Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [288,3,Mod(31,288)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(288, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("288.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 288.o (of order \(6\), degree \(2\), 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: | \(7.84743161358\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | −2.99814 | − | 0.105601i | 0 | −3.20123 | + | 5.54470i | 0 | 6.73381 | − | 3.88777i | 0 | 8.97770 | + | 0.633212i | 0 | ||||||||||
31.2 | 0 | −2.40825 | + | 1.78895i | 0 | 4.14774 | − | 7.18409i | 0 | −7.08016 | + | 4.08773i | 0 | 2.59929 | − | 8.61648i | 0 | ||||||||||
31.3 | 0 | −2.10592 | − | 2.13661i | 0 | −1.25241 | + | 2.16924i | 0 | 0.842935 | − | 0.486669i | 0 | −0.130192 | + | 8.99906i | 0 | ||||||||||
31.4 | 0 | −1.80500 | − | 2.39624i | 0 | 1.69347 | − | 2.93317i | 0 | −8.58804 | + | 4.95831i | 0 | −2.48392 | + | 8.65044i | 0 | ||||||||||
31.5 | 0 | −1.55460 | + | 2.56578i | 0 | −3.70620 | + | 6.41933i | 0 | 0.198210 | − | 0.114437i | 0 | −4.16646 | − | 7.97751i | 0 | ||||||||||
31.6 | 0 | −1.26562 | + | 2.71996i | 0 | 2.31864 | − | 4.01601i | 0 | 1.01137 | − | 0.583914i | 0 | −5.79642 | − | 6.88488i | 0 | ||||||||||
31.7 | 0 | 1.26562 | − | 2.71996i | 0 | 2.31864 | − | 4.01601i | 0 | −1.01137 | + | 0.583914i | 0 | −5.79642 | − | 6.88488i | 0 | ||||||||||
31.8 | 0 | 1.55460 | − | 2.56578i | 0 | −3.70620 | + | 6.41933i | 0 | −0.198210 | + | 0.114437i | 0 | −4.16646 | − | 7.97751i | 0 | ||||||||||
31.9 | 0 | 1.80500 | + | 2.39624i | 0 | 1.69347 | − | 2.93317i | 0 | 8.58804 | − | 4.95831i | 0 | −2.48392 | + | 8.65044i | 0 | ||||||||||
31.10 | 0 | 2.10592 | + | 2.13661i | 0 | −1.25241 | + | 2.16924i | 0 | −0.842935 | + | 0.486669i | 0 | −0.130192 | + | 8.99906i | 0 | ||||||||||
31.11 | 0 | 2.40825 | − | 1.78895i | 0 | 4.14774 | − | 7.18409i | 0 | 7.08016 | − | 4.08773i | 0 | 2.59929 | − | 8.61648i | 0 | ||||||||||
31.12 | 0 | 2.99814 | + | 0.105601i | 0 | −3.20123 | + | 5.54470i | 0 | −6.73381 | + | 3.88777i | 0 | 8.97770 | + | 0.633212i | 0 | ||||||||||
223.1 | 0 | −2.99814 | + | 0.105601i | 0 | −3.20123 | − | 5.54470i | 0 | 6.73381 | + | 3.88777i | 0 | 8.97770 | − | 0.633212i | 0 | ||||||||||
223.2 | 0 | −2.40825 | − | 1.78895i | 0 | 4.14774 | + | 7.18409i | 0 | −7.08016 | − | 4.08773i | 0 | 2.59929 | + | 8.61648i | 0 | ||||||||||
223.3 | 0 | −2.10592 | + | 2.13661i | 0 | −1.25241 | − | 2.16924i | 0 | 0.842935 | + | 0.486669i | 0 | −0.130192 | − | 8.99906i | 0 | ||||||||||
223.4 | 0 | −1.80500 | + | 2.39624i | 0 | 1.69347 | + | 2.93317i | 0 | −8.58804 | − | 4.95831i | 0 | −2.48392 | − | 8.65044i | 0 | ||||||||||
