Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1716,3,Mod(265,1716)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1716, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1716.265");
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.s (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: | \(46.7576133642\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
265.1 | 0 | −1.73205 | 0 | 6.46670 | − | 6.46670i | 0 | 3.09717 | + | 3.09717i | 0 | 3.00000 | 0 | ||||||||||||||
265.2 | 0 | −1.73205 | 0 | −5.88203 | + | 5.88203i | 0 | −8.06727 | − | 8.06727i | 0 | 3.00000 | 0 | ||||||||||||||
265.3 | 0 | −1.73205 | 0 | 5.68026 | − | 5.68026i | 0 | 5.37559 | + | 5.37559i | 0 | 3.00000 | 0 | ||||||||||||||
265.4 | 0 | −1.73205 | 0 | −5.61386 | + | 5.61386i | 0 | 5.25643 | + | 5.25643i | 0 | 3.00000 | 0 | ||||||||||||||
265.5 | 0 | −1.73205 | 0 | −4.83880 | + | 4.83880i | 0 | 7.59021 | + | 7.59021i | 0 | 3.00000 | 0 | ||||||||||||||
265.6 | 0 | −1.73205 | 0 | −3.89541 | + | 3.89541i | 0 | 3.62314 | + | 3.62314i | 0 | 3.00000 | 0 | ||||||||||||||
265.7 | 0 | −1.73205 | 0 | −3.58941 | + | 3.58941i | 0 | −6.11147 | − | 6.11147i | 0 | 3.00000 | 0 | ||||||||||||||
265.8 | 0 | −1.73205 | 0 | −3.20295 | + | 3.20295i | 0 | −4.99272 | − | 4.99272i | 0 | 3.00000 | 0 | ||||||||||||||
265.9 | 0 | −1.73205 | 0 | 2.39752 | − | 2.39752i | 0 | 7.69422 | + | 7.69422i | 0 | 3.00000 | 0 | ||||||||||||||
265.10 | 0 | −1.73205 | 0 | −2.83535 | + | 2.83535i | 0 | 0.860153 | + | 0.860153i | 0 | 3.00000 | 0 | ||||||||||||||
265.11 | 0 | −1.73205 | 0 | 2.13744 | − | 2.13744i | 0 | 5.97512 | + | 5.97512i | 0 | 3.00000 | 0 | ||||||||||||||
265.12 | 0 | −1.73205 | 0 | 2.04087 | − | 2.04087i | 0 | 2.75549 | + | 2.75549i | 0 | 3.00000 | 0 | ||||||||||||||
265.13 | 0 | −1.73205 | 0 | −1.78379 | + | 1.78379i | 0 | −1.82078 | − | 1.82078i | 0 | 3.00000 | 0 | ||||||||||||||
265.14 | 0 | −1.73205 | 0 | −0.614927 | + | 0.614927i | 0 | 8.15993 | + | 8.15993i | 0 | 3.00000 | 0 | ||||||||||||||
265.15 | 0 | −1.73205 | 0 | −0.421357 | + | 0.421357i | 0 | −0.916376 | − | 0.916376i | 0 | 3.00000 | 0 | ||||||||||||||
265.16 | 0 | −1.73205 | 0 | −0.240294 | + | 0.240294i | 0 | 3.09073 | + | 3.09073i | 0 | 3.00000 | 0 | ||||||||||||||
265.17 | 0 | −1.73205 | 0 | −1.19533 | + | 1.19533i | 0 | −4.90121 | − | 4.90121i | 0 | 3.00000 | 0 | ||||||||||||||
265.18 | 0 | −1.73205 | 0 | 2.74539 | − | 2.74539i | 0 | −8.91154 | − | 8.91154i | 0 | 3.00000 | 0 | ||||||||||||||
265.19 | 0 | −1.73205 | 0 | 2.97197 | − | 2.97197i | 0 | −5.14635 | − | 5.14635i | 0 | 3.00000 | 0 | ||||||||||||||
265.20 | 0 | −1.73205 | 0 | 4.44458 | − | 4.44458i | 0 | −5.61255 | − | 5.61255i | 0 | 3.00000 | 0 | ||||||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.d | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1716.3.s.a | ✓ | 96 |
13.d | odd | 4 | 1 | inner | 1716.3.s.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1716.3.s.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
1716.3.s.a | ✓ | 96 | 13.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1716, [\chi])\).