Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [592,2,Mod(31,592)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("592.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 592 = 2^{4} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 592.t (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: | \(4.72714379966\) |
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 | −3.32855 | 0 | −1.47049 | + | 1.47049i | 0 | 4.50934i | 0 | 8.07924 | 0 | ||||||||||||||||
31.2 | 0 | −2.67173 | 0 | 2.55479 | − | 2.55479i | 0 | − | 3.18696i | 0 | 4.13816 | 0 | |||||||||||||||
31.3 | 0 | −2.08948 | 0 | 1.37839 | − | 1.37839i | 0 | 2.77307i | 0 | 1.36594 | 0 | ||||||||||||||||
31.4 | 0 | −1.91097 | 0 | −2.79886 | + | 2.79886i | 0 | 0.284669i | 0 | 0.651811 | 0 | ||||||||||||||||
31.5 | 0 | −1.15308 | 0 | −2.03350 | + | 2.03350i | 0 | − | 3.07110i | 0 | −1.67041 | 0 | |||||||||||||||
31.6 | 0 | −0.595389 | 0 | −0.247425 | + | 0.247425i | 0 | 0.0842208i | 0 | −2.64551 | 0 | ||||||||||||||||
31.7 | 0 | −0.284197 | 0 | 0.617099 | − | 0.617099i | 0 | − | 4.72224i | 0 | −2.91923 | 0 | |||||||||||||||
31.8 | 0 | 0.284197 | 0 | 0.617099 | − | 0.617099i | 0 | 4.72224i | 0 | −2.91923 | 0 | ||||||||||||||||
31.9 | 0 | 0.595389 | 0 | −0.247425 | + | 0.247425i | 0 | − | 0.0842208i | 0 | −2.64551 | 0 | |||||||||||||||
31.10 | 0 | 1.15308 | 0 | −2.03350 | + | 2.03350i | 0 | 3.07110i | 0 | −1.67041 | 0 | ||||||||||||||||
31.11 | 0 | 1.91097 | 0 | −2.79886 | + | 2.79886i | 0 | − | 0.284669i | 0 | 0.651811 | 0 | |||||||||||||||
31.12 | 0 | 2.08948 | 0 | 1.37839 | − | 1.37839i | 0 | − | 2.77307i | 0 | 1.36594 | 0 | |||||||||||||||
31.13 | 0 | 2.67173 | 0 | 2.55479 | − | 2.55479i | 0 | 3.18696i | 0 | 4.13816 | 0 | ||||||||||||||||
31.14 | 0 | 3.32855 | 0 | −1.47049 | + | 1.47049i | 0 | − | 4.50934i | 0 | 8.07924 | 0 | |||||||||||||||
191.1 | 0 | −3.32855 | 0 | −1.47049 | − | 1.47049i | 0 | − | 4.50934i | 0 | 8.07924 | 0 | |||||||||||||||
191.2 | 0 | −2.67173 | 0 | 2.55479 | + | 2.55479i | 0 | 3.18696i | 0 | 4.13816 | 0 | ||||||||||||||||
191.3 | 0 | −2.08948 | 0 | 1.37839 | + | 1.37839i | 0 | − | 2.77307i | 0 | 1.36594 | 0 | |||||||||||||||
191.4 | 0 | −1.91097 | 0 | −2.79886 | − | 2.79886i | 0 | − | 0.284669i | 0 | 0.651811 | 0 | |||||||||||||||
191.5 | 0 | −1.15308 | 0 | −2.03350 | − | 2.03350i | 0 | 3.07110i | 0 | −1.67041 | 0 | ||||||||||||||||
191.6 | 0 | −0.595389 | 0 | −0.247425 | − | 0.247425i | 0 | − | 0.0842208i | 0 | −2.64551 | 0 | |||||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
37.d | odd | 4 | 1 | inner |
148.g | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 592.2.t.d | ✓ | 28 |
4.b | odd | 2 | 1 | inner | 592.2.t.d | ✓ | 28 |
37.d | odd | 4 | 1 | inner | 592.2.t.d | ✓ | 28 |
148.g | even | 4 | 1 | inner | 592.2.t.d | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
592.2.t.d | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
592.2.t.d | ✓ | 28 | 4.b | odd | 2 | 1 | inner |
592.2.t.d | ✓ | 28 | 37.d | odd | 4 | 1 | inner |
592.2.t.d | ✓ | 28 | 148.g | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 28T_{3}^{12} + 288T_{3}^{10} - 1366T_{3}^{8} + 3040T_{3}^{6} - 2796T_{3}^{4} + 801T_{3}^{2} - 48 \) acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\).