Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(67,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([8, 12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.v (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: | \(3.01834519640\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −0.766044 | + | 0.642788i | −1.71753 | − | 0.223844i | 0.173648 | − | 0.984808i | 0.0598945 | − | 0.339678i | 1.45959 | − | 0.932530i | −1.66417 | + | 2.05683i | 0.500000 | + | 0.866025i | 2.89979 | + | 0.768916i | 0.172459 | + | 0.298708i |
67.2 | −0.766044 | + | 0.642788i | −1.42403 | − | 0.985977i | 0.173648 | − | 0.984808i | −0.327106 | + | 1.85511i | 1.72464 | − | 0.160044i | −1.01788 | − | 2.44211i | 0.500000 | + | 0.866025i | 1.05570 | + | 2.80811i | −0.941864 | − | 1.63136i |
67.3 | −0.766044 | + | 0.642788i | −1.38102 | + | 1.04536i | 0.173648 | − | 0.984808i | −0.716530 | + | 4.06364i | 0.385982 | − | 1.68850i | 2.64332 | + | 0.113332i | 0.500000 | + | 0.866025i | 0.814451 | − | 2.88733i | −2.06317 | − | 3.57351i |
67.4 | −0.766044 | + | 0.642788i | −0.921674 | − | 1.46646i | 0.173648 | − | 0.984808i | 0.622926 | − | 3.53279i | 1.64867 | + | 0.530936i | 2.63270 | + | 0.262455i | 0.500000 | + | 0.866025i | −1.30103 | + | 2.70320i | 1.79365 | + | 3.10668i |
67.5 | −0.766044 | + | 0.642788i | −0.829129 | + | 1.52071i | 0.173648 | − | 0.984808i | 0.684323 | − | 3.88099i | −0.342341 | − | 1.69788i | −1.85424 | + | 1.88727i | 0.500000 | + | 0.866025i | −1.62509 | − | 2.52172i | 1.97043 | + | 3.41288i |
67.6 | −0.766044 | + | 0.642788i | −0.402311 | + | 1.68468i | 0.173648 | − | 0.984808i | 0.256458 | − | 1.45445i | −0.774703 | − | 1.54914i | 2.04878 | − | 1.67407i | 0.500000 | + | 0.866025i | −2.67629 | − | 1.35553i | 0.738441 | + | 1.27902i |
67.7 | −0.766044 | + | 0.642788i | 0.0412776 | − | 1.73156i | 0.173648 | − | 0.984808i | −0.194594 | + | 1.10360i | 1.08140 | + | 1.35298i | −0.0465876 | + | 2.64534i | 0.500000 | + | 0.866025i | −2.99659 | − | 0.142949i | −0.560311 | − | 0.970488i |
67.8 | −0.766044 | + | 0.642788i | 0.309985 | + | 1.70409i | 0.173648 | − | 0.984808i | −0.504047 | + | 2.85859i | −1.33283 | − | 1.10615i | −2.58705 | + | 0.554250i | 0.500000 | + | 0.866025i | −2.80782 | + | 1.05648i | −1.45134 | − | 2.51380i |
67.9 | −0.766044 | + | 0.642788i | 0.600510 | − | 1.62462i | 0.173648 | − | 0.984808i | 0.421605 | − | 2.39104i | 0.584268 | + | 1.63053i | −1.84906 | − | 1.89235i | 0.500000 | + | 0.866025i | −2.27878 | − | 1.95120i | 1.21396 | + | 2.10265i |
67.10 | −0.766044 | + | 0.642788i | 1.54135 | − | 0.790081i | 0.173648 | − | 0.984808i | −0.272791 | + | 1.54708i | −0.672891 | + | 1.59600i | 2.24230 | − | 1.40432i | 0.500000 | + | 0.866025i | 1.75154 | − | 2.43559i | −0.785472 | − | 1.36048i |
