Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1728,3,Mod(449,1728)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.449");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1728.q (of order \(6\), 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: | \(47.0845896815\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 288) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
449.1 | 0 | 0 | 0 | −7.38813 | − | 4.26554i | 0 | −1.36942 | − | 2.37191i | 0 | 0 | 0 | ||||||||||||||
449.2 | 0 | 0 | 0 | −7.38813 | − | 4.26554i | 0 | 1.36942 | + | 2.37191i | 0 | 0 | 0 | ||||||||||||||
449.3 | 0 | 0 | 0 | −4.45884 | − | 2.57431i | 0 | −1.35076 | − | 2.33958i | 0 | 0 | 0 | ||||||||||||||
449.4 | 0 | 0 | 0 | −4.45884 | − | 2.57431i | 0 | 1.35076 | + | 2.33958i | 0 | 0 | 0 | ||||||||||||||
449.5 | 0 | 0 | 0 | 0.439631 | + | 0.253821i | 0 | 6.44757 | + | 11.1675i | 0 | 0 | 0 | ||||||||||||||
449.6 | 0 | 0 | 0 | 0.439631 | + | 0.253821i | 0 | −6.44757 | − | 11.1675i | 0 | 0 | 0 | ||||||||||||||
449.7 | 0 | 0 | 0 | 1.15965 | + | 0.669525i | 0 | −0.328661 | − | 0.569258i | 0 | 0 | 0 | ||||||||||||||
449.8 | 0 | 0 | 0 | 1.15965 | + | 0.669525i | 0 | 0.328661 | + | 0.569258i | 0 | 0 | 0 | ||||||||||||||
449.9 | 0 | 0 | 0 | 3.32266 | + | 1.91834i | 0 | −2.70775 | − | 4.68995i | 0 | 0 | 0 | ||||||||||||||
449.10 | 0 | 0 | 0 | 3.32266 | + | 1.91834i | 0 | 2.70775 | + | 4.68995i | 0 | 0 | 0 | ||||||||||||||
449.11 | 0 | 0 | 0 | 6.92504 | + | 3.99817i | 0 | −6.10647 | − | 10.5767i | 0 | 0 | 0 | ||||||||||||||
449.12 | 0 | 0 | 0 | 6.92504 | + | 3.99817i | 0 | 6.10647 | + | 10.5767i | 0 | 0 | 0 | ||||||||||||||
1601.1 | 0 | 0 | 0 | −7.38813 | + | 4.26554i | 0 | −1.36942 | + | 2.37191i | 0 | 0 | 0 | ||||||||||||||
1601.2 | 0 | 0 | 0 | −7.38813 | + | 4.26554i | 0 | 1.36942 | − | 2.37191i | 0 | 0 | 0 | ||||||||||||||
1601.3 | 0 | 0 | 0 | −4.45884 | + | 2.57431i | 0 | −1.35076 | + | 2.33958i | 0 | 0 | 0 | ||||||||||||||
1601.4 | 0 | 0 | 0 | −4.45884 | + | 2.57431i | 0 | 1.35076 | − | 2.33958i | 0 | 0 | 0 | ||||||||||||||
1601.5 | 0 | 0 | 0 | 0.439631 | − | 0.253821i | 0 | 6.44757 | − | 11.1675i | 0 | 0 | 0 | ||||||||||||||
1601.6 | 0 | 0 | 0 | 0.439631 | − | 0.253821i | 0 | −6.44757 | + | 11.1675i | 0 | 0 | 0 | ||||||||||||||
1601.7 | 0 | 0 | 0 | 1.15965 | − | 0.669525i | 0 | −0.328661 | + | 0.569258i | 0 | 0 | 0 | ||||||||||||||
1601.8 | 0 | 0 | 0 | 1.15965 | − | 0.669525i | 0 | 0.328661 | − | 0.569258i | 0 | 0 | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
36.h | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1728.3.q.l | 24 | |
3.b | odd | 2 | 1 | 576.3.q.k | 24 | ||
4.b | odd | 2 | 1 | inner | 1728.3.q.l | 24 | |
8.b | even | 2 | 1 | 864.3.q.b | 24 | ||
8.d | odd | 2 | 1 | 864.3.q.b | 24 | ||
9.c | even | 3 | 1 | 576.3.q.k | 24 | ||
9.d | odd | 6 | 1 | inner | 1728.3.q.l | 24 | |
12.b | even | 2 | 1 | 576.3.q.k | 24 | ||
24.f | even | 2 | 1 | 288.3.q.a | ✓ | 24 | |
24.h | odd | 2 | 1 | 288.3.q.a | ✓ | 24 | |
36.f | odd | 6 | 1 | 576.3.q.k | 24 | ||
36.h | even | 6 | 1 | inner | 1728.3.q.l | 24 | |
72.j | odd | 6 | 1 | 864.3.q.b | 24 | ||
72.j | odd | 6 | 1 | 2592.3.e.j | 24 | ||
72.l | even | 6 | 1 | 864.3.q.b | 24 | ||
72.l | even | 6 | 1 | 2592.3.e.j | 24 | ||
72.n | even | 6 | 1 | 288.3.q.a | ✓ | 24 | |
72.n | even | 6 | 1 | 2592.3.e.j | 24 | ||
72.p | odd | 6 | 1 | 288.3.q.a | ✓ | 24 | |
72.p | odd | 6 | 1 | 2592.3.e.j | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
288.3.q.a | ✓ | 24 | 24.f | even | 2 | 1 | |
288.3.q.a | ✓ | 24 | 24.h | odd | 2 | 1 | |
288.3.q.a | ✓ | 24 | 72.n | even | 6 | 1 | |
288.3.q.a | ✓ | 24 | 72.p | odd | 6 | 1 | |
576.3.q.k | 24 | 3.b | odd | 2 | 1 | ||
576.3.q.k | 24 | 9.c | even | 3 | 1 | ||
576.3.q.k | 24 | 12.b | even | 2 | 1 | ||
576.3.q.k | 24 | 36.f | odd | 6 | 1 | ||
864.3.q.b | 24 | 8.b | even | 2 | 1 | ||
864.3.q.b | 24 | 8.d | odd | 2 | 1 | ||
864.3.q.b | 24 | 72.j | odd | 6 | 1 | ||
864.3.q.b | 24 | 72.l | even | 6 | 1 | ||
1728.3.q.l | 24 | 1.a | even | 1 | 1 | trivial | |
1728.3.q.l | 24 | 4.b | odd | 2 | 1 | inner | |
1728.3.q.l | 24 | 9.d | odd | 6 | 1 | inner | |
1728.3.q.l | 24 | 36.h | even | 6 | 1 | inner | |
2592.3.e.j | 24 | 72.j | odd | 6 | 1 | ||
2592.3.e.j | 24 | 72.l | even | 6 | 1 | ||
2592.3.e.j | 24 | 72.n | even | 6 | 1 | ||
2592.3.e.j | 24 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):
\( T_{5}^{12} - 90 T_{5}^{10} + 6627 T_{5}^{8} - 5292 T_{5}^{7} - 130514 T_{5}^{6} + 159084 T_{5}^{5} + 2092761 T_{5}^{4} - 6522444 T_{5}^{3} + 7884996 T_{5}^{2} - 4056048 T_{5} + 839056 \) |
\( T_{7}^{24} + 360 T_{7}^{22} + 90234 T_{7}^{20} + 11637392 T_{7}^{18} + 1080326187 T_{7}^{16} + 40447183032 T_{7}^{14} + 1071057738570 T_{7}^{12} + 13129538346000 T_{7}^{10} + \cdots + 296043132457216 \) |