Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,3,Mod(559,864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("864.559");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 864.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: | \(23.5422948407\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 72) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
559.1 | 0 | 0 | 0 | −8.07964 | + | 4.66478i | 0 | 4.91220 | + | 2.83606i | 0 | 0 | 0 | ||||||||||||||
559.2 | 0 | 0 | 0 | −6.07342 | + | 3.50649i | 0 | 8.07247 | + | 4.66064i | 0 | 0 | 0 | ||||||||||||||
559.3 | 0 | 0 | 0 | −5.84790 | + | 3.37629i | 0 | 3.50808 | + | 2.02539i | 0 | 0 | 0 | ||||||||||||||
559.4 | 0 | 0 | 0 | −5.42020 | + | 3.12935i | 0 | −5.96345 | − | 3.44300i | 0 | 0 | 0 | ||||||||||||||
559.5 | 0 | 0 | 0 | −5.15803 | + | 2.97799i | 0 | 4.09037 | + | 2.36158i | 0 | 0 | 0 | ||||||||||||||
559.6 | 0 | 0 | 0 | −4.40783 | + | 2.54486i | 0 | −10.9609 | − | 6.32830i | 0 | 0 | 0 | ||||||||||||||
559.7 | 0 | 0 | 0 | −3.84571 | + | 2.22032i | 0 | −0.704321 | − | 0.406640i | 0 | 0 | 0 | ||||||||||||||
559.8 | 0 | 0 | 0 | −1.70411 | + | 0.983869i | 0 | −8.69613 | − | 5.02071i | 0 | 0 | 0 | ||||||||||||||
559.9 | 0 | 0 | 0 | −1.50948 | + | 0.871501i | 0 | −7.93804 | − | 4.58303i | 0 | 0 | 0 | ||||||||||||||
559.10 | 0 | 0 | 0 | −0.0166003 | + | 0.00958419i | 0 | 4.07208 | + | 2.35102i | 0 | 0 | 0 | ||||||||||||||
559.11 | 0 | 0 | 0 | 0.0166003 | − | 0.00958419i | 0 | −4.07208 | − | 2.35102i | 0 | 0 | 0 | ||||||||||||||
559.12 | 0 | 0 | 0 | 1.50948 | − | 0.871501i | 0 | 7.93804 | + | 4.58303i | 0 | 0 | 0 | ||||||||||||||
559.13 | 0 | 0 | 0 | 1.70411 | − | 0.983869i | 0 | 8.69613 | + | 5.02071i | 0 | 0 | 0 | ||||||||||||||
559.14 | 0 | 0 | 0 | 3.84571 | − | 2.22032i | 0 | 0.704321 | + | 0.406640i | 0 | 0 | 0 | ||||||||||||||
559.15 | 0 | 0 | 0 | 4.40783 | − | 2.54486i | 0 | 10.9609 | + | 6.32830i | 0 | 0 | 0 | ||||||||||||||
559.16 | 0 | 0 | 0 | 5.15803 | − | 2.97799i | 0 | −4.09037 | − | 2.36158i | 0 | 0 | 0 | ||||||||||||||
559.17 | 0 | 0 | 0 | 5.42020 | − | 3.12935i | 0 | 5.96345 | + | 3.44300i | 0 | 0 | 0 | ||||||||||||||
559.18 | 0 | 0 | 0 | 5.84790 | − | 3.37629i | 0 | −3.50808 | − | 2.02539i | 0 | 0 | 0 | ||||||||||||||
559.19 | 0 | 0 | 0 | 6.07342 | − | 3.50649i | 0 | −8.07247 | − | 4.66064i | 0 | 0 | 0 | ||||||||||||||
559.20 | 0 | 0 | 0 | 8.07964 | − | 4.66478i | 0 | −4.91220 | − | 2.83606i | 0 | 0 | 0 | ||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
