Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [330,2,Mod(41,330)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(330, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("330.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 330.r (of order \(10\), degree \(4\), 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: | \(2.63506326670\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | 0.809017 | − | 0.587785i | −1.62000 | + | 0.612873i | 0.309017 | − | 0.951057i | −0.587785 | + | 0.809017i | −0.950367 | + | 1.44803i | 1.85454 | + | 0.602576i | −0.309017 | − | 0.951057i | 2.24877 | − | 1.98570i | 1.00000i | ||
41.2 | 0.809017 | − | 0.587785i | −1.60878 | − | 0.641738i | 0.309017 | − | 0.951057i | 0.587785 | − | 0.809017i | −1.67873 | + | 0.426440i | 4.17862 | + | 1.35772i | −0.309017 | − | 0.951057i | 2.17635 | + | 2.06483i | − | 1.00000i | |
41.3 | 0.809017 | − | 0.587785i | −1.55235 | − | 0.768241i | 0.309017 | − | 0.951057i | −0.587785 | + | 0.809017i | −1.70744 | + | 0.290931i | −3.84229 | − | 1.24844i | −0.309017 | − | 0.951057i | 1.81961 | + | 2.38517i | 1.00000i | ||
41.4 | 0.809017 | − | 0.587785i | −0.0409562 | − | 1.73157i | 0.309017 | − | 0.951057i | 0.587785 | − | 0.809017i | −1.05092 | − | 1.37679i | −3.30382 | − | 1.07347i | −0.309017 | − | 0.951057i | −2.99665 | + | 0.141837i | − | 1.00000i | |
41.5 | 0.809017 | − | 0.587785i | 0.838022 | + | 1.51582i | 0.309017 | − | 0.951057i | 0.587785 | − | 0.809017i | 1.56895 | + | 0.733749i | 1.16413 | + | 0.378250i | −0.309017 | − | 0.951057i | −1.59544 | + | 2.54059i | − | 1.00000i | |
41.6 | 0.809017 | − | 0.587785i | 1.26208 | − | 1.18624i | 0.309017 | − | 0.951057i | −0.587785 | + | 0.809017i | 0.323785 | − | 1.70152i | −0.619187 | − | 0.201186i | −0.309017 | − | 0.951057i | 0.185667 | − | 2.99425i | 1.00000i | ||
41.7 | 0.809017 | − | 0.587785i | 1.63151 | + | 0.581536i | 0.309017 | − | 0.951057i | −0.587785 | + | 0.809017i | 1.66174 | − | 0.488504i | 2.60694 | + | 0.847045i | −0.309017 | − | 0.951057i | 2.32363 | + | 1.89756i | 1.00000i | ||
41.8 | 0.809017 | − | 0.587785i | 1.70852 | − | 0.284558i | 0.309017 | − | 0.951057i | 0.587785 | − | 0.809017i | 1.21496 | − | 1.23445i | −2.03894 | − | 0.662490i | −0.309017 | − | 0.951057i | 2.83805 | − | 0.972343i | − | 1.00000i | |
101.1 | −0.309017 | + | 0.951057i | −1.70379 | − | 0.311634i | −0.809017 | − | 0.587785i | −0.951057 | + | 0.309017i | 0.822880 | − | 1.52410i | 1.35300 | − | 1.86225i | 0.809017 | − | 0.587785i | 2.80577 | + | 1.06192i | − | 1.00000i | |
101.2 | −0.309017 | + | 0.951057i | −1.58391 | + | 0.700885i | −0.809017 | − | 0.587785i | −0.951057 | + | 0.309017i | −0.177128 | − | 1.72297i | −2.69542 | + | 3.70992i | 0.809017 | − | 0.587785i | 2.01752 | − | 2.22027i | − | 1.00000i | |
101.3 | −0.309017 | + | 0.951057i | −1.43038 | + | 0.976738i | −0.809017 | − | 0.587785i | 0.951057 | − | 0.309017i | −0.486922 | − | 1.66220i | 1.69090 | − | 2.32732i | 0.809017 | − | 0.587785i | 1.09197 | − | 2.79421i | 1.00000i | ||
