Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1323,2,Mod(521,1323)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1323.521");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1323 = 3^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1323.i (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: | \(10.5642081874\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 441) |
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 | − | 1.17820i | 0 | 0.611843 | 2.16601 | − | 3.75164i | 0 | 0 | − | 3.07728i | 0 | −4.42019 | − | 2.55200i | ||||||||||||
521.2 | 2.57819i | 0 | −4.64709 | −1.16595 | + | 2.01948i | 0 | 0 | − | 6.82470i | 0 | −5.20660 | − | 3.00603i | |||||||||||||
521.3 | − | 0.424201i | 0 | 1.82005 | 1.80381 | − | 3.12430i | 0 | 0 | − | 1.62047i | 0 | −1.32533 | − | 0.765180i | ||||||||||||
521.4 | − | 2.37274i | 0 | −3.62990 | −1.71774 | + | 2.97522i | 0 | 0 | 3.86732i | 0 | 7.05942 | + | 4.07576i | |||||||||||||
521.5 | − | 1.48451i | 0 | −0.203760 | 0.154215 | − | 0.267109i | 0 | 0 | − | 2.66653i | 0 | −0.396525 | − | 0.228934i | ||||||||||||
521.6 | − | 2.70883i | 0 | −5.33776 | −0.601464 | + | 1.04177i | 0 | 0 | 9.04141i | 0 | 2.82197 | + | 1.62926i | |||||||||||||
521.7 | 1.86894i | 0 | −1.49292 | 1.25287 | − | 2.17003i | 0 | 0 | 0.947692i | 0 | 4.05565 | + | 2.34153i | ||||||||||||||
521.8 | − | 0.981621i | 0 | 1.03642 | −0.940599 | + | 1.62916i | 0 | 0 | − | 2.98061i | 0 | 1.59922 | + | 0.923312i | ||||||||||||
521.9 | 0.664297i | 0 | 1.55871 | −0.0141520 | + | 0.0245119i | 0 | 0 | 2.36404i | 0 | −0.0162832 | − | 0.00940110i | ||||||||||||||
521.10 | 0.664297i | 0 | 1.55871 | 0.0141520 | − | 0.0245119i | 0 | 0 | 2.36404i | 0 | 0.0162832 | + | 0.00940110i | ||||||||||||||
521.11 | 0.122344i | 0 | 1.98503 | 0.264715 | − | 0.458500i | 0 | 0 | 0.487547i | 0 | 0.0560949 | + | 0.0323864i | ||||||||||||||
521.12 | 0.122344i | 0 | 1.98503 | −0.264715 | + | 0.458500i | 0 | 0 | 0.487547i | 0 | −0.0560949 | − | 0.0323864i | ||||||||||||||
521.13 | − | 1.48451i | 0 | −0.203760 | −0.154215 | + | 0.267109i | 0 | 0 | − | 2.66653i | 0 | 0.396525 | + | 0.228934i | ||||||||||||
521.14 | 2.57819i | 0 | −4.64709 | 1.16595 | − | 2.01948i | 0 | 0 | − | 6.82470i | 0 | 5.20660 | + | 3.00603i | |||||||||||||
521.15 | 1.83202i | 0 | −1.35631 | 0.322784 | − | 0.559079i | 0 | 0 | 1.17925i | 0 | 1.02425 | + | 0.591348i | ||||||||||||||
521.16 | 2.08430i | 0 | −2.34432 | 1.65233 | − | 2.86191i | 0 | 0 | − | 0.717672i | 0 | 5.96509 | + | 3.44395i | |||||||||||||
521.17 | 1.83202i | 0 | −1.35631 | −0.322784 | + | 0.559079i | 0 | 0 | 1.17925i | 0 | −1.02425 | − | 0.591348i | ||||||||||||||
521.18 | − | 0.981621i | 0 | 1.03642 | 0.940599 | − | 1.62916i | 0 | 0 | − | 2.98061i | 0 | −1.59922 | − | 0.923312i | ||||||||||||
