Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,2,Mod(227,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, 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("882.227");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 882.l (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: | \(7.04280545828\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
227.1 | − | 1.00000i | −1.70306 | + | 0.315583i | −1.00000 | 0.220087 | − | 0.381202i | 0.315583 | + | 1.70306i | 0 | 1.00000i | 2.80082 | − | 1.07491i | −0.381202 | − | 0.220087i | |||||||
227.2 | − | 1.00000i | −1.63594 | − | 0.568941i | −1.00000 | 1.55389 | − | 2.69141i | −0.568941 | + | 1.63594i | 0 | 1.00000i | 2.35261 | + | 1.86151i | −2.69141 | − | 1.55389i | |||||||
227.3 | − | 1.00000i | −1.52614 | − | 0.819074i | −1.00000 | −1.99341 | + | 3.45268i | −0.819074 | + | 1.52614i | 0 | 1.00000i | 1.65823 | + | 2.50005i | 3.45268 | + | 1.99341i | |||||||
227.4 | − | 1.00000i | −0.495324 | − | 1.65972i | −1.00000 | −0.995200 | + | 1.72374i | −1.65972 | + | 0.495324i | 0 | 1.00000i | −2.50931 | + | 1.64419i | 1.72374 | + | 0.995200i | |||||||
227.5 | − | 1.00000i | −0.307440 | − | 1.70455i | −1.00000 | 1.55067 | − | 2.68584i | −1.70455 | + | 0.307440i | 0 | 1.00000i | −2.81096 | + | 1.04809i | −2.68584 | − | 1.55067i | |||||||
227.6 | − | 1.00000i | −0.149173 | + | 1.72562i | −1.00000 | 0.948881 | − | 1.64351i | 1.72562 | + | 0.149173i | 0 | 1.00000i | −2.95549 | − | 0.514831i | −1.64351 | − | 0.948881i | |||||||
227.7 | − | 1.00000i | 0.149173 | − | 1.72562i | −1.00000 | −0.948881 | + | 1.64351i | −1.72562 | − | 0.149173i | 0 | 1.00000i | −2.95549 | − | 0.514831i | 1.64351 | + | 0.948881i | |||||||
227.8 | − | 1.00000i | 0.307440 | + | 1.70455i | −1.00000 | −1.55067 | + | 2.68584i | 1.70455 | − | 0.307440i | 0 | 1.00000i | −2.81096 | + | 1.04809i | 2.68584 | + | 1.55067i | |||||||
227.9 | − | 1.00000i | 0.495324 | + | 1.65972i | −1.00000 | 0.995200 | − | 1.72374i | 1.65972 | − | 0.495324i | 0 | 1.00000i | −2.50931 | + | 1.64419i | −1.72374 | − | 0.995200i | |||||||
227.10 | − | 1.00000i | 1.52614 | + | 0.819074i | −1.00000 | 1.99341 | − | 3.45268i | 0.819074 | − | 1.52614i | 0 | 1.00000i | 1.65823 | + | 2.50005i | −3.45268 | − | 1.99341i | |||||||
227.11 | − | 1.00000i | 1.63594 | + | 0.568941i | −1.00000 | −1.55389 | + | 2.69141i | 0.568941 | − | 1.63594i | 0 | 1.00000i | 2.35261 | + | 1.86151i | 2.69141 | + | 1.55389i | |||||||
227.12 | − | 1.00000i | 1.70306 | − | 0.315583i | −1.00000 | −0.220087 | + | 0.381202i | −0.315583 | − | 1.70306i | 0 | 1.00000i | 2.80082 | − | 1.07491i | 0.381202 | + | 0.220087i | |||||||
227.13 | 1.00000i | −1.71914 | − | 0.211098i | −1.00000 | 0.584859 | − | 1.01301i | 0.211098 | − | 1.71914i | 0 | − | 1.00000i | 2.91088 | + | 0.725813i | 1.01301 | + | 0.584859i | |||||||
227.14 | 1.00000i | −1.66366 | − | 0.481898i | −1.00000 | −0.712984 | + | 1.23492i | 0.481898 | − | 1.66366i | 0 | − | 1.00000i | 2.53555 | + | 1.60343i | −1.23492 | − | 0.712984i | |||||||
227.15 | 1.00000i | −1.60962 | − | 0.639623i | −1.00000 | −1.35026 | + | 2.33872i | 0.639623 | − | 1.60962i | 0 | − | 1.00000i | 2.18177 | + | 2.05910i | −2.33872 | − | 1.35026i | |||||||
