Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,6,Mod(447,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.447");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.f (of order \(2\), degree \(1\), 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: | \(71.8519512762\) |
| Analytic rank: | \(0\) |
| Dimension: | \(40\) |
| Twist minimal: | no (minimal twist has level 224) |
| 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 447.1 | 0 | −27.7905 | 0 | − | 20.1045i | 0 | 129.633 | + | 1.51903i | 0 | 529.310 | 0 | |||||||||||||||
| 447.2 | 0 | −27.7905 | 0 | 20.1045i | 0 | 129.633 | − | 1.51903i | 0 | 529.310 | 0 | ||||||||||||||||
| 447.3 | 0 | −27.7186 | 0 | 36.5226i | 0 | −50.6715 | + | 119.329i | 0 | 525.323 | 0 | ||||||||||||||||
| 447.4 | 0 | −27.7186 | 0 | − | 36.5226i | 0 | −50.6715 | − | 119.329i | 0 | 525.323 | 0 | |||||||||||||||
| 447.5 | 0 | −21.2903 | 0 | 65.9484i | 0 | −127.920 | − | 21.0574i | 0 | 210.277 | 0 | ||||||||||||||||
| 447.6 | 0 | −21.2903 | 0 | − | 65.9484i | 0 | −127.920 | + | 21.0574i | 0 | 210.277 | 0 | |||||||||||||||
| 447.7 | 0 | −18.5627 | 0 | − | 108.712i | 0 | 120.535 | + | 47.7324i | 0 | 101.574 | 0 | |||||||||||||||
| 447.8 | 0 | −18.5627 | 0 | 108.712i | 0 | 120.535 | − | 47.7324i | 0 | 101.574 | 0 | ||||||||||||||||
| 447.9 | 0 | −17.7948 | 0 | 50.5025i | 0 | −69.6313 | + | 109.355i | 0 | 73.6556 | 0 | ||||||||||||||||
| 447.10 | 0 | −17.7948 | 0 | − | 50.5025i | 0 | −69.6313 | − | 109.355i | 0 | 73.6556 | 0 | |||||||||||||||
| 447.11 | 0 | −16.5590 | 0 | 75.2305i | 0 | 14.1834 | − | 128.864i | 0 | 31.2014 | 0 | ||||||||||||||||
| 447.12 | 0 | −16.5590 | 0 | − | 75.2305i | 0 | 14.1834 | + | 128.864i | 0 | 31.2014 | 0 | |||||||||||||||
| 447.13 | 0 | −14.3879 | 0 | 10.7035i | 0 | 41.4774 | − | 122.828i | 0 | −35.9896 | 0 | ||||||||||||||||
| 447.14 | 0 | −14.3879 | 0 | − | 10.7035i | 0 | 41.4774 | + | 122.828i | 0 | −35.9896 | 0 | |||||||||||||||
| 447.15 | 0 | −9.88778 | 0 | − | 56.9139i | 0 | 10.4912 | − | 129.217i | 0 | −145.232 | 0 | |||||||||||||||
| 447.16 | 0 | −9.88778 | 0 | 56.9139i | 0 | 10.4912 | + | 129.217i | 0 | −145.232 | 0 | ||||||||||||||||
| 447.17 | 0 | −1.98158 | 0 | − | 18.1344i | 0 | 116.731 | − | 56.3997i | 0 | −239.073 | 0 | |||||||||||||||
| 447.18 | 0 | −1.98158 | 0 | 18.1344i | 0 | 116.731 | + | 56.3997i | 0 | −239.073 | 0 | ||||||||||||||||
| 447.19 | 0 | −1.39727 | 0 | − | 79.6094i | 0 | −119.955 | + | 49.1700i | 0 | −241.048 | 0 | |||||||||||||||
| 447.20 | 0 | −1.39727 | 0 | 79.6094i | 0 | −119.955 | − | 49.1700i | 0 | −241.048 | 0 | ||||||||||||||||
| See all 40 embeddings | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 448.6.f.e | 40 | |
| 4.b | odd | 2 | 1 | inner | 448.6.f.e | 40 | |
| 7.b | odd | 2 | 1 | inner | 448.6.f.e | 40 | |
| 8.b | even | 2 | 1 | 224.6.f.a | ✓ | 40 | |
| 8.d | odd | 2 | 1 | 224.6.f.a | ✓ | 40 | |
| 28.d | even | 2 | 1 | inner | 448.6.f.e | 40 | |
| 56.e | even | 2 | 1 | 224.6.f.a | ✓ | 40 | |
| 56.h | odd | 2 | 1 | 224.6.f.a | ✓ | 40 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.6.f.a | ✓ | 40 | 8.b | even | 2 | 1 | |
| 224.6.f.a | ✓ | 40 | 8.d | odd | 2 | 1 | |
| 224.6.f.a | ✓ | 40 | 56.e | even | 2 | 1 | |
| 224.6.f.a | ✓ | 40 | 56.h | odd | 2 | 1 | |
| 448.6.f.e | 40 | 1.a | even | 1 | 1 | trivial | |
| 448.6.f.e | 40 | 4.b | odd | 2 | 1 | inner | |
| 448.6.f.e | 40 | 7.b | odd | 2 | 1 | inner | |
| 448.6.f.e | 40 | 28.d | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} - 3240 T_{3}^{18} + 4379368 T_{3}^{16} - 3221886720 T_{3}^{14} + 1410844739712 T_{3}^{12} + \cdots + 12\!\cdots\!12 \)
acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\).