Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [666,2,Mod(127,666)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(666, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("666.127");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 666.x (of order \(9\), degree \(6\), 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: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
127.1 | −0.939693 | − | 0.342020i | 0 | 0.766044 | + | 0.642788i | −0.648124 | + | 3.67569i | 0 | 0.0344122 | − | 0.195161i | −0.500000 | − | 0.866025i | 0 | 1.86620 | − | 3.23235i | ||||||
127.2 | −0.939693 | − | 0.342020i | 0 | 0.766044 | + | 0.642788i | 0.0551662 | − | 0.312863i | 0 | −0.563253 | + | 3.19437i | −0.500000 | − | 0.866025i | 0 | −0.158845 | + | 0.275127i | ||||||
127.3 | −0.939693 | − | 0.342020i | 0 | 0.766044 | + | 0.642788i | 0.255073 | − | 1.44659i | 0 | 0.767063 | − | 4.35023i | −0.500000 | − | 0.866025i | 0 | −0.734452 | + | 1.27211i | ||||||
127.4 | −0.939693 | − | 0.342020i | 0 | 0.766044 | + | 0.642788i | 0.685182 | − | 3.88586i | 0 | −0.238222 | + | 1.35102i | −0.500000 | − | 0.866025i | 0 | −1.97290 | + | 3.41717i | ||||||
145.1 | 0.766044 | + | 0.642788i | 0 | 0.173648 | + | 0.984808i | −3.63447 | − | 1.32284i | 0 | 1.85534 | + | 0.675287i | −0.500000 | + | 0.866025i | 0 | −1.93386 | − | 3.34955i | ||||||
145.2 | 0.766044 | + | 0.642788i | 0 | 0.173648 | + | 0.984808i | −1.56809 | − | 0.570738i | 0 | −3.64428 | − | 1.32641i | −0.500000 | + | 0.866025i | 0 | −0.834363 | − | 1.44516i | ||||||
145.3 | 0.766044 | + | 0.642788i | 0 | 0.173648 | + | 0.984808i | 1.01783 | + | 0.370460i | 0 | −0.173626 | − | 0.0631945i | −0.500000 | + | 0.866025i | 0 | 0.541576 | + | 0.938038i | ||||||
145.4 | 0.766044 | + | 0.642788i | 0 | 0.173648 | + | 0.984808i | 2.30535 | + | 0.839077i | 0 | 1.96257 | + | 0.714317i | −0.500000 | + | 0.866025i | 0 | 1.22665 | + | 2.12462i | ||||||
181.1 | 0.173648 | − | 0.984808i | 0 | −0.939693 | − | 0.342020i | −2.41229 | + | 2.02415i | 0 | 1.53678 | − | 1.28951i | −0.500000 | + | 0.866025i | 0 | 1.57451 | + | 2.72713i | ||||||
181.2 | 0.173648 | − | 0.984808i | 0 | −0.939693 | − | 0.342020i | −0.491477 | + | 0.412398i | 0 | −1.28775 | + | 1.08055i | −0.500000 | + | 0.866025i | 0 | 0.320789 | + | 0.555622i | ||||||
181.3 | 0.173648 | − | 0.984808i | 0 | −0.939693 | − | 0.342020i | 1.96856 | − | 1.65182i | 0 | −3.79061 | + | 3.18070i | −0.500000 | + | 0.866025i | 0 | −1.28488 | − | 2.22549i | ||||||
181.4 | 0.173648 | − | 0.984808i | 0 | −0.939693 | − | 0.342020i | 2.46730 | − | 2.07031i | 0 | 3.54159 | − | 2.97174i | −0.500000 | + | 0.866025i | 0 | −1.61041 | − | 2.78932i | ||||||
271.1 | 0.766044 | − | 0.642788i | 0 | 0.173648 | − | 0.984808i | −3.63447 | + | 1.32284i | 0 | 1.85534 | − | 0.675287i | −0.500000 | − | 0.866025i | 0 | −1.93386 | + | 3.34955i | ||||||
271.2 | 0.766044 | − | 0.642788i | 0 | 0.173648 | − | 0.984808i | −1.56809 | + | 0.570738i | 0 | −3.64428 | + | 1.32641i | −0.500000 | − | 0.866025i | 0 | −0.834363 | + | 1.44516i | ||||||
271.3 | 0.766044 | − | 0.642788i | 0 | 0.173648 | − | 0.984808i | 1.01783 | − | 0.370460i | 0 | −0.173626 | + | 0.0631945i | −0.500000 | − | 0.866025i | 0 | 0.541576 | − | 0.938038i | ||||||
271.4 | 0.766044 | − | 0.642788i | 0 | 0.173648 | − | 0.984808i | 2.30535 | − | 0.839077i | 0 | 1.96257 | − | 0.714317i | −0.500000 | − | 0.866025i | 0 | 1.22665 | − | 2.12462i | ||||||
379.1 | 0.173648 | + | 0.984808i | 0 | −0.939693 | + | 0.342020i | −2.41229 | − | 2.02415i | 0 | 1.53678 | + | 1.28951i | −0.500000 | − | 0.866025i | 0 | 1.57451 | − | 2.72713i | ||||||
379.2 | 0.173648 | + | 0.984808i | 0 | −0.939693 | + | 0.342020i | −0.491477 | − | 0.412398i | 0 | −1.28775 | − | 1.08055i | −0.500000 | − | 0.866025i | 0 | 0.320789 | − | 0.555622i | ||||||
379.3 | 0.173648 | + | 0.984808i | 0 | −0.939693 | + | 0.342020i | 1.96856 | + | 1.65182i | 0 | −3.79061 | − | 3.18070i | −0.500000 | − | 0.866025i | 0 | −1.28488 | + | 2.22549i | ||||||
379.4 | 0.173648 | + | 0.984808i | 0 | −0.939693 | + | 0.342020i | 2.46730 | + | 2.07031i | 0 | 3.54159 | + | 2.97174i | −0.500000 | − | 0.866025i | 0 | −1.61041 | + | 2.78932i | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.f | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 666.2.x.h | ✓ | 24 |
3.b | odd | 2 | 1 | 666.2.x.i | yes | 24 | |
37.f | even | 9 | 1 | inner | 666.2.x.h | ✓ | 24 |
111.p | odd | 18 | 1 | 666.2.x.i | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
666.2.x.h | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
666.2.x.h | ✓ | 24 | 37.f | even | 9 | 1 | inner |
666.2.x.i | yes | 24 | 3.b | odd | 2 | 1 | |
666.2.x.i | yes | 24 | 111.p | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 6 T_{5}^{22} + 26 T_{5}^{21} - 57 T_{5}^{20} + 84 T_{5}^{19} + 2000 T_{5}^{18} + \cdots + 3884841 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\).