101.4 | −0.309017 | + | 0.951057i | −1.17047 | − | 1.27671i | −0.809017 | − | 0.587785i | 0.951057 | − | 0.309017i | 1.57592 | − | 0.718661i | −2.51864 | + | 3.46661i | 0.809017 | − | 0.587785i | −0.259982 | + | 2.98871i | 1.00000i | ||
101.5 | −0.309017 | + | 0.951057i | −0.112210 | − | 1.72841i | −0.809017 | − | 0.587785i | −0.951057 | + | 0.309017i | 1.67849 | + | 0.427391i | 1.01702 | − | 1.39980i | 0.809017 | − | 0.587785i | −2.97482 | + | 0.387890i | − | 1.00000i | |
101.6 | −0.309017 | + | 0.951057i | 1.04818 | + | 1.37889i | −0.809017 | − | 0.587785i | 0.951057 | − | 0.309017i | −1.63530 | + | 0.570775i | −1.31068 | + | 1.80400i | 0.809017 | − | 0.587785i | −0.802655 | + | 2.89063i | 1.00000i | ||
101.7 | −0.309017 | + | 0.951057i | 1.63983 | − | 0.557641i | −0.809017 | − | 0.587785i | −0.951057 | + | 0.309017i | 0.0236135 | + | 1.73189i | 0.325400 | − | 0.447874i | 0.809017 | − | 0.587785i | 2.37807 | − | 1.82887i | − | 1.00000i | |
101.8 | −0.309017 | + | 0.951057i | 1.69472 | − | 0.357682i | −0.809017 | − | 0.587785i | 0.951057 | − | 0.309017i | −0.183521 | + | 1.72230i | 2.13842 | − | 2.94328i | 0.809017 | − | 0.587785i | 2.74413 | − | 1.21234i | 1.00000i | ||
161.1 | 0.809017 | + | 0.587785i | −1.62000 | − | 0.612873i | 0.309017 | + | 0.951057i | −0.587785 | − | 0.809017i | −0.950367 | − | 1.44803i | 1.85454 | − | 0.602576i | −0.309017 | + | 0.951057i | 2.24877 | + | 1.98570i | − | 1.00000i | |
161.2 | 0.809017 | + | 0.587785i | −1.60878 | + | 0.641738i | 0.309017 | + | 0.951057i | 0.587785 | + | 0.809017i | −1.67873 | − | 0.426440i | 4.17862 | − | 1.35772i | −0.309017 | + | 0.951057i | 2.17635 | − | 2.06483i | 1.00000i | ||
161.3 | 0.809017 | + | 0.587785i | −1.55235 | + | 0.768241i | 0.309017 | + | 0.951057i | −0.587785 | − | 0.809017i | −1.70744 | − | 0.290931i | −3.84229 | + | 1.24844i | −0.309017 | + | 0.951057i | 1.81961 | − | 2.38517i | − | 1.00000i | |
161.4 | 0.809017 | + | 0.587785i | −0.0409562 | + | 1.73157i | 0.309017 | + | 0.951057i | 0.587785 | + | 0.809017i | −1.05092 | + | 1.37679i | −3.30382 | + | 1.07347i | −0.309017 | + | 0.951057i | −2.99665 | − | 0.141837i | 1.00000i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
33.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 330.2.r.b | yes | 32 |
3.b | odd | 2 | 1 | 330.2.r.a | ✓ | 32 | |
11.d | odd | 10 | 1 | 330.2.r.a | ✓ | 32 | |
33.f | even | 10 | 1 | inner | 330.2.r.b | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
330.2.r.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
330.2.r.a | ✓ | 32 | 11.d | odd | 10 | 1 | |
330.2.r.b | yes | 32 | 1.a | even | 1 | 1 | trivial |
330.2.r.b | yes | 32 | 33.f | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{32} - 14 T_{17}^{31} + 189 T_{17}^{30} - 1366 T_{17}^{29} + 10764 T_{17}^{28} + \cdots + 9922903204096 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\).