Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1728,2,Mod(431,1728)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.431");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1728.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: | \(13.7981494693\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 432) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
431.1 | 0 | 0 | 0 | −3.06203 | − | 3.06203i | 0 | −1.75946 | 0 | 0 | 0 | ||||||||||||||||
431.2 | 0 | 0 | 0 | −2.15382 | − | 2.15382i | 0 | −4.43758 | 0 | 0 | 0 | ||||||||||||||||
431.3 | 0 | 0 | 0 | −1.84591 | − | 1.84591i | 0 | 0.0355882 | 0 | 0 | 0 | ||||||||||||||||
431.4 | 0 | 0 | 0 | −1.75903 | − | 1.75903i | 0 | 4.05756 | 0 | 0 | 0 | ||||||||||||||||
431.5 | 0 | 0 | 0 | −1.29039 | − | 1.29039i | 0 | 3.83003 | 0 | 0 | 0 | ||||||||||||||||
431.6 | 0 | 0 | 0 | −1.26575 | − | 1.26575i | 0 | −1.47880 | 0 | 0 | 0 | ||||||||||||||||
431.7 | 0 | 0 | 0 | −0.463925 | − | 0.463925i | 0 | 1.85883 | 0 | 0 | 0 | ||||||||||||||||
431.8 | 0 | 0 | 0 | −0.0308139 | − | 0.0308139i | 0 | −2.10616 | 0 | 0 | 0 | ||||||||||||||||
431.9 | 0 | 0 | 0 | 0.0308139 | + | 0.0308139i | 0 | −2.10616 | 0 | 0 | 0 | ||||||||||||||||
431.10 | 0 | 0 | 0 | 0.463925 | + | 0.463925i | 0 | 1.85883 | 0 | 0 | 0 | ||||||||||||||||
431.11 | 0 | 0 | 0 | 1.26575 | + | 1.26575i | 0 | −1.47880 | 0 | 0 | 0 | ||||||||||||||||
431.12 | 0 | 0 | 0 | 1.29039 | + | 1.29039i | 0 | 3.83003 | 0 | 0 | 0 | ||||||||||||||||
431.13 | 0 | 0 | 0 | 1.75903 | + | 1.75903i | 0 | 4.05756 | 0 | 0 | 0 | ||||||||||||||||
431.14 | 0 | 0 | 0 | 1.84591 | + | 1.84591i | 0 | 0.0355882 | 0 | 0 | 0 | ||||||||||||||||
431.15 | 0 | 0 | 0 | 2.15382 | + | 2.15382i | 0 | −4.43758 | 0 | 0 | 0 | ||||||||||||||||
431.16 | 0 | 0 | 0 | 3.06203 | + | 3.06203i | 0 | −1.75946 | 0 | 0 | 0 | ||||||||||||||||
1295.1 | 0 | 0 | 0 | −3.06203 | + | 3.06203i | 0 | −1.75946 | 0 | 0 | 0 | ||||||||||||||||
1295.2 | 0 | 0 | 0 | −2.15382 | + | 2.15382i | 0 | −4.43758 | 0 | 0 | 0 | ||||||||||||||||
1295.3 | 0 | 0 | 0 | −1.84591 | + | 1.84591i | 0 | 0.0355882 | 0 | 0 | 0 | ||||||||||||||||
1295.4 | 0 | 0 | 0 | −1.75903 | + | 1.75903i | 0 | 4.05756 | 0 | 0 | 0 | ||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1728.2.l.b | 32 | |
3.b | odd | 2 | 1 | inner | 1728.2.l.b | 32 | |
4.b | odd | 2 | 1 | 432.2.l.b | ✓ | 32 | |
12.b | even | 2 | 1 | 432.2.l.b | ✓ | 32 | |
16.e | even | 4 | 1 | 432.2.l.b | ✓ | 32 | |
16.f | odd | 4 | 1 | inner | 1728.2.l.b | 32 | |
48.i | odd | 4 | 1 | 432.2.l.b | ✓ | 32 | |
48.k | even | 4 | 1 | inner | 1728.2.l.b | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
432.2.l.b | ✓ | 32 | 4.b | odd | 2 | 1 | |
432.2.l.b | ✓ | 32 | 12.b | even | 2 | 1 | |
432.2.l.b | ✓ | 32 | 16.e | even | 4 | 1 | |
432.2.l.b | ✓ | 32 | 48.i | odd | 4 | 1 | |
1728.2.l.b | 32 | 1.a | even | 1 | 1 | trivial | |
1728.2.l.b | 32 | 3.b | odd | 2 | 1 | inner | |
1728.2.l.b | 32 | 16.f | odd | 4 | 1 | inner | |
1728.2.l.b | 32 | 48.k | even | 4 | 1 | inner |