Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1152,3,Mod(161,1152)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1152.161");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1152 = 2^{7} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1152.j (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: | \(31.3897264543\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 144) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 | 0 | 0 | 0 | −6.08688 | − | 6.08688i | 0 | − | 9.40026i | 0 | 0 | 0 | |||||||||||||||
161.2 | 0 | 0 | 0 | −5.52702 | − | 5.52702i | 0 | 7.79421i | 0 | 0 | 0 | ||||||||||||||||
161.3 | 0 | 0 | 0 | −5.14405 | − | 5.14405i | 0 | − | 7.48880i | 0 | 0 | 0 | |||||||||||||||
161.4 | 0 | 0 | 0 | −3.90343 | − | 3.90343i | 0 | 0.778757i | 0 | 0 | 0 | ||||||||||||||||
161.5 | 0 | 0 | 0 | −1.92848 | − | 1.92848i | 0 | 2.43162i | 0 | 0 | 0 | ||||||||||||||||
161.6 | 0 | 0 | 0 | −1.84570 | − | 1.84570i | 0 | 0.226665i | 0 | 0 | 0 | ||||||||||||||||
161.7 | 0 | 0 | 0 | −1.66372 | − | 1.66372i | 0 | 13.3224i | 0 | 0 | 0 | ||||||||||||||||
161.8 | 0 | 0 | 0 | −0.900179 | − | 0.900179i | 0 | − | 7.66460i | 0 | 0 | 0 | |||||||||||||||
161.9 | 0 | 0 | 0 | 0.900179 | + | 0.900179i | 0 | − | 7.66460i | 0 | 0 | 0 | |||||||||||||||
161.10 | 0 | 0 | 0 | 1.66372 | + | 1.66372i | 0 | 13.3224i | 0 | 0 | 0 | ||||||||||||||||
161.11 | 0 | 0 | 0 | 1.84570 | + | 1.84570i | 0 | 0.226665i | 0 | 0 | 0 | ||||||||||||||||
161.12 | 0 | 0 | 0 | 1.92848 | + | 1.92848i | 0 | 2.43162i | 0 | 0 | 0 | ||||||||||||||||
161.13 | 0 | 0 | 0 | 3.90343 | + | 3.90343i | 0 | 0.778757i | 0 | 0 | 0 | ||||||||||||||||
161.14 | 0 | 0 | 0 | 5.14405 | + | 5.14405i | 0 | − | 7.48880i | 0 | 0 | 0 | |||||||||||||||
161.15 | 0 | 0 | 0 | 5.52702 | + | 5.52702i | 0 | 7.79421i | 0 | 0 | 0 | ||||||||||||||||
161.16 | 0 | 0 | 0 | 6.08688 | + | 6.08688i | 0 | − | 9.40026i | 0 | 0 | 0 | |||||||||||||||
737.1 | 0 | 0 | 0 | −6.08688 | + | 6.08688i | 0 | 9.40026i | 0 | 0 | 0 | ||||||||||||||||
737.2 | 0 | 0 | 0 | −5.52702 | + | 5.52702i | 0 | − | 7.79421i | 0 | 0 | 0 | |||||||||||||||
737.3 | 0 | 0 | 0 | −5.14405 | + | 5.14405i | 0 | 7.48880i | 0 | 0 | 0 | ||||||||||||||||
737.4 | 0 | 0 | 0 | −3.90343 | + | 3.90343i | 0 | − | 0.778757i | 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.e | even | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1152.3.j.b | 32 | |
3.b | odd | 2 | 1 | inner | 1152.3.j.b | 32 | |
4.b | odd | 2 | 1 | 1152.3.j.a | 32 | ||
8.b | even | 2 | 1 | 576.3.j.a | 32 | ||
8.d | odd | 2 | 1 | 144.3.j.a | ✓ | 32 | |
12.b | even | 2 | 1 | 1152.3.j.a | 32 | ||
16.e | even | 4 | 1 | 576.3.j.a | 32 | ||
16.e | even | 4 | 1 | inner | 1152.3.j.b | 32 | |
16.f | odd | 4 | 1 | 144.3.j.a | ✓ | 32 | |
16.f | odd | 4 | 1 | 1152.3.j.a | 32 | ||
24.f | even | 2 | 1 | 144.3.j.a | ✓ | 32 | |
24.h | odd | 2 | 1 | 576.3.j.a | 32 | ||
48.i | odd | 4 | 1 | 576.3.j.a | 32 | ||
48.i | odd | 4 | 1 | inner | 1152.3.j.b | 32 | |
48.k | even | 4 | 1 | 144.3.j.a | ✓ | 32 | |
48.k | even | 4 | 1 | 1152.3.j.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
144.3.j.a | ✓ | 32 | 8.d | odd | 2 | 1 | |
144.3.j.a | ✓ | 32 | 16.f | odd | 4 | 1 | |
144.3.j.a | ✓ | 32 | 24.f | even | 2 | 1 | |
144.3.j.a | ✓ | 32 | 48.k | even | 4 | 1 | |
576.3.j.a | 32 | 8.b | even | 2 | 1 | ||
576.3.j.a | 32 | 16.e | even | 4 | 1 | ||
576.3.j.a | 32 | 24.h | odd | 2 | 1 | ||
576.3.j.a | 32 | 48.i | odd | 4 | 1 | ||
1152.3.j.a | 32 | 4.b | odd | 2 | 1 | ||
1152.3.j.a | 32 | 12.b | even | 2 | 1 | ||
1152.3.j.a | 32 | 16.f | odd | 4 | 1 | ||
1152.3.j.a | 32 | 48.k | even | 4 | 1 | ||
1152.3.j.b | 32 | 1.a | even | 1 | 1 | trivial | |
1152.3.j.b | 32 | 3.b | odd | 2 | 1 | inner | |
1152.3.j.b | 32 | 16.e | even | 4 | 1 | inner | |
1152.3.j.b | 32 | 48.i | odd | 4 | 1 | inner |