Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [720,2,Mod(251,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("720.251");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.bl (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: | \(5.74922894553\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | −1.40272 | − | 0.179908i | 0 | 1.93527 | + | 0.504724i | 0.707107 | − | 0.707107i | 0 | 1.61527 | −2.62384 | − | 1.05616i | 0 | −1.11909 | + | 0.864661i | ||||||||
251.2 | −1.16580 | + | 0.800573i | 0 | 0.718165 | − | 1.86661i | 0.707107 | − | 0.707107i | 0 | −4.13654 | 0.657124 | + | 2.75103i | 0 | −0.258252 | + | 1.39043i | ||||||||
251.3 | −0.904816 | + | 1.08688i | 0 | −0.362616 | − | 1.96685i | −0.707107 | + | 0.707107i | 0 | −2.69352 | 2.46583 | + | 1.38552i | 0 | −0.128739 | − | 1.40834i | ||||||||
251.4 | −0.827528 | − | 1.14682i | 0 | −0.630396 | + | 1.89805i | −0.707107 | + | 0.707107i | 0 | −0.527405 | 2.69840 | − | 0.847739i | 0 | 1.39608 | + | 0.225774i | ||||||||
251.5 | −0.348666 | + | 1.37056i | 0 | −1.75686 | − | 0.955734i | −0.707107 | + | 0.707107i | 0 | 4.49261 | 1.92245 | − | 2.07465i | 0 | −0.722588 | − | 1.21568i | ||||||||
251.6 | −0.219596 | + | 1.39706i | 0 | −1.90356 | − | 0.613577i | 0.707107 | − | 0.707107i | 0 | −0.750417 | 1.27522 | − | 2.52464i | 0 | 0.832593 | + | 1.14315i | ||||||||
251.7 | 0.219596 | − | 1.39706i | 0 | −1.90356 | − | 0.613577i | −0.707107 | + | 0.707107i | 0 | −0.750417 | −1.27522 | + | 2.52464i | 0 | 0.832593 | + | 1.14315i | ||||||||
251.8 | 0.348666 | − | 1.37056i | 0 | −1.75686 | − | 0.955734i | 0.707107 | − | 0.707107i | 0 | 4.49261 | −1.92245 | + | 2.07465i | 0 | −0.722588 | − | 1.21568i | ||||||||
251.9 | 0.827528 | + | 1.14682i | 0 | −0.630396 | + | 1.89805i | 0.707107 | − | 0.707107i | 0 | −0.527405 | −2.69840 | + | 0.847739i | 0 | 1.39608 | + | 0.225774i | ||||||||
251.10 | 0.904816 | − | 1.08688i | 0 | −0.362616 | − | 1.96685i | 0.707107 | − | 0.707107i | 0 | −2.69352 | −2.46583 | − | 1.38552i | 0 | −0.128739 | − | 1.40834i | ||||||||
251.11 | 1.16580 | − | 0.800573i | 0 | 0.718165 | − | 1.86661i | −0.707107 | + | 0.707107i | 0 | −4.13654 | −0.657124 | − | 2.75103i | 0 | −0.258252 | + | 1.39043i | ||||||||
251.12 | 1.40272 | + | 0.179908i | 0 | 1.93527 | + | 0.504724i | −0.707107 | + | 0.707107i | 0 | 1.61527 | 2.62384 | + | 1.05616i | 0 | −1.11909 | + | 0.864661i | ||||||||
611.1 | −1.40272 | + | 0.179908i | 0 | 1.93527 | − | 0.504724i | 0.707107 | + | 0.707107i | 0 | 1.61527 | −2.62384 | + | 1.05616i | 0 | −1.11909 | − | 0.864661i | ||||||||
611.2 | −1.16580 | − | 0.800573i | 0 | 0.718165 | + | 1.86661i | 0.707107 | + | 0.707107i | 0 | −4.13654 | 0.657124 | − | 2.75103i | 0 | −0.258252 | − | 1.39043i | ||||||||
611.3 | −0.904816 | − | 1.08688i | 0 | −0.362616 | + | 1.96685i | −0.707107 | − | 0.707107i | 0 | −2.69352 | 2.46583 | − | 1.38552i | 0 | −0.128739 | + | 1.40834i | ||||||||
611.4 | −0.827528 | + | 1.14682i | 0 | −0.630396 | − | 1.89805i | −0.707107 | − | 0.707107i | 0 | −0.527405 | 2.69840 | + | 0.847739i | 0 | 1.39608 | − | 0.225774i | ||||||||
611.5 | −0.348666 | − | 1.37056i | 0 | −1.75686 | + | 0.955734i | −0.707107 | − | 0.707107i | 0 | 4.49261 | 1.92245 | + | 2.07465i | 0 | −0.722588 | + | 1.21568i | ||||||||
611.6 | −0.219596 | − | 1.39706i | 0 | −1.90356 | + | 0.613577i | 0.707107 | + | 0.707107i | 0 | −0.750417 | 1.27522 | + | 2.52464i | 0 | 0.832593 | − | 1.14315i | ||||||||
611.7 | 0.219596 | + | 1.39706i | 0 | −1.90356 | + | 0.613577i | −0.707107 | − | 0.707107i | 0 | −0.750417 | −1.27522 | − | 2.52464i | 0 | 0.832593 | − | 1.14315i | ||||||||
611.8 | 0.348666 | + | 1.37056i | 0 | −1.75686 | + | 0.955734i | 0.707107 | + | 0.707107i | 0 | 4.49261 | −1.92245 | − | 2.07465i | 0 | −0.722588 | + | 1.21568i | ||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 720.2.bl.b | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 720.2.bl.b | ✓ | 24 |
4.b | odd | 2 | 1 | 2880.2.bl.b | 24 | ||
12.b | even | 2 | 1 | 2880.2.bl.b | 24 | ||
16.e | even | 4 | 1 | 2880.2.bl.b | 24 | ||
16.f | odd | 4 | 1 | inner | 720.2.bl.b | ✓ | 24 |
48.i | odd | 4 | 1 | 2880.2.bl.b | 24 | ||
48.k | even | 4 | 1 | inner | 720.2.bl.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.2.bl.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
720.2.bl.b | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
720.2.bl.b | ✓ | 24 | 16.f | odd | 4 | 1 | inner |
720.2.bl.b | ✓ | 24 | 48.k | even | 4 | 1 | inner |
2880.2.bl.b | 24 | 4.b | odd | 2 | 1 | ||
2880.2.bl.b | 24 | 12.b | even | 2 | 1 | ||
2880.2.bl.b | 24 | 16.e | even | 4 | 1 | ||
2880.2.bl.b | 24 | 48.i | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 2T_{7}^{5} - 22T_{7}^{4} - 48T_{7}^{3} + 48T_{7}^{2} + 96T_{7} + 32 \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).