Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5148,2,Mod(2465,5148)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 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("5148.2465");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 5148.ba (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: | \(41.1069869606\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2465.1 | 0 | 0 | 0 | −2.67976 | + | 2.67976i | 0 | 2.08327 | − | 2.08327i | 0 | 0 | 0 | ||||||||||||||
2465.2 | 0 | 0 | 0 | −2.60062 | + | 2.60062i | 0 | 2.65451 | − | 2.65451i | 0 | 0 | 0 | ||||||||||||||
2465.3 | 0 | 0 | 0 | −2.42828 | + | 2.42828i | 0 | −1.87701 | + | 1.87701i | 0 | 0 | 0 | ||||||||||||||
2465.4 | 0 | 0 | 0 | −2.30501 | + | 2.30501i | 0 | −1.30717 | + | 1.30717i | 0 | 0 | 0 | ||||||||||||||
2465.5 | 0 | 0 | 0 | −1.71821 | + | 1.71821i | 0 | −2.09939 | + | 2.09939i | 0 | 0 | 0 | ||||||||||||||
2465.6 | 0 | 0 | 0 | −1.50302 | + | 1.50302i | 0 | −1.64397 | + | 1.64397i | 0 | 0 | 0 | ||||||||||||||
2465.7 | 0 | 0 | 0 | −1.17572 | + | 1.17572i | 0 | −2.53642 | + | 2.53642i | 0 | 0 | 0 | ||||||||||||||
2465.8 | 0 | 0 | 0 | −1.07369 | + | 1.07369i | 0 | −0.820398 | + | 0.820398i | 0 | 0 | 0 | ||||||||||||||
2465.9 | 0 | 0 | 0 | −0.829290 | + | 0.829290i | 0 | 2.29670 | − | 2.29670i | 0 | 0 | 0 | ||||||||||||||
2465.10 | 0 | 0 | 0 | −0.632743 | + | 0.632743i | 0 | 2.17694 | − | 2.17694i | 0 | 0 | 0 | ||||||||||||||
2465.11 | 0 | 0 | 0 | −0.107295 | + | 0.107295i | 0 | 1.07295 | − | 1.07295i | 0 | 0 | 0 | ||||||||||||||
2465.12 | 0 | 0 | 0 | 0.107295 | − | 0.107295i | 0 | 1.07295 | − | 1.07295i | 0 | 0 | 0 | ||||||||||||||
2465.13 | 0 | 0 | 0 | 0.632743 | − | 0.632743i | 0 | 2.17694 | − | 2.17694i | 0 | 0 | 0 | ||||||||||||||
2465.14 | 0 | 0 | 0 | 0.829290 | − | 0.829290i | 0 | 2.29670 | − | 2.29670i | 0 | 0 | 0 | ||||||||||||||
2465.15 | 0 | 0 | 0 | 1.07369 | − | 1.07369i | 0 | −0.820398 | + | 0.820398i | 0 | 0 | 0 | ||||||||||||||
2465.16 | 0 | 0 | 0 | 1.17572 | − | 1.17572i | 0 | −2.53642 | + | 2.53642i | 0 | 0 | 0 | ||||||||||||||
2465.17 | 0 | 0 | 0 | 1.50302 | − | 1.50302i | 0 | −1.64397 | + | 1.64397i | 0 | 0 | 0 | ||||||||||||||
2465.18 | 0 | 0 | 0 | 1.71821 | − | 1.71821i | 0 | −2.09939 | + | 2.09939i | 0 | 0 | 0 | ||||||||||||||
2465.19 | 0 | 0 | 0 | 2.30501 | − | 2.30501i | 0 | −1.30717 | + | 1.30717i | 0 | 0 | 0 | ||||||||||||||
2465.20 | 0 | 0 | 0 | 2.42828 | − | 2.42828i | 0 | −1.87701 | + | 1.87701i | 0 | 0 | 0 | ||||||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
13.d | odd | 4 | 1 | inner |
39.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 5148.2.ba.b | ✓ | 44 |
3.b | odd | 2 | 1 | inner | 5148.2.ba.b | ✓ | 44 |
13.d | odd | 4 | 1 | inner | 5148.2.ba.b | ✓ | 44 |
39.f | even | 4 | 1 | inner | 5148.2.ba.b | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5148.2.ba.b | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
5148.2.ba.b | ✓ | 44 | 3.b | odd | 2 | 1 | inner |
5148.2.ba.b | ✓ | 44 | 13.d | odd | 4 | 1 | inner |
5148.2.ba.b | ✓ | 44 | 39.f | even | 4 | 1 | inner |