223.5 | 0 | −1.55460 | − | 2.56578i | 0 | −3.70620 | − | 6.41933i | 0 | 0.198210 | + | 0.114437i | 0 | −4.16646 | + | 7.97751i | 0 | ||||||||||
223.6 | 0 | −1.26562 | − | 2.71996i | 0 | 2.31864 | + | 4.01601i | 0 | 1.01137 | + | 0.583914i | 0 | −5.79642 | + | 6.88488i | 0 | ||||||||||
223.7 | 0 | 1.26562 | + | 2.71996i | 0 | 2.31864 | + | 4.01601i | 0 | −1.01137 | − | 0.583914i | 0 | −5.79642 | + | 6.88488i | 0 | ||||||||||
223.8 | 0 | 1.55460 | + | 2.56578i | 0 | −3.70620 | − | 6.41933i | 0 | −0.198210 | − | 0.114437i | 0 | −4.16646 | + | 7.97751i | 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.c | even | 3 | 1 | inner |
36.f | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 288.3.o.c | ✓ | 24 |
3.b | odd | 2 | 1 | 864.3.o.c | 24 | ||
4.b | odd | 2 | 1 | inner | 288.3.o.c | ✓ | 24 |
8.b | even | 2 | 1 | 576.3.o.i | 24 | ||
8.d | odd | 2 | 1 | 576.3.o.i | 24 | ||
9.c | even | 3 | 1 | inner | 288.3.o.c | ✓ | 24 |
9.c | even | 3 | 1 | 2592.3.g.k | 12 | ||
9.d | odd | 6 | 1 | 864.3.o.c | 24 | ||
9.d | odd | 6 | 1 | 2592.3.g.l | 12 | ||
12.b | even | 2 | 1 | 864.3.o.c | 24 | ||
24.f | even | 2 | 1 | 1728.3.o.i | 24 | ||
24.h | odd | 2 | 1 | 1728.3.o.i | 24 | ||
36.f | odd | 6 | 1 | inner | 288.3.o.c | ✓ | 24 |
36.f | odd | 6 | 1 | 2592.3.g.k | 12 | ||
36.h | even | 6 | 1 | 864.3.o.c | 24 | ||
36.h | even | 6 | 1 | 2592.3.g.l | 12 | ||
72.j | odd | 6 | 1 | 1728.3.o.i | 24 | ||
72.l | even | 6 | 1 | 1728.3.o.i | 24 | ||
72.n | even | 6 | 1 | 576.3.o.i | 24 | ||
72.p | odd | 6 | 1 | 576.3.o.i | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
288.3.o.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
288.3.o.c | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
288.3.o.c | ✓ | 24 | 9.c | even | 3 | 1 | inner |
288.3.o.c | ✓ | 24 | 36.f | odd | 6 | 1 | inner |
576.3.o.i | 24 | 8.b | even | 2 | 1 | ||
576.3.o.i | 24 | 8.d | odd | 2 | 1 | ||
576.3.o.i | 24 | 72.n | even | 6 | 1 | ||
576.3.o.i | 24 | 72.p | odd | 6 | 1 | ||
864.3.o.c | 24 | 3.b | odd | 2 | 1 | ||
864.3.o.c | 24 | 9.d | odd | 6 | 1 | ||
864.3.o.c | 24 | 12.b | even | 2 | 1 | ||
864.3.o.c | 24 | 36.h | even | 6 | 1 | ||
1728.3.o.i | 24 | 24.f | even | 2 | 1 | ||
1728.3.o.i | 24 | 24.h | odd | 2 | 1 | ||
1728.3.o.i | 24 | 72.j | odd | 6 | 1 | ||
1728.3.o.i | 24 | 72.l | even | 6 | 1 | ||
2592.3.g.k | 12 | 9.c | even | 3 | 1 | ||
2592.3.g.k | 12 | 36.f | odd | 6 | 1 | ||
2592.3.g.l | 12 | 9.d | odd | 6 | 1 | ||
2592.3.g.l | 12 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 102 T_{5}^{10} + 16 T_{5}^{9} + 7719 T_{5}^{8} - 96 T_{5}^{7} + 242958 T_{5}^{6} - 207528 T_{5}^{5} + 5622153 T_{5}^{4} - 2696528 T_{5}^{3} + 42417024 T_{5}^{2} + 14125056 T_{5} + 239878144 \)
acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\).