72.p | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 864.3.t.b | 40 | |
3.b | odd | 2 | 1 | 288.3.t.b | 40 | ||
4.b | odd | 2 | 1 | 216.3.p.b | 40 | ||
8.b | even | 2 | 1 | 216.3.p.b | 40 | ||
8.d | odd | 2 | 1 | inner | 864.3.t.b | 40 | |
9.c | even | 3 | 1 | inner | 864.3.t.b | 40 | |
9.c | even | 3 | 1 | 2592.3.b.f | 20 | ||
9.d | odd | 6 | 1 | 288.3.t.b | 40 | ||
9.d | odd | 6 | 1 | 2592.3.b.e | 20 | ||
12.b | even | 2 | 1 | 72.3.p.b | ✓ | 40 | |
24.f | even | 2 | 1 | 288.3.t.b | 40 | ||
24.h | odd | 2 | 1 | 72.3.p.b | ✓ | 40 | |
36.f | odd | 6 | 1 | 216.3.p.b | 40 | ||
36.f | odd | 6 | 1 | 648.3.b.e | 20 | ||
36.h | even | 6 | 1 | 72.3.p.b | ✓ | 40 | |
36.h | even | 6 | 1 | 648.3.b.f | 20 | ||
72.j | odd | 6 | 1 | 72.3.p.b | ✓ | 40 | |
72.j | odd | 6 | 1 | 648.3.b.f | 20 | ||
72.l | even | 6 | 1 | 288.3.t.b | 40 | ||
72.l | even | 6 | 1 | 2592.3.b.e | 20 | ||
72.n | even | 6 | 1 | 216.3.p.b | 40 | ||
72.n | even | 6 | 1 | 648.3.b.e | 20 | ||
72.p | odd | 6 | 1 | inner | 864.3.t.b | 40 | |
72.p | odd | 6 | 1 | 2592.3.b.f | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.3.p.b | ✓ | 40 | 12.b | even | 2 | 1 | |
72.3.p.b | ✓ | 40 | 24.h | odd | 2 | 1 | |
72.3.p.b | ✓ | 40 | 36.h | even | 6 | 1 | |
72.3.p.b | ✓ | 40 | 72.j | odd | 6 | 1 | |
216.3.p.b | 40 | 4.b | odd | 2 | 1 | ||
216.3.p.b | 40 | 8.b | even | 2 | 1 | ||
216.3.p.b | 40 | 36.f | odd | 6 | 1 | ||
216.3.p.b | 40 | 72.n | even | 6 | 1 | ||
288.3.t.b | 40 | 3.b | odd | 2 | 1 | ||
288.3.t.b | 40 | 9.d | odd | 6 | 1 | ||
288.3.t.b | 40 | 24.f | even | 2 | 1 | ||
288.3.t.b | 40 | 72.l | even | 6 | 1 | ||
648.3.b.e | 20 | 36.f | odd | 6 | 1 | ||
648.3.b.e | 20 | 72.n | even | 6 | 1 | ||
648.3.b.f | 20 | 36.h | even | 6 | 1 | ||
648.3.b.f | 20 | 72.j | odd | 6 | 1 | ||
864.3.t.b | 40 | 1.a | even | 1 | 1 | trivial | |
864.3.t.b | 40 | 8.d | odd | 2 | 1 | inner | |
864.3.t.b | 40 | 9.c | even | 3 | 1 | inner | |
864.3.t.b | 40 | 72.p | odd | 6 | 1 | inner | |
2592.3.b.e | 20 | 9.d | odd | 6 | 1 | ||
2592.3.b.e | 20 | 72.l | even | 6 | 1 | ||
2592.3.b.f | 20 | 9.c | even | 3 | 1 | ||
2592.3.b.f | 20 | 72.p | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{40} - 309 T_{5}^{38} + 55716 T_{5}^{36} - 6712983 T_{5}^{34} + 603811641 T_{5}^{32} - 41961529638 T_{5}^{30} + 2327328612189 T_{5}^{28} - 104016298280787 T_{5}^{26} + \cdots + 35\!\cdots\!00 \)
acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\).