Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(71,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.71");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.cw (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: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
71.1 | −2.63139 | + | 0.705080i | 0 | 4.69504 | − | 2.71068i | 0.707107 | + | 0.707107i | 0 | 0.0146491 | − | 0.0546712i | −6.59062 | + | 6.59062i | 0 | −2.35924 | − | 1.36211i | ||||||
71.2 | −2.51328 | + | 0.673430i | 0 | 4.13100 | − | 2.38503i | −0.707107 | − | 0.707107i | 0 | 0.0422882 | − | 0.157822i | −5.09649 | + | 5.09649i | 0 | 2.25334 | + | 1.30097i | ||||||
71.3 | −1.45112 | + | 0.388827i | 0 | 0.222519 | − | 0.128471i | −0.707107 | − | 0.707107i | 0 | −0.546011 | + | 2.03774i | 1.85164 | − | 1.85164i | 0 | 1.30104 | + | 0.751156i | ||||||
71.4 | −0.838600 | + | 0.224702i | 0 | −1.07929 | + | 0.623130i | 0.707107 | + | 0.707107i | 0 | 0.215420 | − | 0.803959i | 1.99287 | − | 1.99287i | 0 | −0.751868 | − | 0.434091i | ||||||
71.5 | −0.494406 | + | 0.132476i | 0 | −1.50516 | + | 0.869006i | 0.707107 | + | 0.707107i | 0 | −0.592372 | + | 2.21076i | 1.35290 | − | 1.35290i | 0 | −0.443272 | − | 0.255923i | ||||||
71.6 | 0.494406 | − | 0.132476i | 0 | −1.50516 | + | 0.869006i | −0.707107 | − | 0.707107i | 0 | −0.592372 | + | 2.21076i | −1.35290 | + | 1.35290i | 0 | −0.443272 | − | 0.255923i | ||||||
71.7 | 0.838600 | − | 0.224702i | 0 | −1.07929 | + | 0.623130i | −0.707107 | − | 0.707107i | 0 | 0.215420 | − | 0.803959i | −1.99287 | + | 1.99287i | 0 | −0.751868 | − | 0.434091i | ||||||
71.8 | 1.45112 | − | 0.388827i | 0 | 0.222519 | − | 0.128471i | 0.707107 | + | 0.707107i | 0 | −0.546011 | + | 2.03774i | −1.85164 | + | 1.85164i | 0 | 1.30104 | + | 0.751156i | ||||||
71.9 | 2.51328 | − | 0.673430i | 0 | 4.13100 | − | 2.38503i | 0.707107 | + | 0.707107i | 0 | 0.0422882 | − | 0.157822i | 5.09649 | − | 5.09649i | 0 | 2.25334 | + | 1.30097i | ||||||
71.10 | 2.63139 | − | 0.705080i | 0 | 4.69504 | − | 2.71068i | −0.707107 | − | 0.707107i | 0 | 0.0146491 | − | 0.0546712i | 6.59062 | − | 6.59062i | 0 | −2.35924 | − | 1.36211i | ||||||
206.1 | −2.63139 | − | 0.705080i | 0 | 4.69504 | + | 2.71068i | 0.707107 | − | 0.707107i | 0 | 0.0146491 | + | 0.0546712i | −6.59062 | − | 6.59062i | 0 | −2.35924 | + | 1.36211i | ||||||
206.2 | −2.51328 | − | 0.673430i | 0 | 4.13100 | + | 2.38503i | −0.707107 | + | 0.707107i | 0 | 0.0422882 | + | 0.157822i | −5.09649 | − | 5.09649i | 0 | 2.25334 | − | 1.30097i | ||||||
206.3 | −1.45112 | − | 0.388827i | 0 | 0.222519 | + | 0.128471i | −0.707107 | + | 0.707107i | 0 | −0.546011 | − | 2.03774i | 1.85164 | + | 1.85164i | 0 | 1.30104 | − | 0.751156i | ||||||
206.4 | −0.838600 | − | 0.224702i | 0 | −1.07929 | − | 0.623130i | 0.707107 | − | 0.707107i | 0 | 0.215420 | + | 0.803959i | 1.99287 | + | 1.99287i | 0 | −0.751868 | + | 0.434091i | ||||||
206.5 | −0.494406 | − | 0.132476i | 0 | −1.50516 | − | 0.869006i | 0.707107 | − | 0.707107i | 0 | −0.592372 | − | 2.21076i | 1.35290 | + | 1.35290i | 0 | −0.443272 | + | 0.255923i | ||||||
206.6 | 0.494406 | + | 0.132476i | 0 | −1.50516 | − | 0.869006i | −0.707107 | + | 0.707107i | 0 | −0.592372 | − | 2.21076i | −1.35290 | − | 1.35290i | 0 | −0.443272 | + | 0.255923i | ||||||
206.7 | 0.838600 | + | 0.224702i | 0 | −1.07929 | − | 0.623130i | −0.707107 | + | 0.707107i | 0 | 0.215420 | + | 0.803959i | −1.99287 | − | 1.99287i | 0 | −0.751868 | + | 0.434091i | ||||||
206.8 | 1.45112 | + | 0.388827i | 0 | 0.222519 | + | 0.128471i | 0.707107 | − | 0.707107i | 0 | −0.546011 | − | 2.03774i | −1.85164 | − | 1.85164i | 0 | 1.30104 | − | 0.751156i | ||||||
206.9 | 2.51328 | + | 0.673430i | 0 | 4.13100 | + | 2.38503i | 0.707107 | − | 0.707107i | 0 | 0.0422882 | + | 0.157822i | 5.09649 | + | 5.09649i | 0 | 2.25334 | − | 1.30097i | ||||||
206.10 | 2.63139 | + | 0.705080i | 0 | 4.69504 | + | 2.71068i | −0.707107 | + | 0.707107i | 0 | 0.0146491 | + | 0.0546712i | 6.59062 | + | 6.59062i | 0 | −2.35924 | + | 1.36211i | ||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
39.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.cw.b | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 585.2.cw.b | ✓ | 40 |
13.f | odd | 12 | 1 | inner | 585.2.cw.b | ✓ | 40 |
39.k | even | 12 | 1 | inner | 585.2.cw.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.cw.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
585.2.cw.b | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
585.2.cw.b | ✓ | 40 | 13.f | odd | 12 | 1 | inner |
585.2.cw.b | ✓ | 40 | 39.k | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 12 T_{2}^{38} + 576 T_{2}^{34} - 284 T_{2}^{32} - 17916 T_{2}^{30} + 15552 T_{2}^{28} + \cdots + 1296 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).