Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [575,2,Mod(68,575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(575, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("575.68");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 575 = 5^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 575.e (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: | \(4.59139811622\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
68.1 | −1.97358 | + | 1.97358i | −2.17182 | − | 2.17182i | − | 5.79000i | 0 | 8.57249 | 0 | 7.47985 | + | 7.47985i | 6.43359i | 0 | |||||||||||
68.2 | −1.74373 | + | 1.74373i | 2.39185 | + | 2.39185i | − | 4.08117i | 0 | −8.34147 | 0 | 3.62899 | + | 3.62899i | 8.44190i | 0 | |||||||||||
68.3 | −1.72013 | + | 1.72013i | 0.738445 | + | 0.738445i | − | 3.91772i | 0 | −2.54045 | 0 | 3.29873 | + | 3.29873i | − | 1.90940i | 0 | ||||||||||
68.4 | −1.26741 | + | 1.26741i | −2.06693 | − | 2.06693i | − | 1.21267i | 0 | 5.23931 | 0 | −0.997876 | − | 0.997876i | 5.54441i | 0 | |||||||||||
68.5 | −0.706163 | + | 0.706163i | −0.104887 | − | 0.104887i | 1.00267i | 0 | 0.148135 | 0 | −2.12037 | − | 2.12037i | − | 2.97800i | 0 | |||||||||||
68.6 | −0.0235935 | + | 0.0235935i | 1.65341 | + | 1.65341i | 1.99889i | 0 | −0.0780193 | 0 | −0.0943478 | − | 0.0943478i | 2.46750i | 0 | ||||||||||||
68.7 | 0.0235935 | − | 0.0235935i | −1.65341 | − | 1.65341i | 1.99889i | 0 | −0.0780193 | 0 | 0.0943478 | + | 0.0943478i | 2.46750i | 0 | ||||||||||||
68.8 | 0.706163 | − | 0.706163i | 0.104887 | + | 0.104887i | 1.00267i | 0 | 0.148135 | 0 | 2.12037 | + | 2.12037i | − | 2.97800i | 0 | |||||||||||
68.9 | 1.26741 | − | 1.26741i | 2.06693 | + | 2.06693i | − | 1.21267i | 0 | 5.23931 | 0 | 0.997876 | + | 0.997876i | 5.54441i | 0 | |||||||||||
68.10 | 1.72013 | − | 1.72013i | −0.738445 | − | 0.738445i | − | 3.91772i | 0 | −2.54045 | 0 | −3.29873 | − | 3.29873i | − | 1.90940i | 0 | ||||||||||
68.11 | 1.74373 | − | 1.74373i | −2.39185 | − | 2.39185i | − | 4.08117i | 0 | −8.34147 | 0 | −3.62899 | − | 3.62899i | 8.44190i | 0 | |||||||||||
68.12 | 1.97358 | − | 1.97358i | 2.17182 | + | 2.17182i | − | 5.79000i | 0 | 8.57249 | 0 | −7.47985 | − | 7.47985i | 6.43359i | 0 | |||||||||||
482.1 | −1.97358 | − | 1.97358i | −2.17182 | + | 2.17182i | 5.79000i | 0 | 8.57249 | 0 | 7.47985 | − | 7.47985i | − | 6.43359i | 0 | |||||||||||
482.2 | −1.74373 | − | 1.74373i | 2.39185 | − | 2.39185i | 4.08117i | 0 | −8.34147 | 0 | 3.62899 | − | 3.62899i | − | 8.44190i | 0 | |||||||||||
482.3 | −1.72013 | − | 1.72013i | 0.738445 | − | 0.738445i | 3.91772i | 0 | −2.54045 | 0 | 3.29873 | − | 3.29873i | 1.90940i | 0 | ||||||||||||
482.4 | −1.26741 | − | 1.26741i | −2.06693 | + | 2.06693i | 1.21267i | 0 | 5.23931 | 0 | −0.997876 | + | 0.997876i | − | 5.54441i | 0 | |||||||||||
482.5 | −0.706163 | − | 0.706163i | −0.104887 | + | 0.104887i | − | 1.00267i | 0 | 0.148135 | 0 | −2.12037 | + | 2.12037i | 2.97800i | 0 | |||||||||||
482.6 | −0.0235935 | − | 0.0235935i | 1.65341 | − | 1.65341i | − | 1.99889i | 0 | −0.0780193 | 0 | −0.0943478 | + | 0.0943478i | − | 2.46750i | 0 | ||||||||||
482.7 | 0.0235935 | + | 0.0235935i | −1.65341 | + | 1.65341i | − | 1.99889i | 0 | −0.0780193 | 0 | 0.0943478 | − | 0.0943478i | − | 2.46750i | 0 | ||||||||||
482.8 | 0.706163 | + | 0.706163i | 0.104887 | − | 0.104887i | − | 1.00267i | 0 | 0.148135 | 0 | 2.12037 | − | 2.12037i | 2.97800i | 0 | |||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
115.c | odd | 2 | 1 | inner |
115.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 575.2.e.e | ✓ | 24 |
5.b | even | 2 | 1 | inner | 575.2.e.e | ✓ | 24 |
5.c | odd | 4 | 2 | inner | 575.2.e.e | ✓ | 24 |
23.b | odd | 2 | 1 | CM | 575.2.e.e | ✓ | 24 |
115.c | odd | 2 | 1 | inner | 575.2.e.e | ✓ | 24 |
115.e | even | 4 | 2 | inner | 575.2.e.e | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
575.2.e.e | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
575.2.e.e | ✓ | 24 | 5.b | even | 2 | 1 | inner |
575.2.e.e | ✓ | 24 | 5.c | odd | 4 | 2 | inner |
575.2.e.e | ✓ | 24 | 23.b | odd | 2 | 1 | CM |
575.2.e.e | ✓ | 24 | 115.c | odd | 2 | 1 | inner |
575.2.e.e | ✓ | 24 | 115.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{24} + 144T_{2}^{20} + 7176T_{2}^{16} + 144047T_{2}^{12} + 947448T_{2}^{8} + 806808T_{2}^{4} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\).