Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(255,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.255");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.p (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: | \(26.4328556826\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 224) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
255.1 | 0 | −4.91971 | + | 8.52120i | 0 | 4.22368 | − | 2.43854i | 0 | 18.4862 | + | 1.12186i | 0 | −34.9072 | − | 60.4610i | 0 | ||||||||||
255.2 | 0 | −4.28593 | + | 7.42345i | 0 | −13.1506 | + | 7.59249i | 0 | −18.3478 | + | 2.52164i | 0 | −23.2384 | − | 40.2500i | 0 | ||||||||||
255.3 | 0 | −4.15093 | + | 7.18962i | 0 | −10.5034 | + | 6.06415i | 0 | 0.898453 | − | 18.4985i | 0 | −20.9604 | − | 36.3045i | 0 | ||||||||||
255.4 | 0 | −3.65183 | + | 6.32516i | 0 | 12.0402 | − | 6.95142i | 0 | −1.67411 | + | 18.4444i | 0 | −13.1717 | − | 22.8141i | 0 | ||||||||||
255.5 | 0 | −3.12400 | + | 5.41092i | 0 | 0.225941 | − | 0.130447i | 0 | −14.3061 | + | 11.7616i | 0 | −6.01870 | − | 10.4247i | 0 | ||||||||||
255.6 | 0 | −3.02960 | + | 5.24743i | 0 | 12.5389 | − | 7.23935i | 0 | −0.978923 | − | 18.4944i | 0 | −4.85698 | − | 8.41254i | 0 | ||||||||||
255.7 | 0 | −2.69336 | + | 4.66504i | 0 | −12.5403 | + | 7.24016i | 0 | 18.3374 | + | 2.59641i | 0 | −1.00837 | − | 1.74655i | 0 | ||||||||||
255.8 | 0 | −1.86026 | + | 3.22206i | 0 | 4.38985 | − | 2.53448i | 0 | −18.5198 | + | 0.132997i | 0 | 6.57889 | + | 11.3950i | 0 | ||||||||||
255.9 | 0 | −1.62864 | + | 2.82088i | 0 | 16.4571 | − | 9.50153i | 0 | 15.1537 | − | 10.6473i | 0 | 8.19508 | + | 14.1943i | 0 | ||||||||||
255.10 | 0 | −1.35521 | + | 2.34729i | 0 | −3.53108 | + | 2.03867i | 0 | 13.7196 | + | 12.4408i | 0 | 9.82683 | + | 17.0206i | 0 | ||||||||||
255.11 | 0 | −0.692224 | + | 1.19897i | 0 | 4.20674 | − | 2.42876i | 0 | −6.91230 | − | 17.1820i | 0 | 12.5417 | + | 21.7228i | 0 | ||||||||||
255.12 | 0 | −0.490265 | + | 0.849164i | 0 | −14.3571 | + | 8.28906i | 0 | 6.74534 | − | 17.2482i | 0 | 13.0193 | + | 22.5501i | 0 | ||||||||||
255.13 | 0 | 0.490265 | − | 0.849164i | 0 | −14.3571 | + | 8.28906i | 0 | −6.74534 | + | 17.2482i | 0 | 13.0193 | + | 22.5501i | 0 | ||||||||||
255.14 | 0 | 0.692224 | − | 1.19897i | 0 | 4.20674 | − | 2.42876i | 0 | 6.91230 | + | 17.1820i | 0 | 12.5417 | + | 21.7228i | 0 | ||||||||||
255.15 | 0 | 1.35521 | − | 2.34729i | 0 | −3.53108 | + | 2.03867i | 0 | −13.7196 | − | 12.4408i | 0 | 9.82683 | + | 17.0206i | 0 | ||||||||||
255.16 | 0 | 1.62864 | − | 2.82088i | 0 | 16.4571 | − | 9.50153i | 0 | −15.1537 | + | 10.6473i | 0 | 8.19508 | + | 14.1943i | 0 | ||||||||||
255.17 | 0 | 1.86026 | − | 3.22206i | 0 | 4.38985 | − | 2.53448i | 0 | 18.5198 | − | 0.132997i | 0 | 6.57889 | + | 11.3950i | 0 | ||||||||||
255.18 | 0 | 2.69336 | − | 4.66504i | 0 | −12.5403 | + | 7.24016i | 0 | −18.3374 | − | 2.59641i | 0 | −1.00837 | − | 1.74655i | 0 | ||||||||||
255.19 | 0 | 3.02960 | − | 5.24743i | 0 | 12.5389 | − | 7.23935i | 0 | 0.978923 | + | 18.4944i | 0 | −4.85698 | − | 8.41254i | 0 | ||||||||||
255.20 | 0 | 3.12400 | − | 5.41092i | 0 | 0.225941 | − | 0.130447i | 0 | 14.3061 | − | 11.7616i | 0 | −6.01870 | − | 10.4247i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
28.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 448.4.p.i | 48 | |
4.b | odd | 2 | 1 | inner | 448.4.p.i | 48 | |
7.d | odd | 6 | 1 | inner | 448.4.p.i | 48 | |
8.b | even | 2 | 1 | 224.4.p.a | ✓ | 48 | |
8.d | odd | 2 | 1 | 224.4.p.a | ✓ | 48 | |
28.f | even | 6 | 1 | inner | 448.4.p.i | 48 | |
56.j | odd | 6 | 1 | 224.4.p.a | ✓ | 48 | |
56.m | even | 6 | 1 | 224.4.p.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.4.p.a | ✓ | 48 | 8.b | even | 2 | 1 | |
224.4.p.a | ✓ | 48 | 8.d | odd | 2 | 1 | |
224.4.p.a | ✓ | 48 | 56.j | odd | 6 | 1 | |
224.4.p.a | ✓ | 48 | 56.m | even | 6 | 1 | |
448.4.p.i | 48 | 1.a | even | 1 | 1 | trivial | |
448.4.p.i | 48 | 4.b | odd | 2 | 1 | inner | |
448.4.p.i | 48 | 7.d | odd | 6 | 1 | inner | |
448.4.p.i | 48 | 28.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{48} + 432 T_{3}^{46} + 106534 T_{3}^{44} + 17868192 T_{3}^{42} + 2257274037 T_{3}^{40} + 223414613312 T_{3}^{38} + 17851857344466 T_{3}^{36} + \cdots + 46\!\cdots\!21 \)
acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\).