | 4.b | odd | 2 | 1 | ||
384.7.l.a | 48 | 16.e | even | 4 | 1 | ||
384.7.l.b | 48 | 1.a | even | 1 | 1 | trivial | |
384.7.l.b | 48 | 16.f | odd | 4 | 1 | inner |