Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(31,448)]
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, 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("448.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 448.q (of order \(6\), degree \(2\), 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: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0 | −8.78932 | − | 5.07452i | 0 | 8.37041 | + | 14.4980i | 0 | −0.901615 | + | 18.4983i | 0 | 38.0015 | + | 65.8205i | 0 | ||||||||||
31.2 | 0 | −6.84052 | − | 3.94938i | 0 | 9.21856 | + | 15.9670i | 0 | 14.3277 | − | 11.7353i | 0 | 17.6951 | + | 30.6489i | 0 | ||||||||||
31.3 | 0 | −6.23656 | − | 3.60068i | 0 | −3.36366 | − | 5.82602i | 0 | −4.34160 | + | 18.0042i | 0 | 12.4298 | + | 21.5290i | 0 | ||||||||||
31.4 | 0 | −5.65076 | − | 3.26247i | 0 | −8.63222 | − | 14.9514i | 0 | 17.0056 | + | 7.33551i | 0 | 7.78736 | + | 13.4881i | 0 | ||||||||||
31.5 | 0 | −4.17280 | − | 2.40917i | 0 | −3.69991 | − | 6.40843i | 0 | −13.0239 | − | 13.1673i | 0 | −1.89183 | − | 3.27674i | 0 | ||||||||||
31.6 | 0 | −3.34210 | − | 1.92956i | 0 | −3.71621 | − | 6.43666i | 0 | −6.86804 | − | 17.1997i | 0 | −6.05359 | − | 10.4851i | 0 | ||||||||||
31.7 | 0 | −1.64284 | − | 0.948492i | 0 | 0.518458 | + | 0.897996i | 0 | −18.1631 | + | 3.61980i | 0 | −11.7007 | − | 20.2663i | 0 | ||||||||||
31.8 | 0 | −1.35962 | − | 0.784974i | 0 | 7.30456 | + | 12.6519i | 0 | 14.7369 | − | 11.2171i | 0 | −12.2676 | − | 21.2482i | 0 | ||||||||||
31.9 | 0 | 1.35962 | + | 0.784974i | 0 | 7.30456 | + | 12.6519i | 0 | −14.7369 | + | 11.2171i | 0 | −12.2676 | − | 21.2482i | 0 | ||||||||||
31.10 | 0 | 1.64284 | + | 0.948492i | 0 | 0.518458 | + | 0.897996i | 0 | 18.1631 | − | 3.61980i | 0 | −11.7007 | − | 20.2663i | 0 | ||||||||||
31.11 | 0 | 3.34210 | + | 1.92956i | 0 | −3.71621 | − | 6.43666i | 0 | 6.86804 | + | 17.1997i | 0 | −6.05359 | − | 10.4851i | 0 | ||||||||||
31.12 | 0 | 4.17280 | + | 2.40917i | 0 | −3.69991 | − | 6.40843i | 0 | 13.0239 | + | 13.1673i | 0 | −1.89183 | − | 3.27674i | 0 | ||||||||||
31.13 | 0 | 5.65076 | + | 3.26247i | 0 | −8.63222 | − | 14.9514i | 0 | −17.0056 | − | 7.33551i | 0 | 7.78736 | + | 13.4881i | 0 | ||||||||||
31.14 | 0 | 6.23656 | + | 3.60068i | 0 | −3.36366 | − | 5.82602i | 0 | 4.34160 | − | 18.0042i | 0 | 12.4298 | + | 21.5290i | 0 | ||||||||||
31.15 | 0 | 6.84052 | + | 3.94938i | 0 | 9.21856 | + | 15.9670i | 0 | −14.3277 | + | 11.7353i | 0 | 17.6951 | + | 30.6489i | 0 | ||||||||||
31.16 | 0 | 8.78932 | + | 5.07452i | 0 | 8.37041 | + | 14.4980i | 0 | 0.901615 | − | 18.4983i | 0 | 38.0015 | + | 65.8205i | 0 | ||||||||||
159.1 | 0 | −8.78932 | + | 5.07452i | 0 | 8.37041 | − | 14.4980i | 0 | −0.901615 | − | 18.4983i | 0 | 38.0015 | − | 65.8205i | 0 | ||||||||||
159.2 | 0 | −6.84052 | + | 3.94938i | 0 | 9.21856 | − | 15.9670i | 0 | 14.3277 | + | 11.7353i | 0 | 17.6951 | − | 30.6489i | 0 | ||||||||||
159.3 | 0 | −6.23656 | + | 3.60068i | 0 | −3.36366 | + | 5.82602i | 0 | −4.34160 | − | 18.0042i | 0 | 12.4298 | − | 21.5290i | 0 | ||||||||||
159.4 | 0 | −5.65076 | + | 3.26247i | 0 | −8.63222 | + | 14.9514i | 0 | 17.0056 | − | 7.33551i | 0 | 7.78736 | − | 13.4881i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
56.j | odd | 6 | 1 | inner |
56.m | 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.q.c | yes | 32 |
4.b | odd | 2 | 1 | inner | 448.4.q.c | yes | 32 |
7.d | odd | 6 | 1 | 448.4.q.a | ✓ | 32 | |
8.b | even | 2 | 1 | 448.4.q.a | ✓ | 32 | |
8.d | odd | 2 | 1 | 448.4.q.a | ✓ | 32 | |
28.f | even | 6 | 1 | 448.4.q.a | ✓ | 32 | |
56.j | odd | 6 | 1 | inner | 448.4.q.c | yes | 32 |
56.m | even | 6 | 1 | inner | 448.4.q.c | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
448.4.q.a | ✓ | 32 | 7.d | odd | 6 | 1 | |
448.4.q.a | ✓ | 32 | 8.b | even | 2 | 1 | |
448.4.q.a | ✓ | 32 | 8.d | odd | 2 | 1 | |
448.4.q.a | ✓ | 32 | 28.f | even | 6 | 1 | |
448.4.q.c | yes | 32 | 1.a | even | 1 | 1 | trivial |
448.4.q.c | yes | 32 | 4.b | odd | 2 | 1 | inner |
448.4.q.c | yes | 32 | 56.j | odd | 6 | 1 | inner |
448.4.q.c | yes | 32 | 56.m | even | 6 | 1 | inner |