Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,3,Mod(721,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.721");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 722.b (of order \(2\), degree \(1\), 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: | \(19.6730750868\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 38) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
721.1 | − | 1.41421i | − | 5.84230i | −2.00000 | −5.64041 | −8.26227 | 1.98059 | 2.82843i | −25.1325 | 7.97674i | ||||||||||||||||
721.2 | − | 1.41421i | − | 5.05892i | −2.00000 | 3.90971 | −7.15439 | −3.55148 | 2.82843i | −16.5927 | − | 5.52916i | |||||||||||||||
721.3 | − | 1.41421i | − | 4.14954i | −2.00000 | −1.13449 | −5.86834 | 11.1120 | 2.82843i | −8.21868 | 1.60442i | ||||||||||||||||
721.4 | − | 1.41421i | − | 3.70454i | −2.00000 | 0.388748 | −5.23901 | 5.92863 | 2.82843i | −4.72359 | − | 0.549773i | |||||||||||||||
721.5 | − | 1.41421i | − | 0.423806i | −2.00000 | −8.50146 | −0.599352 | 5.17090 | 2.82843i | 8.82039 | 12.0229i | ||||||||||||||||
721.6 | − | 1.41421i | − | 0.175771i | −2.00000 | 7.89135 | −0.248578 | 7.65909 | 2.82843i | 8.96910 | − | 11.1601i | |||||||||||||||
721.7 | − | 1.41421i | 1.64949i | −2.00000 | 4.64041 | 2.33273 | 4.36176 | 2.82843i | 6.27919 | − | 6.56253i | ||||||||||||||||
721.8 | − | 1.41421i | 2.42931i | −2.00000 | 0.134495 | 3.43557 | −9.58683 | 2.82843i | 3.09843 | − | 0.190204i | ||||||||||||||||
721.9 | − | 1.41421i | 2.44574i | −2.00000 | −4.90971 | 3.45880 | −1.16621 | 2.82843i | 3.01834 | 6.94338i | |||||||||||||||||
721.10 | − | 1.41421i | 3.63432i | −2.00000 | 7.50146 | 5.13971 | −7.59704 | 2.82843i | −4.20829 | − | 10.6087i | ||||||||||||||||
721.11 | − | 1.41421i | 3.88163i | −2.00000 | −8.89135 | 5.48946 | −5.10658 | 2.82843i | −6.06708 | 12.5743i | |||||||||||||||||
721.12 | − | 1.41421i | 5.31438i | −2.00000 | −1.38875 | 7.51567 | 8.79517 | 2.82843i | −19.2426 | 1.96399i | |||||||||||||||||
721.13 | 1.41421i | − | 5.31438i | −2.00000 | −1.38875 | 7.51567 | 8.79517 | − | 2.82843i | −19.2426 | − | 1.96399i | |||||||||||||||
721.14 | 1.41421i | − | 3.88163i | −2.00000 | −8.89135 | 5.48946 | −5.10658 | − | 2.82843i | −6.06708 | − | 12.5743i | |||||||||||||||
721.15 | 1.41421i | − | 3.63432i | −2.00000 | 7.50146 | 5.13971 | −7.59704 | − | 2.82843i | −4.20829 | 10.6087i | ||||||||||||||||
721.16 | 1.41421i | − | 2.44574i | −2.00000 | −4.90971 | 3.45880 | −1.16621 | − | 2.82843i | 3.01834 | − | 6.94338i | |||||||||||||||
721.17 | 1.41421i | − | 2.42931i | −2.00000 | 0.134495 | 3.43557 | −9.58683 | − | 2.82843i | 3.09843 | 0.190204i | ||||||||||||||||
721.18 | 1.41421i | − | 1.64949i | −2.00000 | 4.64041 | 2.33273 | 4.36176 | − | 2.82843i | 6.27919 | 6.56253i | ||||||||||||||||
721.19 | 1.41421i | 0.175771i | −2.00000 | 7.89135 | −0.248578 | 7.65909 | − | 2.82843i | 8.96910 | 11.1601i | |||||||||||||||||
721.20 | 1.41421i | 0.423806i | −2.00000 | −8.50146 | −0.599352 | 5.17090 | − | 2.82843i | 8.82039 | − | 12.0229i | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 722.3.b.f | 24 | |
19.b | odd | 2 | 1 | inner | 722.3.b.f | 24 | |
19.e | even | 9 | 1 | 38.3.f.a | ✓ | 24 | |
19.f | odd | 18 | 1 | 38.3.f.a | ✓ | 24 | |
57.j | even | 18 | 1 | 342.3.z.b | 24 | ||
57.l | odd | 18 | 1 | 342.3.z.b | 24 | ||
76.k | even | 18 | 1 | 304.3.z.c | 24 | ||
76.l | odd | 18 | 1 | 304.3.z.c | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.3.f.a | ✓ | 24 | 19.e | even | 9 | 1 | |
38.3.f.a | ✓ | 24 | 19.f | odd | 18 | 1 | |
304.3.z.c | 24 | 76.k | even | 18 | 1 | ||
304.3.z.c | 24 | 76.l | odd | 18 | 1 | ||
342.3.z.b | 24 | 57.j | even | 18 | 1 | ||
342.3.z.b | 24 | 57.l | odd | 18 | 1 | ||
722.3.b.f | 24 | 1.a | even | 1 | 1 | trivial | |
722.3.b.f | 24 | 19.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{24} + 162 T_{3}^{22} + 11331 T_{3}^{20} + 449416 T_{3}^{18} + 11159685 T_{3}^{16} + 180632898 T_{3}^{14} + 1921819843 T_{3}^{12} + 13207043862 T_{3}^{10} + 55876820409 T_{3}^{8} + \cdots + 618367689 \)
acting on \(S_{3}^{\mathrm{new}}(722, [\chi])\).