Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(19,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 10, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.by (of order \(12\), 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: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | −0.965926 | − | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | −2.19856 | + | 0.589103i | 0.258819 | − | 0.965926i | 1.83682 | − | 1.90423i | −0.707107 | − | 0.707107i | −1.00000 | 2.27612 | ||||||
19.2 | −0.965926 | − | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | −1.43558 | + | 0.384662i | 0.258819 | − | 0.965926i | 1.74877 | + | 1.98539i | −0.707107 | − | 0.707107i | −1.00000 | 1.48622 | ||||||
19.3 | −0.965926 | − | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | −0.106603 | + | 0.0285643i | 0.258819 | − | 0.965926i | −1.53649 | + | 2.15388i | −0.707107 | − | 0.707107i | −1.00000 | 0.110364 | ||||||
19.4 | −0.965926 | − | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | 3.74075 | − | 1.00233i | 0.258819 | − | 0.965926i | −2.20451 | + | 1.46293i | −0.707107 | − | 0.707107i | −1.00000 | −3.87271 | ||||||
19.5 | 0.965926 | + | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | −2.85793 | + | 0.765779i | −0.258819 | + | 0.965926i | 0.565013 | + | 2.58472i | 0.707107 | + | 0.707107i | −1.00000 | −2.95874 | ||||||
19.6 | 0.965926 | + | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | 0.128673 | − | 0.0344778i | −0.258819 | + | 0.965926i | −2.64307 | + | 0.119073i | 0.707107 | + | 0.707107i | −1.00000 | 0.133212 | ||||||
19.7 | 0.965926 | + | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | 0.421063 | − | 0.112824i | −0.258819 | + | 0.965926i | 2.44466 | + | 1.01175i | 0.707107 | + | 0.707107i | −1.00000 | 0.435917 | ||||||
19.8 | 0.965926 | + | 0.258819i | 1.00000i | 0.866025 | + | 0.500000i | 2.30819 | − | 0.618478i | −0.258819 | + | 0.965926i | 1.78879 | − | 1.94942i | 0.707107 | + | 0.707107i | −1.00000 | 2.38962 | ||||||
115.1 | −0.965926 | + | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | −2.19856 | − | 0.589103i | 0.258819 | + | 0.965926i | 1.83682 | + | 1.90423i | −0.707107 | + | 0.707107i | −1.00000 | 2.27612 | |||||
115.2 | −0.965926 | + | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | −1.43558 | − | 0.384662i | 0.258819 | + | 0.965926i | 1.74877 | − | 1.98539i | −0.707107 | + | 0.707107i | −1.00000 | 1.48622 | |||||
115.3 | −0.965926 | + | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | −0.106603 | − | 0.0285643i | 0.258819 | + | 0.965926i | −1.53649 | − | 2.15388i | −0.707107 | + | 0.707107i | −1.00000 | 0.110364 | |||||
115.4 | −0.965926 | + | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | 3.74075 | + | 1.00233i | 0.258819 | + | 0.965926i | −2.20451 | − | 1.46293i | −0.707107 | + | 0.707107i | −1.00000 | −3.87271 | |||||
115.5 | 0.965926 | − | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | −2.85793 | − | 0.765779i | −0.258819 | − | 0.965926i | 0.565013 | − | 2.58472i | 0.707107 | − | 0.707107i | −1.00000 | −2.95874 | |||||
115.6 | 0.965926 | − | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | 0.128673 | + | 0.0344778i | −0.258819 | − | 0.965926i | −2.64307 | − | 0.119073i | 0.707107 | − | 0.707107i | −1.00000 | 0.133212 | |||||
115.7 | 0.965926 | − | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | 0.421063 | + | 0.112824i | −0.258819 | − | 0.965926i | 2.44466 | − | 1.01175i | 0.707107 | − | 0.707107i | −1.00000 | 0.435917 | |||||
115.8 | 0.965926 | − | 0.258819i | − | 1.00000i | 0.866025 | − | 0.500000i | 2.30819 | + | 0.618478i | −0.258819 | − | 0.965926i | 1.78879 | + | 1.94942i | 0.707107 | − | 0.707107i | −1.00000 | 2.38962 | |||||
397.1 | −0.258819 | + | 0.965926i | − | 1.00000i | −0.866025 | − | 0.500000i | −0.995376 | − | 3.71479i | 0.965926 | + | 0.258819i | 2.54668 | − | 0.717240i | 0.707107 | − | 0.707107i | −1.00000 | 3.84584 | |||||
397.2 | −0.258819 | + | 0.965926i | − | 1.00000i | −0.866025 | − | 0.500000i | −0.166844 | − | 0.622671i | 0.965926 | + | 0.258819i | −2.38407 | + | 1.14726i | 0.707107 | − | 0.707107i | −1.00000 | 0.644637 | |||||
397.3 | −0.258819 | + | 0.965926i | − | 1.00000i | −0.866025 | − | 0.500000i | 0.379698 | + | 1.41705i | 0.965926 | + | 0.258819i | 1.63697 | − | 2.07854i | 0.707107 | − | 0.707107i | −1.00000 | −1.46704 | |||||
397.4 | −0.258819 | + | 0.965926i | − | 1.00000i | −0.866025 | − | 0.500000i | 0.782522 | + | 2.92041i | 0.965926 | + | 0.258819i | 1.58056 | + | 2.12175i | 0.707107 | − | 0.707107i | −1.00000 | −3.02343 | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.w | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.by.a | ✓ | 32 |
7.d | odd | 6 | 1 | 546.2.cg.a | yes | 32 | |
13.f | odd | 12 | 1 | 546.2.cg.a | yes | 32 | |
91.w | even | 12 | 1 | inner | 546.2.by.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.by.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
546.2.by.a | ✓ | 32 | 91.w | even | 12 | 1 | inner |
546.2.cg.a | yes | 32 | 7.d | odd | 6 | 1 | |
546.2.cg.a | yes | 32 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 12 T_{5}^{30} - 36 T_{5}^{29} - 156 T_{5}^{28} - 124 T_{5}^{27} - 1800 T_{5}^{26} + \cdots + 9409 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).