Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,4,Mod(31,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.bc (of order \(4\), 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: | \(36.8171918436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(28\) |
| Relative dimension: | \(14\) over \(\Q(i)\) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | − | 3.00000i | 0 | −13.9138 | + | 13.9138i | 0 | −3.70358 | + | 3.70358i | 0 | −9.00000 | 0 | |||||||||||||
| 31.2 | 0 | − | 3.00000i | 0 | −11.5719 | + | 11.5719i | 0 | −25.2676 | + | 25.2676i | 0 | −9.00000 | 0 | |||||||||||||
| 31.3 | 0 | − | 3.00000i | 0 | −8.48307 | + | 8.48307i | 0 | 5.05973 | − | 5.05973i | 0 | −9.00000 | 0 | |||||||||||||
| 31.4 | 0 | − | 3.00000i | 0 | −6.77018 | + | 6.77018i | 0 | 21.6794 | − | 21.6794i | 0 | −9.00000 | 0 | |||||||||||||
| 31.5 | 0 | − | 3.00000i | 0 | −6.10630 | + | 6.10630i | 0 | 23.8576 | − | 23.8576i | 0 | −9.00000 | 0 | |||||||||||||
| 31.6 | 0 | − | 3.00000i | 0 | −2.84589 | + | 2.84589i | 0 | −9.35646 | + | 9.35646i | 0 | −9.00000 | 0 | |||||||||||||
| 31.7 | 0 | − | 3.00000i | 0 | −1.88620 | + | 1.88620i | 0 | 9.36349 | − | 9.36349i | 0 | −9.00000 | 0 | |||||||||||||
| 31.8 | 0 | − | 3.00000i | 0 | −0.273393 | + | 0.273393i | 0 | −21.0333 | + | 21.0333i | 0 | −9.00000 | 0 | |||||||||||||
| 31.9 | 0 | − | 3.00000i | 0 | 5.36394 | − | 5.36394i | 0 | −11.4083 | + | 11.4083i | 0 | −9.00000 | 0 | |||||||||||||
| 31.10 | 0 | − | 3.00000i | 0 | 7.68384 | − | 7.68384i | 0 | 9.93516 | − | 9.93516i | 0 | −9.00000 | 0 | |||||||||||||
| 31.11 | 0 | − | 3.00000i | 0 | 7.87509 | − | 7.87509i | 0 | −15.6298 | + | 15.6298i | 0 | −9.00000 | 0 | |||||||||||||
| 31.12 | 0 | − | 3.00000i | 0 | 8.84364 | − | 8.84364i | 0 | −3.20396 | + | 3.20396i | 0 | −9.00000 | 0 | |||||||||||||
| 31.13 | 0 | − | 3.00000i | 0 | 10.3664 | − | 10.3664i | 0 | 6.70327 | − | 6.70327i | 0 | −9.00000 | 0 | |||||||||||||
| 31.14 | 0 | − | 3.00000i | 0 | 13.7178 | − | 13.7178i | 0 | 9.00439 | − | 9.00439i | 0 | −9.00000 | 0 | |||||||||||||
| 463.1 | 0 | 3.00000i | 0 | −13.9138 | − | 13.9138i | 0 | −3.70358 | − | 3.70358i | 0 | −9.00000 | 0 | ||||||||||||||
| 463.2 | 0 | 3.00000i | 0 | −11.5719 | − | 11.5719i | 0 | −25.2676 | − | 25.2676i | 0 | −9.00000 | 0 | ||||||||||||||
| 463.3 | 0 | 3.00000i | 0 | −8.48307 | − | 8.48307i | 0 | 5.05973 | + | 5.05973i | 0 | −9.00000 | 0 | ||||||||||||||
| 463.4 | 0 | 3.00000i | 0 | −6.77018 | − | 6.77018i | 0 | 21.6794 | + | 21.6794i | 0 | −9.00000 | 0 | ||||||||||||||
| 463.5 | 0 | 3.00000i | 0 | −6.10630 | − | 6.10630i | 0 | 23.8576 | + | 23.8576i | 0 | −9.00000 | 0 | ||||||||||||||
| 463.6 | 0 | 3.00000i | 0 | −2.84589 | − | 2.84589i | 0 | −9.35646 | − | 9.35646i | 0 | −9.00000 | 0 | ||||||||||||||
| See all 28 embeddings | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 52.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.4.bc.c | ✓ | 28 |
| 4.b | odd | 2 | 1 | 624.4.bc.d | yes | 28 | |
| 13.d | odd | 4 | 1 | 624.4.bc.d | yes | 28 | |
| 52.f | even | 4 | 1 | inner | 624.4.bc.c | ✓ | 28 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 624.4.bc.c | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
| 624.4.bc.c | ✓ | 28 | 52.f | even | 4 | 1 | inner |
| 624.4.bc.d | yes | 28 | 4.b | odd | 2 | 1 | |
| 624.4.bc.d | yes | 28 | 13.d | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5}^{28} - 4 T_{5}^{27} + 8 T_{5}^{26} + 72 T_{5}^{25} + 250468 T_{5}^{24} - 1122080 T_{5}^{23} + \cdots + 18\!\cdots\!44 \)
|
|
\( T_{7}^{28} + 8 T_{7}^{27} + 32 T_{7}^{26} - 6464 T_{7}^{25} + 2466704 T_{7}^{24} + 15378944 T_{7}^{23} + \cdots + 39\!\cdots\!64 \)
|