Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,3,Mod(117,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.117");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 392.j (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: | \(10.6812263629\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 56) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
117.1 | −1.97030 | + | 0.343404i | 1.94818 | + | 3.37434i | 3.76415 | − | 1.35322i | 4.42985 | − | 7.67272i | −4.99725 | − | 5.97944i | 0 | −6.95179 | + | 3.95886i | −3.09078 | + | 5.35338i | −6.09327 | + | 16.6388i | ||
117.2 | −1.87135 | − | 0.705725i | 0.126628 | + | 0.219326i | 3.00390 | + | 2.64132i | −1.78589 | + | 3.09325i | −0.0821813 | − | 0.499801i | 0 | −3.75731 | − | 7.06276i | 4.46793 | − | 7.73868i | 5.52501 | − | 4.52821i | ||
117.3 | −1.70738 | + | 1.04157i | −2.78005 | − | 4.81519i | 1.83027 | − | 3.55670i | 1.52921 | − | 2.64866i | 9.76195 | + | 5.32573i | 0 | 0.579586 | + | 7.97898i | −10.9574 | + | 18.9787i | 0.147833 | + | 6.11504i | ||
117.4 | −1.61426 | + | 1.18075i | 0.455431 | + | 0.788830i | 1.21166 | − | 3.81207i | −3.17251 | + | 5.49495i | −1.66660 | − | 0.735624i | 0 | 2.54518 | + | 7.58433i | 4.08516 | − | 7.07571i | −1.36692 | − | 12.6162i | ||
117.5 | −1.33557 | − | 1.48871i | −1.70138 | − | 2.94687i | −0.432496 | + | 3.97655i | 2.15858 | − | 3.73877i | −2.11472 | + | 6.46862i | 0 | 6.49755 | − | 4.66711i | −1.28938 | + | 2.23327i | −8.44888 | + | 1.77990i | ||
117.6 | −0.215431 | + | 1.98836i | −0.455431 | − | 0.788830i | −3.90718 | − | 0.856711i | 3.17251 | − | 5.49495i | 1.66660 | − | 0.735624i | 0 | 2.54518 | − | 7.58433i | 4.08516 | − | 7.07571i | 10.2425 | + | 7.49189i | ||
117.7 | −0.212190 | − | 1.98871i | 1.16781 | + | 2.02271i | −3.90995 | + | 0.843971i | −1.55055 | + | 2.68563i | 3.77480 | − | 2.75165i | 0 | 2.50807 | + | 7.59668i | 1.77242 | − | 3.06992i | 5.66995 | + | 2.51373i | ||
117.8 | −0.0483365 | + | 1.99942i | 2.78005 | + | 4.81519i | −3.99533 | − | 0.193289i | −1.52921 | + | 2.64866i | −9.76195 | + | 5.32573i | 0 | 0.579586 | − | 7.97898i | −10.9574 | + | 18.9787i | −5.22186 | − | 3.18554i | ||
117.9 | 0.611223 | − | 1.90431i | −1.93494 | − | 3.35141i | −3.25281 | − | 2.32792i | −2.33882 | + | 4.05096i | −7.56482 | + | 1.63627i | 0 | −6.42128 | + | 4.77149i | −2.98798 | + | 5.17534i | 6.28474 | + | 6.92988i | ||
117.10 | 0.687752 | + | 1.87803i | −1.94818 | − | 3.37434i | −3.05399 | + | 2.58324i | −4.42985 | + | 7.67272i | 4.99725 | − | 5.97944i | 0 | −6.95179 | − | 3.95886i | −3.09078 | + | 5.35338i | −17.4562 | − | 3.04246i | ||
117.11 | 1.34357 | − | 1.48149i | 1.93494 | + | 3.35141i | −0.389632 | − | 3.98098i | 2.33882 | − | 4.05096i | 7.56482 | + | 1.63627i | 0 | −6.42128 | − | 4.77149i | −2.98798 | + | 5.17534i | −2.85908 | − | 8.90769i | ||
117.12 | 1.54685 | + | 1.26777i | −0.126628 | − | 0.219326i | 0.785498 | + | 3.92212i | 1.78589 | − | 3.09325i | 0.0821813 | − | 0.499801i | 0 | −3.75731 | + | 7.06276i | 4.46793 | − | 7.73868i | 6.68405 | − | 2.52070i | ||
117.13 | 1.82837 | − | 0.810594i | −1.16781 | − | 2.02271i | 2.68588 | − | 2.96413i | 1.55055 | − | 2.68563i | −3.77480 | − | 2.75165i | 0 | 2.50807 | − | 7.59668i | 1.77242 | − | 3.06992i | 0.658023 | − | 6.16719i | ||
117.14 | 1.95704 | + | 0.412286i | 1.70138 | + | 2.94687i | 3.66004 | + | 1.61372i | −2.15858 | + | 3.73877i | 2.11472 | + | 6.46862i | 0 | 6.49755 | + | 4.66711i | −1.28938 | + | 2.23327i | −5.76588 | + | 6.42699i | ||
