Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,4,Mod(623,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.623");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 624.n (of order \(2\), degree \(1\), 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: | \(24\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
623.1 | 0 | −4.94584 | − | 1.59332i | 0 | −2.47960 | 0 | 21.7201 | 0 | 21.9227 | + | 15.7606i | 0 | ||||||||||||||
623.2 | 0 | −4.94584 | − | 1.59332i | 0 | 2.47960 | 0 | −21.7201 | 0 | 21.9227 | + | 15.7606i | 0 | ||||||||||||||
623.3 | 0 | −4.94584 | + | 1.59332i | 0 | −2.47960 | 0 | 21.7201 | 0 | 21.9227 | − | 15.7606i | 0 | ||||||||||||||
623.4 | 0 | −4.94584 | + | 1.59332i | 0 | 2.47960 | 0 | −21.7201 | 0 | 21.9227 | − | 15.7606i | 0 | ||||||||||||||
623.5 | 0 | −4.38900 | − | 2.78149i | 0 | −18.3912 | 0 | 7.53270 | 0 | 11.5266 | + | 24.4159i | 0 | ||||||||||||||
623.6 | 0 | −4.38900 | − | 2.78149i | 0 | 18.3912 | 0 | −7.53270 | 0 | 11.5266 | + | 24.4159i | 0 | ||||||||||||||
623.7 | 0 | −4.38900 | + | 2.78149i | 0 | −18.3912 | 0 | 7.53270 | 0 | 11.5266 | − | 24.4159i | 0 | ||||||||||||||
623.8 | 0 | −4.38900 | + | 2.78149i | 0 | 18.3912 | 0 | −7.53270 | 0 | 11.5266 | − | 24.4159i | 0 | ||||||||||||||
623.9 | 0 | −2.74324 | − | 4.41301i | 0 | −14.3044 | 0 | −18.8811 | 0 | −11.9493 | + | 24.2119i | 0 | ||||||||||||||
623.10 | 0 | −2.74324 | − | 4.41301i | 0 | 14.3044 | 0 | 18.8811 | 0 | −11.9493 | + | 24.2119i | 0 | ||||||||||||||
623.11 | 0 | −2.74324 | + | 4.41301i | 0 | −14.3044 | 0 | −18.8811 | 0 | −11.9493 | − | 24.2119i | 0 | ||||||||||||||
623.12 | 0 | −2.74324 | + | 4.41301i | 0 | 14.3044 | 0 | 18.8811 | 0 | −11.9493 | − | 24.2119i | 0 | ||||||||||||||
623.13 | 0 | 2.74324 | − | 4.41301i | 0 | −14.3044 | 0 | 18.8811 | 0 | −11.9493 | − | 24.2119i | 0 | ||||||||||||||
623.14 | 0 | 2.74324 | − | 4.41301i | 0 | 14.3044 | 0 | −18.8811 | 0 | −11.9493 | − | 24.2119i | 0 | ||||||||||||||
623.15 | 0 | 2.74324 | + | 4.41301i | 0 | −14.3044 | 0 | 18.8811 | 0 | −11.9493 | + | 24.2119i | 0 | ||||||||||||||
623.16 | 0 | 2.74324 | + | 4.41301i | 0 | 14.3044 | 0 | −18.8811 | 0 | −11.9493 | + | 24.2119i | 0 | ||||||||||||||
623.17 | 0 | 4.38900 | − | 2.78149i | 0 | −18.3912 | 0 | −7.53270 | 0 | 11.5266 | − | 24.4159i | 0 | ||||||||||||||
623.18 | 0 | 4.38900 | − | 2.78149i | 0 | 18.3912 | 0 | 7.53270 | 0 | 11.5266 | − | 24.4159i | 0 | ||||||||||||||
623.19 | 0 | 4.38900 | + | 2.78149i | 0 | −18.3912 | 0 | −7.53270 | 0 | 11.5266 | + | 24.4159i | 0 | ||||||||||||||
623.20 | 0 | 4.38900 | + | 2.78149i | 0 | 18.3912 | 0 | 7.53270 | 0 | 11.5266 | + | 24.4159i | 0 | ||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
39.d | odd | 2 | 1 | inner |
52.b | odd | 2 | 1 | inner |
156.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 624.4.n.c | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
12.b | even | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
13.b | even | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
39.d | odd | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
52.b | odd | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
156.h | even | 2 | 1 | inner | 624.4.n.c | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
624.4.n.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
624.4.n.c | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
624.4.n.c | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
624.4.n.c | ✓ | 24 | 12.b | even | 2 | 1 | inner |
624.4.n.c | ✓ | 24 | 13.b | even | 2 | 1 | inner |
624.4.n.c | ✓ | 24 | 39.d | odd | 2 | 1 | inner |
624.4.n.c | ✓ | 24 | 52.b | odd | 2 | 1 | inner |
624.4.n.c | ✓ | 24 | 156.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):
\( T_{5}^{6} - 549T_{5}^{4} + 72546T_{5}^{2} - 425520 \) |
\( T_{7}^{6} - 885T_{7}^{4} + 215178T_{7}^{2} - 9542880 \) |