521.19 | − | 0.424201i | 0 | 1.82005 | −1.80381 | + | 3.12430i | 0 | 0 | − | 1.62047i | 0 | 1.32533 | + | 0.765180i | ||||||||||||
521.20 | − | 1.17820i | 0 | 0.611843 | −2.16601 | + | 3.75164i | 0 | 0 | − | 3.07728i | 0 | 4.42019 | + | 2.55200i | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
63.i | even | 6 | 1 | inner |
63.j | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1323.2.i.d | 48 | |
3.b | odd | 2 | 1 | 441.2.i.d | 48 | ||
7.b | odd | 2 | 1 | inner | 1323.2.i.d | 48 | |
7.c | even | 3 | 1 | 1323.2.o.e | 48 | ||
7.c | even | 3 | 1 | 1323.2.s.d | 48 | ||
7.d | odd | 6 | 1 | 1323.2.o.e | 48 | ||
7.d | odd | 6 | 1 | 1323.2.s.d | 48 | ||
9.c | even | 3 | 1 | 441.2.s.d | 48 | ||
9.d | odd | 6 | 1 | 1323.2.s.d | 48 | ||
21.c | even | 2 | 1 | 441.2.i.d | 48 | ||
21.g | even | 6 | 1 | 441.2.o.e | ✓ | 48 | |
21.g | even | 6 | 1 | 441.2.s.d | 48 | ||
21.h | odd | 6 | 1 | 441.2.o.e | ✓ | 48 | |
21.h | odd | 6 | 1 | 441.2.s.d | 48 | ||
63.g | even | 3 | 1 | 441.2.o.e | ✓ | 48 | |
63.h | even | 3 | 1 | 441.2.i.d | 48 | ||
63.i | even | 6 | 1 | inner | 1323.2.i.d | 48 | |
63.j | odd | 6 | 1 | inner | 1323.2.i.d | 48 | |
63.k | odd | 6 | 1 | 441.2.o.e | ✓ | 48 | |
63.l | odd | 6 | 1 | 441.2.s.d | 48 | ||
63.n | odd | 6 | 1 | 1323.2.o.e | 48 | ||
63.o | even | 6 | 1 | 1323.2.s.d | 48 | ||
63.s | even | 6 | 1 | 1323.2.o.e | 48 | ||
63.t | odd | 6 | 1 | 441.2.i.d | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
441.2.i.d | 48 | 3.b | odd | 2 | 1 | ||
441.2.i.d | 48 | 21.c | even | 2 | 1 | ||
441.2.i.d | 48 | 63.h | even | 3 | 1 | ||
441.2.i.d | 48 | 63.t | odd | 6 | 1 | ||
441.2.o.e | ✓ | 48 | 21.g | even | 6 | 1 | |
441.2.o.e | ✓ | 48 | 21.h | odd | 6 | 1 | |
441.2.o.e | ✓ | 48 | 63.g | even | 3 | 1 | |
441.2.o.e | ✓ | 48 | 63.k | odd | 6 | 1 | |
441.2.s.d | 48 | 9.c | even | 3 | 1 | ||
441.2.s.d | 48 | 21.g | even | 6 | 1 | ||
441.2.s.d | 48 | 21.h | odd | 6 | 1 | ||
441.2.s.d | 48 | 63.l | odd | 6 | 1 | ||
1323.2.i.d | 48 | 1.a | even | 1 | 1 | trivial | |
1323.2.i.d | 48 | 7.b | odd | 2 | 1 | inner | |
1323.2.i.d | 48 | 63.i | even | 6 | 1 | inner | |
1323.2.i.d | 48 | 63.j | odd | 6 | 1 | inner | |
1323.2.o.e | 48 | 7.c | even | 3 | 1 | ||
1323.2.o.e | 48 | 7.d | odd | 6 | 1 | ||
1323.2.o.e | 48 | 63.n | odd | 6 | 1 | ||
1323.2.o.e | 48 | 63.s | even | 6 | 1 | ||
1323.2.s.d | 48 | 7.c | even | 3 | 1 | ||
1323.2.s.d | 48 | 7.d | odd | 6 | 1 | ||
1323.2.s.d | 48 | 9.d | odd | 6 | 1 | ||
1323.2.s.d | 48 | 63.o | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 36 T_{2}^{22} + 558 T_{2}^{20} + 4884 T_{2}^{18} + 26613 T_{2}^{16} + 93876 T_{2}^{14} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\).