Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1716,3,Mod(1429,1716)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1716.1429");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1716.g (of order \(2\), degree \(1\), 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: | \(46.7576133642\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1429.1 | 0 | −1.73205 | 0 | 8.59197i | 0 | 10.7767 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.2 | 0 | −1.73205 | 0 | − | 8.59197i | 0 | −10.7767 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.3 | 0 | −1.73205 | 0 | − | 4.88103i | 0 | −11.6125 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.4 | 0 | −1.73205 | 0 | 4.88103i | 0 | 11.6125 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.5 | 0 | −1.73205 | 0 | 2.83041i | 0 | −10.0694 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.6 | 0 | −1.73205 | 0 | − | 2.83041i | 0 | 10.0694 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.7 | 0 | −1.73205 | 0 | 8.88743i | 0 | 1.96517 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.8 | 0 | −1.73205 | 0 | − | 8.88743i | 0 | −1.96517 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.9 | 0 | −1.73205 | 0 | 2.19640i | 0 | −6.40882 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.10 | 0 | −1.73205 | 0 | − | 2.19640i | 0 | 6.40882 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.11 | 0 | −1.73205 | 0 | − | 3.60524i | 0 | −4.41724 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.12 | 0 | −1.73205 | 0 | 3.60524i | 0 | 4.41724 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.13 | 0 | −1.73205 | 0 | 4.63273i | 0 | −1.27920 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.14 | 0 | −1.73205 | 0 | − | 4.63273i | 0 | 1.27920 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.15 | 0 | −1.73205 | 0 | − | 4.63273i | 0 | −1.27920 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.16 | 0 | −1.73205 | 0 | 4.63273i | 0 | 1.27920 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.17 | 0 | −1.73205 | 0 | 3.60524i | 0 | −4.41724 | 0 | 3.00000 | 0 | ||||||||||||||||||
1429.18 | 0 | −1.73205 | 0 | − | 3.60524i | 0 | 4.41724 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.19 | 0 | −1.73205 | 0 | − | 2.19640i | 0 | −6.40882 | 0 | 3.00000 | 0 | |||||||||||||||||
1429.20 | 0 | −1.73205 | 0 | 2.19640i | 0 | 6.40882 | 0 | 3.00000 | 0 | ||||||||||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
143.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1716.3.g.a | ✓ | 56 |
11.b | odd | 2 | 1 | inner | 1716.3.g.a | ✓ | 56 |
13.b | even | 2 | 1 | inner | 1716.3.g.a | ✓ | 56 |
143.d | odd | 2 | 1 | inner | 1716.3.g.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1716.3.g.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
1716.3.g.a | ✓ | 56 | 11.b | odd | 2 | 1 | inner |
1716.3.g.a | ✓ | 56 | 13.b | even | 2 | 1 | inner |
1716.3.g.a | ✓ | 56 | 143.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1716, [\chi])\).