Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,4,Mod(17,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.17");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.bc (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: | \(19.8246417619\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 168) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −5.18895 | − | 0.273513i | 0 | 2.97936 | + | 5.16040i | 0 | 17.8140 | − | 5.06584i | 0 | 26.8504 | + | 2.83849i | 0 | ||||||||||
17.2 | 0 | −5.12717 | − | 0.843898i | 0 | −8.29692 | − | 14.3707i | 0 | −9.36640 | − | 15.9772i | 0 | 25.5757 | + | 8.65361i | 0 | ||||||||||
17.3 | 0 | −4.99832 | − | 1.42013i | 0 | −6.11124 | − | 10.5850i | 0 | −4.39164 | + | 17.9920i | 0 | 22.9665 | + | 14.1965i | 0 | ||||||||||
17.4 | 0 | −4.88874 | + | 1.76075i | 0 | 6.04693 | + | 10.4736i | 0 | −15.9632 | − | 9.39028i | 0 | 20.7995 | − | 17.2157i | 0 | ||||||||||
17.5 | 0 | −3.72903 | − | 3.61861i | 0 | 6.11124 | + | 10.5850i | 0 | −4.39164 | + | 17.9920i | 0 | 0.811307 | + | 26.9878i | 0 | ||||||||||
17.6 | 0 | −3.62731 | + | 3.72057i | 0 | 0.263312 | + | 0.456070i | 0 | 16.6747 | + | 8.05944i | 0 | −0.685212 | − | 26.9913i | 0 | ||||||||||
17.7 | 0 | −3.29442 | − | 4.01831i | 0 | 8.29692 | + | 14.3707i | 0 | −9.36640 | − | 15.9772i | 0 | −5.29359 | + | 26.4760i | 0 | ||||||||||
17.8 | 0 | −3.24856 | + | 4.05547i | 0 | 3.20701 | + | 5.55470i | 0 | −14.8103 | + | 11.1200i | 0 | −5.89369 | − | 26.3489i | 0 | ||||||||||
17.9 | 0 | −2.83134 | − | 4.35701i | 0 | −2.97936 | − | 5.16040i | 0 | 17.8140 | − | 5.06584i | 0 | −10.9670 | + | 24.6724i | 0 | ||||||||||
17.10 | 0 | −2.36622 | + | 4.62612i | 0 | −9.22876 | − | 15.9847i | 0 | 6.70316 | − | 17.2646i | 0 | −15.8020 | − | 21.8928i | 0 | ||||||||||
17.11 | 0 | −0.919514 | − | 5.11415i | 0 | −6.04693 | − | 10.4736i | 0 | −15.9632 | − | 9.39028i | 0 | −25.3090 | + | 9.40505i | 0 | ||||||||||
17.12 | 0 | 0.328966 | + | 5.18573i | 0 | −7.91382 | − | 13.7071i | 0 | −13.3812 | + | 12.8040i | 0 | −26.7836 | + | 3.41186i | 0 | ||||||||||
17.13 | 0 | 0.726622 | + | 5.14510i | 0 | 9.08545 | + | 15.7365i | 0 | 18.4348 | − | 1.77707i | 0 | −25.9440 | + | 7.47709i | 0 | ||||||||||
17.14 | 0 | 1.35065 | + | 5.01754i | 0 | 2.40532 | + | 4.16613i | 0 | 3.40309 | − | 18.2049i | 0 | −23.3515 | + | 13.5539i | 0 | ||||||||||
17.15 | 0 | 1.40845 | − | 5.00163i | 0 | −0.263312 | − | 0.456070i | 0 | 16.6747 | + | 8.05944i | 0 | −23.0325 | − | 14.0891i | 0 | ||||||||||
17.16 | 0 | 1.88786 | − | 4.84107i | 0 | −3.20701 | − | 5.55470i | 0 | −14.8103 | + | 11.1200i | 0 | −19.8720 | − | 18.2785i | 0 | ||||||||||
17.17 | 0 | 2.82323 | − | 4.36227i | 0 | 9.22876 | + | 15.9847i | 0 | 6.70316 | − | 17.2646i | 0 | −11.0587 | − | 24.6314i | 0 | ||||||||||
17.18 | 0 | 3.06041 | + | 4.19928i | 0 | 0.746155 | + | 1.29238i | 0 | −15.9029 | − | 9.49205i | 0 | −8.26784 | + | 25.7030i | 0 | ||||||||||
17.19 | 0 | 4.27838 | + | 2.94880i | 0 | −5.57507 | − | 9.65631i | 0 | 7.78587 | + | 16.8042i | 0 | 9.60915 | + | 25.2322i | 0 | ||||||||||
17.20 | 0 | 4.65546 | − | 2.30797i | 0 | 7.91382 | + | 13.7071i | 0 | −13.3812 | + | 12.8040i | 0 | 16.3465 | − | 21.4893i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 336.4.bc.f | 48 | |
3.b | odd | 2 | 1 | inner | 336.4.bc.f | 48 | |
4.b | odd | 2 | 1 | 168.4.u.a | ✓ | 48 | |
7.d | odd | 6 | 1 | inner | 336.4.bc.f | 48 | |
12.b | even | 2 | 1 | 168.4.u.a | ✓ | 48 | |
21.g | even | 6 | 1 | inner | 336.4.bc.f | 48 | |
28.f | even | 6 | 1 | 168.4.u.a | ✓ | 48 | |
84.j | odd | 6 | 1 | 168.4.u.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.4.u.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
168.4.u.a | ✓ | 48 | 12.b | even | 2 | 1 | |
168.4.u.a | ✓ | 48 | 28.f | even | 6 | 1 | |
168.4.u.a | ✓ | 48 | 84.j | odd | 6 | 1 | |
336.4.bc.f | 48 | 1.a | even | 1 | 1 | trivial | |
336.4.bc.f | 48 | 3.b | odd | 2 | 1 | inner | |
336.4.bc.f | 48 | 7.d | odd | 6 | 1 | inner | |
336.4.bc.f | 48 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):
\( T_{5}^{48} + 1719 T_{5}^{46} + 1690641 T_{5}^{44} + 1132587728 T_{5}^{42} + 572282844510 T_{5}^{40} + \cdots + 19\!\cdots\!16 \) |
\( T_{13}^{24} + 27165 T_{13}^{22} + 309247695 T_{13}^{20} + 1920428280007 T_{13}^{18} + \cdots + 13\!\cdots\!64 \) |