Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,4,Mod(521,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.521");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1764.t (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: | \(104.079369250\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | −10.7460 | − | 18.6127i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.2 | 0 | 0 | 0 | −10.7460 | − | 18.6127i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.3 | 0 | 0 | 0 | −7.13585 | − | 12.3596i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.4 | 0 | 0 | 0 | −7.13585 | − | 12.3596i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.5 | 0 | 0 | 0 | −6.78515 | − | 11.7522i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.6 | 0 | 0 | 0 | −6.78515 | − | 11.7522i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.7 | 0 | 0 | 0 | −4.41277 | − | 7.64315i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.8 | 0 | 0 | 0 | −4.41277 | − | 7.64315i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.9 | 0 | 0 | 0 | −3.27916 | − | 5.67967i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.10 | 0 | 0 | 0 | −3.27916 | − | 5.67967i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.11 | 0 | 0 | 0 | −0.582251 | − | 1.00849i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.12 | 0 | 0 | 0 | −0.582251 | − | 1.00849i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.13 | 0 | 0 | 0 | 0.582251 | + | 1.00849i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.14 | 0 | 0 | 0 | 0.582251 | + | 1.00849i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.15 | 0 | 0 | 0 | 3.27916 | + | 5.67967i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.16 | 0 | 0 | 0 | 3.27916 | + | 5.67967i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.17 | 0 | 0 | 0 | 4.41277 | + | 7.64315i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.18 | 0 | 0 | 0 | 4.41277 | + | 7.64315i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.19 | 0 | 0 | 0 | 6.78515 | + | 11.7522i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
521.20 | 0 | 0 | 0 | 6.78515 | + | 11.7522i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||
See all 48 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 | 1764.4.t.c | 48 | |
3.b | odd | 2 | 1 | inner | 1764.4.t.c | 48 | |
7.b | odd | 2 | 1 | inner | 1764.4.t.c | 48 | |
7.c | even | 3 | 1 | 1764.4.f.b | ✓ | 24 | |
7.c | even | 3 | 1 | inner | 1764.4.t.c | 48 | |
7.d | odd | 6 | 1 | 1764.4.f.b | ✓ | 24 | |
7.d | odd | 6 | 1 | inner | 1764.4.t.c | 48 | |
21.c | even | 2 | 1 | inner | 1764.4.t.c | 48 | |
21.g | even | 6 | 1 | 1764.4.f.b | ✓ | 24 | |
21.g | even | 6 | 1 | inner | 1764.4.t.c | 48 | |
21.h | odd | 6 | 1 | 1764.4.f.b | ✓ | 24 | |
21.h | odd | 6 | 1 | inner | 1764.4.t.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1764.4.f.b | ✓ | 24 | 7.c | even | 3 | 1 | |
1764.4.f.b | ✓ | 24 | 7.d | odd | 6 | 1 | |
1764.4.f.b | ✓ | 24 | 21.g | even | 6 | 1 | |
1764.4.f.b | ✓ | 24 | 21.h | odd | 6 | 1 | |
1764.4.t.c | 48 | 1.a | even | 1 | 1 | trivial | |
1764.4.t.c | 48 | 3.b | odd | 2 | 1 | inner | |
1764.4.t.c | 48 | 7.b | odd | 2 | 1 | inner | |
1764.4.t.c | 48 | 7.c | even | 3 | 1 | inner | |
1764.4.t.c | 48 | 7.d | odd | 6 | 1 | inner | |
1764.4.t.c | 48 | 21.c | even | 2 | 1 | inner | |
1764.4.t.c | 48 | 21.g | even | 6 | 1 | inner | |
1764.4.t.c | 48 | 21.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{24} + 972 T_{5}^{22} + 620730 T_{5}^{20} + 221373352 T_{5}^{18} + 56634544824 T_{5}^{16} + 9623457152688 T_{5}^{14} + \cdots + 61\!\cdots\!04 \)
acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\).