325.1 | −1.97030 | − | 0.343404i | 1.94818 | − | 3.37434i | 3.76415 | + | 1.35322i | 4.42985 | + | 7.67272i | −4.99725 | + | 5.97944i | 0 | −6.95179 | − | 3.95886i | −3.09078 | − | 5.35338i | −6.09327 | − | 16.6388i | ||
325.2 | −1.87135 | + | 0.705725i | 0.126628 | − | 0.219326i | 3.00390 | − | 2.64132i | −1.78589 | − | 3.09325i | −0.0821813 | + | 0.499801i | 0 | −3.75731 | + | 7.06276i | 4.46793 | + | 7.73868i | 5.52501 | + | 4.52821i | ||
325.3 | −1.70738 | − | 1.04157i | −2.78005 | + | 4.81519i | 1.83027 | + | 3.55670i | 1.52921 | + | 2.64866i | 9.76195 | − | 5.32573i | 0 | 0.579586 | − | 7.97898i | −10.9574 | − | 18.9787i | 0.147833 | − | 6.11504i | ||
325.4 | −1.61426 | − | 1.18075i | 0.455431 | − | 0.788830i | 1.21166 | + | 3.81207i | −3.17251 | − | 5.49495i | −1.66660 | + | 0.735624i | 0 | 2.54518 | − | 7.58433i | 4.08516 | + | 7.07571i | −1.36692 | + | 12.6162i | ||
325.5 | −1.33557 | + | 1.48871i | −1.70138 | + | 2.94687i | −0.432496 | − | 3.97655i | 2.15858 | + | 3.73877i | −2.11472 | − | 6.46862i | 0 | 6.49755 | + | 4.66711i | −1.28938 | − | 2.23327i | −8.44888 | − | 1.77990i | ||
325.6 | −0.215431 | − | 1.98836i | −0.455431 | + | 0.788830i | −3.90718 | + | 0.856711i | 3.17251 | + | 5.49495i | 1.66660 | + | 0.735624i | 0 | 2.54518 | + | 7.58433i | 4.08516 | + | 7.07571i | 10.2425 | − | 7.49189i | ||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
8.b | even | 2 | 1 | inner |
56.j | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 392.3.j.e | 28 | |
7.b | odd | 2 | 1 | 56.3.j.a | ✓ | 28 | |
7.c | even | 3 | 1 | 56.3.j.a | ✓ | 28 | |
7.c | even | 3 | 1 | 392.3.h.a | 28 | ||
7.d | odd | 6 | 1 | 392.3.h.a | 28 | ||
7.d | odd | 6 | 1 | inner | 392.3.j.e | 28 | |
8.b | even | 2 | 1 | inner | 392.3.j.e | 28 | |
28.d | even | 2 | 1 | 224.3.n.a | 28 | ||
28.f | even | 6 | 1 | 1568.3.h.a | 28 | ||
28.g | odd | 6 | 1 | 224.3.n.a | 28 | ||
28.g | odd | 6 | 1 | 1568.3.h.a | 28 | ||
56.e | even | 2 | 1 | 224.3.n.a | 28 | ||
56.h | odd | 2 | 1 | 56.3.j.a | ✓ | 28 | |
56.j | odd | 6 | 1 | 392.3.h.a | 28 | ||
56.j | odd | 6 | 1 | inner | 392.3.j.e | 28 | |
56.k | odd | 6 | 1 | 224.3.n.a | 28 | ||
56.k | odd | 6 | 1 | 1568.3.h.a | 28 | ||
56.m | even | 6 | 1 | 1568.3.h.a | 28 | ||
56.p | even | 6 | 1 | 56.3.j.a | ✓ | 28 | |
56.p | even | 6 | 1 | 392.3.h.a | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
56.3.j.a | ✓ | 28 | 7.b | odd | 2 | 1 | |
56.3.j.a | ✓ | 28 | 7.c | even | 3 | 1 | |
56.3.j.a | ✓ | 28 | 56.h | odd | 2 | 1 | |
56.3.j.a | ✓ | 28 | 56.p | even | 6 | 1 | |
224.3.n.a | 28 | 28.d | even | 2 | 1 | ||
224.3.n.a | 28 | 28.g | odd | 6 | 1 | ||
224.3.n.a | 28 | 56.e | even | 2 | 1 | ||
224.3.n.a | 28 | 56.k | odd | 6 | 1 | ||
392.3.h.a | 28 | 7.c | even | 3 | 1 | ||
392.3.h.a | 28 | 7.d | odd | 6 | 1 | ||
392.3.h.a | 28 | 56.j | odd | 6 | 1 | ||
392.3.h.a | 28 | 56.p | even | 6 | 1 | ||
392.3.j.e | 28 | 1.a | even | 1 | 1 | trivial | |
392.3.j.e | 28 | 7.d | odd | 6 | 1 | inner | |
392.3.j.e | 28 | 8.b | even | 2 | 1 | inner | |
392.3.j.e | 28 | 56.j | odd | 6 | 1 | inner | |
1568.3.h.a | 28 | 28.f | even | 6 | 1 | ||
1568.3.h.a | 28 | 28.g | odd | 6 | 1 | ||
1568.3.h.a | 28 | 56.k | odd | 6 | 1 | ||
1568.3.h.a | 28 | 56.m | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{28} + 79 T_{3}^{26} + 3908 T_{3}^{24} + 118973 T_{3}^{22} + 2641713 T_{3}^{20} + 41993874 T_{3}^{18} + 504732141 T_{3}^{16} + 4350754377 T_{3}^{14} + 27476030709 T_{3}^{12} + \cdots + 558140625 \)
acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).