Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [927,2,Mod(19,927)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(927, base_ring=CyclotomicField(102))
chi = DirichletCharacter(H, H._module([0, 80]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("927.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 927 = 3^{2} \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 927.ba (of order \(51\), degree \(32\), 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: | \(7.40213226737\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{51}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | 0 | 0 | 1.99621 | + | 0.123122i | 0 | 0 | 2.75651 | + | 1.26819i | 0 | 0 | 0 | ||||||||||||||
28.1 | 0 | 0 | 0.779572 | + | 1.84181i | 0 | 0 | −1.10983 | + | 3.14946i | 0 | 0 | 0 | ||||||||||||||
55.1 | 0 | 0 | 0.0615901 | + | 1.99905i | 0 | 0 | −0.940317 | − | 4.29362i | 0 | 0 | 0 | ||||||||||||||
82.1 | 0 | 0 | 1.90588 | + | 0.606305i | 0 | 0 | −2.53900 | + | 3.83174i | 0 | 0 | 0 | ||||||||||||||
91.1 | 0 | 0 | −0.427866 | + | 1.95370i | 0 | 0 | 5.22320 | − | 0.322156i | 0 | 0 | 0 | ||||||||||||||
118.1 | 0 | 0 | 0.0615901 | − | 1.99905i | 0 | 0 | −0.940317 | + | 4.29362i | 0 | 0 | 0 | ||||||||||||||
136.1 | 0 | 0 | −1.30124 | − | 1.51881i | 0 | 0 | 4.71507 | − | 1.18589i | 0 | 0 | 0 | ||||||||||||||
163.1 | 0 | 0 | −0.427866 | − | 1.95370i | 0 | 0 | 5.22320 | + | 0.322156i | 0 | 0 | 0 | ||||||||||||||
208.1 | 0 | 0 | −1.98484 | + | 0.245777i | 0 | 0 | −3.35907 | + | 3.92073i | 0 | 0 | 0 | ||||||||||||||
235.1 | 0 | 0 | −1.63239 | − | 1.15555i | 0 | 0 | −2.06121 | − | 4.86979i | 0 | 0 | 0 | ||||||||||||||
244.1 | 0 | 0 | 1.99621 | − | 0.123122i | 0 | 0 | 2.75651 | − | 1.26819i | 0 | 0 | 0 | ||||||||||||||
289.1 | 0 | 0 | 1.55816 | − | 1.25385i | 0 | 0 | −0.0673181 | − | 2.18497i | 0 | 0 | 0 | ||||||||||||||
298.1 | 0 | 0 | 0.779572 | − | 1.84181i | 0 | 0 | −1.10983 | − | 3.14946i | 0 | 0 | 0 | ||||||||||||||
316.1 | 0 | 0 | 0.306783 | + | 1.97633i | 0 | 0 | −1.11300 | − | 0.597669i | 0 | 0 | 0 | ||||||||||||||
325.1 | 0 | 0 | −1.76202 | + | 0.946187i | 0 | 0 | 1.55422 | − | 0.494432i | 0 | 0 | 0 | ||||||||||||||
334.1 | 0 | 0 | −1.30124 | + | 1.51881i | 0 | 0 | 4.71507 | + | 1.18589i | 0 | 0 | 0 | ||||||||||||||
361.1 | 0 | 0 | −1.98484 | − | 0.245777i | 0 | 0 | −3.35907 | − | 3.92073i | 0 | 0 | 0 | ||||||||||||||
406.1 | 0 | 0 | −0.664710 | + | 1.88631i | 0 | 0 | −1.75994 | − | 1.81500i | 0 | 0 | 0 | ||||||||||||||
532.1 | 0 | 0 | −0.664710 | − | 1.88631i | 0 | 0 | −1.75994 | + | 1.81500i | 0 | 0 | 0 | ||||||||||||||
541.1 | 0 | 0 | −1.93959 | − | 0.487827i | 0 | 0 | −0.104034 | + | 0.670197i | 0 | 0 | 0 | ||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
103.g | even | 51 | 1 | inner |
309.n | odd | 102 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 927.2.ba.a | ✓ | 32 |
3.b | odd | 2 | 1 | CM | 927.2.ba.a | ✓ | 32 |
103.g | even | 51 | 1 | inner | 927.2.ba.a | ✓ | 32 |
309.n | odd | 102 | 1 | inner | 927.2.ba.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
927.2.ba.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
927.2.ba.a | ✓ | 32 | 3.b | odd | 2 | 1 | CM |
927.2.ba.a | ✓ | 32 | 103.g | even | 51 | 1 | inner |
927.2.ba.a | ✓ | 32 | 309.n | odd | 102 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\).