227.16 | 1.00000i | −1.45331 | + | 0.942282i | −1.00000 | 0.724499 | − | 1.25487i | −0.942282 | − | 1.45331i | 0 | − | 1.00000i | 1.22421 | − | 2.73885i | 1.25487 | + | 0.724499i | |||||||
227.17 | 1.00000i | −0.848829 | + | 1.50980i | −1.00000 | 1.96067 | − | 3.39598i | −1.50980 | − | 0.848829i | 0 | − | 1.00000i | −1.55898 | − | 2.56312i | 3.39598 | + | 1.96067i | |||||||
227.18 | 1.00000i | −0.765075 | − | 1.55392i | −1.00000 | −0.474556 | + | 0.821956i | 1.55392 | − | 0.765075i | 0 | − | 1.00000i | −1.82932 | + | 2.37773i | −0.821956 | − | 0.474556i | |||||||
227.19 | 1.00000i | 0.765075 | + | 1.55392i | −1.00000 | 0.474556 | − | 0.821956i | −1.55392 | + | 0.765075i | 0 | − | 1.00000i | −1.82932 | + | 2.37773i | 0.821956 | + | 0.474556i | |||||||
227.20 | 1.00000i | 0.848829 | − | 1.50980i | −1.00000 | −1.96067 | + | 3.39598i | 1.50980 | + | 0.848829i | 0 | − | 1.00000i | −1.55898 | − | 2.56312i | −3.39598 | − | 1.96067i | |||||||
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 | 882.2.l.c | 48 | |
3.b | odd | 2 | 1 | 2646.2.l.c | 48 | ||
7.b | odd | 2 | 1 | inner | 882.2.l.c | 48 | |
7.c | even | 3 | 1 | 882.2.m.c | ✓ | 48 | |
7.c | even | 3 | 1 | 882.2.t.c | 48 | ||
7.d | odd | 6 | 1 | 882.2.m.c | ✓ | 48 | |
7.d | odd | 6 | 1 | 882.2.t.c | 48 | ||
9.c | even | 3 | 1 | 2646.2.t.c | 48 | ||
9.d | odd | 6 | 1 | 882.2.t.c | 48 | ||
21.c | even | 2 | 1 | 2646.2.l.c | 48 | ||
21.g | even | 6 | 1 | 2646.2.m.c | 48 | ||
21.g | even | 6 | 1 | 2646.2.t.c | 48 | ||
21.h | odd | 6 | 1 | 2646.2.m.c | 48 | ||
21.h | odd | 6 | 1 | 2646.2.t.c | 48 | ||
63.g | even | 3 | 1 | 2646.2.m.c | 48 | ||
63.h | even | 3 | 1 | 2646.2.l.c | 48 | ||
63.i | even | 6 | 1 | inner | 882.2.l.c | 48 | |
63.j | odd | 6 | 1 | inner | 882.2.l.c | 48 | |
63.k | odd | 6 | 1 | 2646.2.m.c | 48 | ||
63.l | odd | 6 | 1 | 2646.2.t.c | 48 | ||
63.n | odd | 6 | 1 | 882.2.m.c | ✓ | 48 | |
63.o | even | 6 | 1 | 882.2.t.c | 48 | ||
63.s | even | 6 | 1 | 882.2.m.c | ✓ | 48 | |
63.t | odd | 6 | 1 | 2646.2.l.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
882.2.l.c | 48 | 1.a | even | 1 | 1 | trivial | |
882.2.l.c | 48 | 7.b | odd | 2 | 1 | inner | |
882.2.l.c | 48 | 63.i | even | 6 | 1 | inner | |
882.2.l.c | 48 | 63.j | odd | 6 | 1 | inner | |
882.2.m.c | ✓ | 48 | 7.c | even | 3 | 1 | |
882.2.m.c | ✓ | 48 | 7.d | odd | 6 | 1 | |
882.2.m.c | ✓ | 48 | 63.n | odd | 6 | 1 | |
882.2.m.c | ✓ | 48 | 63.s | even | 6 | 1 | |
882.2.t.c | 48 | 7.c | even | 3 | 1 | ||
882.2.t.c | 48 | 7.d | odd | 6 | 1 | ||
882.2.t.c | 48 | 9.d | odd | 6 | 1 | ||
882.2.t.c | 48 | 63.o | even | 6 | 1 | ||
2646.2.l.c | 48 | 3.b | odd | 2 | 1 | ||
2646.2.l.c | 48 | 21.c | even | 2 | 1 | ||
2646.2.l.c | 48 | 63.h | even | 3 | 1 | ||
2646.2.l.c | 48 | 63.t | odd | 6 | 1 | ||
2646.2.m.c | 48 | 21.g | even | 6 | 1 | ||
2646.2.m.c | 48 | 21.h | odd | 6 | 1 | ||
2646.2.m.c | 48 | 63.g | even | 3 | 1 | ||
2646.2.m.c | 48 | 63.k | odd | 6 | 1 | ||
2646.2.t.c | 48 | 9.c | even | 3 | 1 | ||
2646.2.t.c | 48 | 21.g | even | 6 | 1 | ||
2646.2.t.c | 48 | 21.h | odd | 6 | 1 | ||
2646.2.t.c | 48 | 63.l | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 72 T_{5}^{46} + 2976 T_{5}^{44} + 83216 T_{5}^{42} + 1745790 T_{5}^{40} + \cdots + 5801854959616 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\).