Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3528,2,Mod(521,3528)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3528, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3528.521");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3528.bl (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: | \(28.1712218331\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
521.1 | 0 | 0 | 0 | −2.09315 | − | 3.62544i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.2 | 0 | 0 | 0 | −2.09315 | − | 3.62544i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.3 | 0 | 0 | 0 | −1.82841 | − | 3.16690i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.4 | 0 | 0 | 0 | −1.82841 | − | 3.16690i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.5 | 0 | 0 | 0 | −1.28721 | − | 2.22952i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.6 | 0 | 0 | 0 | −1.28721 | − | 2.22952i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.7 | 0 | 0 | 0 | −0.786588 | − | 1.36241i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.8 | 0 | 0 | 0 | −0.786588 | − | 1.36241i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.9 | 0 | 0 | 0 | 0.786588 | + | 1.36241i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.10 | 0 | 0 | 0 | 0.786588 | + | 1.36241i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.11 | 0 | 0 | 0 | 1.28721 | + | 2.22952i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.12 | 0 | 0 | 0 | 1.28721 | + | 2.22952i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.13 | 0 | 0 | 0 | 1.82841 | + | 3.16690i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.14 | 0 | 0 | 0 | 1.82841 | + | 3.16690i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.15 | 0 | 0 | 0 | 2.09315 | + | 3.62544i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.16 | 0 | 0 | 0 | 2.09315 | + | 3.62544i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
1097.1 | 0 | 0 | 0 | −2.09315 | + | 3.62544i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
1097.2 | 0 | 0 | 0 | −2.09315 | + | 3.62544i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
1097.3 | 0 | 0 | 0 | −1.82841 | + | 3.16690i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
1097.4 | 0 | 0 | 0 | −1.82841 | + | 3.16690i | 0 | 0 | 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 |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.c | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
21.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3528.2.bl.d | 32 | |
3.b | odd | 2 | 1 | inner | 3528.2.bl.d | 32 | |
7.b | odd | 2 | 1 | inner | 3528.2.bl.d | 32 | |
7.c | even | 3 | 1 | 3528.2.k.c | ✓ | 16 | |
7.c | even | 3 | 1 | inner | 3528.2.bl.d | 32 | |
7.d | odd | 6 | 1 | 3528.2.k.c | ✓ | 16 | |
7.d | odd | 6 | 1 | inner | 3528.2.bl.d | 32 | |
21.c | even | 2 | 1 | inner | 3528.2.bl.d | 32 | |
21.g | even | 6 | 1 | 3528.2.k.c | ✓ | 16 | |
21.g | even | 6 | 1 | inner | 3528.2.bl.d | 32 | |
21.h | odd | 6 | 1 | 3528.2.k.c | ✓ | 16 | |
21.h | odd | 6 | 1 | inner | 3528.2.bl.d | 32 | |
28.f | even | 6 | 1 | 7056.2.k.i | 16 | ||
28.g | odd | 6 | 1 | 7056.2.k.i | 16 | ||
84.j | odd | 6 | 1 | 7056.2.k.i | 16 | ||
84.n | even | 6 | 1 | 7056.2.k.i | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3528.2.k.c | ✓ | 16 | 7.c | even | 3 | 1 | |
3528.2.k.c | ✓ | 16 | 7.d | odd | 6 | 1 | |
3528.2.k.c | ✓ | 16 | 21.g | even | 6 | 1 | |
3528.2.k.c | ✓ | 16 | 21.h | odd | 6 | 1 | |
3528.2.bl.d | 32 | 1.a | even | 1 | 1 | trivial | |
3528.2.bl.d | 32 | 3.b | odd | 2 | 1 | inner | |
3528.2.bl.d | 32 | 7.b | odd | 2 | 1 | inner | |
3528.2.bl.d | 32 | 7.c | even | 3 | 1 | inner | |
3528.2.bl.d | 32 | 7.d | odd | 6 | 1 | inner | |
3528.2.bl.d | 32 | 21.c | even | 2 | 1 | inner | |
3528.2.bl.d | 32 | 21.g | even | 6 | 1 | inner | |
3528.2.bl.d | 32 | 21.h | odd | 6 | 1 | inner | |
7056.2.k.i | 16 | 28.f | even | 6 | 1 | ||
7056.2.k.i | 16 | 28.g | odd | 6 | 1 | ||
7056.2.k.i | 16 | 84.j | odd | 6 | 1 | ||
7056.2.k.i | 16 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 40 T_{5}^{14} + 1068 T_{5}^{12} + 16000 T_{5}^{10} + 173580 T_{5}^{8} + 1096960 T_{5}^{6} + \cdots + 14776336 \) acting on \(S_{2}^{\mathrm{new}}(3528, [\chi])\).