Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [666,2,Mod(17,666)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(666, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([18, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("666.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 666.bs (of order \(36\), degree \(12\), 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: | \(5.31803677462\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −0.573576 | + | 0.819152i | 0 | −0.342020 | − | 0.939693i | −0.980582 | + | 0.0857898i | 0 | 1.16963 | + | 0.981439i | 0.965926 | + | 0.258819i | 0 | 0.492164 | − | 0.852453i | ||||||
17.2 | −0.573576 | + | 0.819152i | 0 | −0.342020 | − | 0.939693i | −0.718088 | + | 0.0628246i | 0 | −2.07081 | − | 1.73762i | 0.965926 | + | 0.258819i | 0 | 0.360415 | − | 0.624258i | ||||||
17.3 | −0.573576 | + | 0.819152i | 0 | −0.342020 | − | 0.939693i | 3.66079 | − | 0.320278i | 0 | −1.19170 | − | 0.999952i | 0.965926 | + | 0.258819i | 0 | −1.83739 | + | 3.18245i | ||||||
17.4 | 0.573576 | − | 0.819152i | 0 | −0.342020 | − | 0.939693i | −3.66079 | + | 0.320278i | 0 | −1.19170 | − | 0.999952i | −0.965926 | − | 0.258819i | 0 | −1.83739 | + | 3.18245i | ||||||
17.5 | 0.573576 | − | 0.819152i | 0 | −0.342020 | − | 0.939693i | 0.718088 | − | 0.0628246i | 0 | −2.07081 | − | 1.73762i | −0.965926 | − | 0.258819i | 0 | 0.360415 | − | 0.624258i | ||||||
17.6 | 0.573576 | − | 0.819152i | 0 | −0.342020 | − | 0.939693i | 0.980582 | − | 0.0857898i | 0 | 1.16963 | + | 0.981439i | −0.965926 | − | 0.258819i | 0 | 0.492164 | − | 0.852453i | ||||||
35.1 | −0.996195 | − | 0.0871557i | 0 | 0.984808 | + | 0.173648i | −0.753592 | + | 1.61608i | 0 | 4.63847 | − | 1.68826i | −0.965926 | − | 0.258819i | 0 | 0.891575 | − | 1.54425i | ||||||
35.2 | −0.996195 | − | 0.0871557i | 0 | 0.984808 | + | 0.173648i | −0.567891 | + | 1.21785i | 0 | −0.300852 | + | 0.109501i | −0.965926 | − | 0.258819i | 0 | 0.671872 | − | 1.16372i | ||||||
35.3 | −0.996195 | − | 0.0871557i | 0 | 0.984808 | + | 0.173648i | 0.778175 | − | 1.66880i | 0 | −1.77033 | + | 0.644346i | −0.965926 | − | 0.258819i | 0 | −0.920660 | + | 1.59463i | ||||||
35.4 | 0.996195 | + | 0.0871557i | 0 | 0.984808 | + | 0.173648i | −0.778175 | + | 1.66880i | 0 | −1.77033 | + | 0.644346i | 0.965926 | + | 0.258819i | 0 | −0.920660 | + | 1.59463i | ||||||
35.5 | 0.996195 | + | 0.0871557i | 0 | 0.984808 | + | 0.173648i | 0.567891 | − | 1.21785i | 0 | −0.300852 | + | 0.109501i | 0.965926 | + | 0.258819i | 0 | 0.671872 | − | 1.16372i | ||||||
35.6 | 0.996195 | + | 0.0871557i | 0 | 0.984808 | + | 0.173648i | 0.753592 | − | 1.61608i | 0 | 4.63847 | − | 1.68826i | 0.965926 | + | 0.258819i | 0 | 0.891575 | − | 1.54425i | ||||||
89.1 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.27330 | + | 1.59178i | 0 | 0.235550 | − | 1.33587i | −0.258819 | + | 0.965926i | 0 | 1.38759 | − | 2.40338i | ||||||
89.2 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 0.733836 | − | 0.513837i | 0 | −0.650195 | + | 3.68744i | −0.258819 | + | 0.965926i | 0 | −0.447924 | + | 0.775827i | ||||||
89.3 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 2.09980 | − | 1.47030i | 0 | 0.541765 | − | 3.07250i | −0.258819 | + | 0.965926i | 0 | −1.28169 | + | 2.21995i | ||||||
89.4 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.09980 | + | 1.47030i | 0 | 0.541765 | − | 3.07250i | 0.258819 | − | 0.965926i | 0 | −1.28169 | + | 2.21995i | ||||||
89.5 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | −0.733836 | + | 0.513837i | 0 | −0.650195 | + | 3.68744i | 0.258819 | − | 0.965926i | 0 | −0.447924 | + | 0.775827i | ||||||
89.6 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | 2.27330 | − | 1.59178i | 0 | 0.235550 | − | 1.33587i | 0.258819 | − | 0.965926i | 0 | 1.38759 | − | 2.40338i | ||||||
143.1 | −0.422618 | + | 0.906308i | 0 | −0.642788 | − | 0.766044i | −2.39204 | + | 3.41619i | 0 | −0.629322 | − | 3.56906i | 0.965926 | − | 0.258819i | 0 | −2.08520 | − | 3.61167i | ||||||
143.2 | −0.422618 | + | 0.906308i | 0 | −0.642788 | − | 0.766044i | 0.411306 | − | 0.587405i | 0 | 0.509688 | + | 2.89058i | 0.965926 | − | 0.258819i | 0 | 0.358545 | + | 0.621018i | ||||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
37.i | odd | 36 | 1 | inner |
111.q | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 666.2.bs.a | ✓ | 72 |
3.b | odd | 2 | 1 | inner | 666.2.bs.a | ✓ | 72 |
37.i | odd | 36 | 1 | inner | 666.2.bs.a | ✓ | 72 |
111.q | even | 36 | 1 | inner | 666.2.bs.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
666.2.bs.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
666.2.bs.a | ✓ | 72 | 3.b | odd | 2 | 1 | inner |
666.2.bs.a | ✓ | 72 | 37.i | odd | 36 | 1 | inner |
666.2.bs.a | ✓ | 72 | 111.q | even | 36 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{72} + 327 T_{5}^{68} - 6192 T_{5}^{66} + 28017 T_{5}^{64} - 932364 T_{5}^{62} + 6643081 T_{5}^{60} + 6446736 T_{5}^{58} - 698278833 T_{5}^{56} + 10632994704 T_{5}^{54} + \cdots + 570386655107361 \)
acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\).