67.11 | −0.766044 | + | 0.642788i | 1.59651 | + | 0.671670i | 0.173648 | − | 0.984808i | −0.153204 | + | 0.868862i | −1.65474 | + | 0.511691i | 0.258184 | + | 2.63312i | 0.500000 | + | 0.866025i | 2.09772 | + | 2.14466i | −0.441133 | − | 0.764065i |
67.12 | −0.766044 | + | 0.642788i | 1.64636 | + | 0.538065i | 0.173648 | − | 0.984808i | 0.470361 | − | 2.66755i | −1.60704 | + | 0.646075i | −1.97994 | − | 1.75495i | 0.500000 | + | 0.866025i | 2.42097 | + | 1.77169i | 1.35435 | + | 2.34580i |
79.1 | −0.766044 | − | 0.642788i | −1.71753 | + | 0.223844i | 0.173648 | + | 0.984808i | 0.0598945 | + | 0.339678i | 1.45959 | + | 0.932530i | −1.66417 | − | 2.05683i | 0.500000 | − | 0.866025i | 2.89979 | − | 0.768916i | 0.172459 | − | 0.298708i |
79.2 | −0.766044 | − | 0.642788i | −1.42403 | + | 0.985977i | 0.173648 | + | 0.984808i | −0.327106 | − | 1.85511i | 1.72464 | + | 0.160044i | −1.01788 | + | 2.44211i | 0.500000 | − | 0.866025i | 1.05570 | − | 2.80811i | −0.941864 | + | 1.63136i |
79.3 | −0.766044 | − | 0.642788i | −1.38102 | − | 1.04536i | 0.173648 | + | 0.984808i | −0.716530 | − | 4.06364i | 0.385982 | + | 1.68850i | 2.64332 | − | 0.113332i | 0.500000 | − | 0.866025i | 0.814451 | + | 2.88733i | −2.06317 | + | 3.57351i |
79.4 | −0.766044 | − | 0.642788i | −0.921674 | + | 1.46646i | 0.173648 | + | 0.984808i | 0.622926 | + | 3.53279i | 1.64867 | − | 0.530936i | 2.63270 | − | 0.262455i | 0.500000 | − | 0.866025i | −1.30103 | − | 2.70320i | 1.79365 | − | 3.10668i |
79.5 | −0.766044 | − | 0.642788i | −0.829129 | − | 1.52071i | 0.173648 | + | 0.984808i | 0.684323 | + | 3.88099i | −0.342341 | + | 1.69788i | −1.85424 | − | 1.88727i | 0.500000 | − | 0.866025i | −1.62509 | + | 2.52172i | 1.97043 | − | 3.41288i |
79.6 | −0.766044 | − | 0.642788i | −0.402311 | − | 1.68468i | 0.173648 | + | 0.984808i | 0.256458 | + | 1.45445i | −0.774703 | + | 1.54914i | 2.04878 | + | 1.67407i | 0.500000 | − | 0.866025i | −2.67629 | + | 1.35553i | 0.738441 | − | 1.27902i |
79.7 | −0.766044 | − | 0.642788i | 0.0412776 | + | 1.73156i | 0.173648 | + | 0.984808i | −0.194594 | − | 1.10360i | 1.08140 | − | 1.35298i | −0.0465876 | − | 2.64534i | 0.500000 | − | 0.866025i | −2.99659 | + | 0.142949i | −0.560311 | + | 0.970488i |
79.8 | −0.766044 | − | 0.642788i | 0.309985 | − | 1.70409i | 0.173648 | + | 0.984808i | −0.504047 | − | 2.85859i | −1.33283 | + | 1.10615i | −2.58705 | − | 0.554250i | 0.500000 | − | 0.866025i | −2.80782 | − | 1.05648i | −1.45134 | + | 2.51380i |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
189.u | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.v.a | ✓ | 72 |
7.c | even | 3 | 1 | 378.2.w.b | yes | 72 | |
27.e | even | 9 | 1 | 378.2.w.b | yes | 72 | |
189.u | even | 9 | 1 | inner | 378.2.v.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.v.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
378.2.v.a | ✓ | 72 | 189.u | even | 9 | 1 | inner |
378.2.w.b | yes | 72 | 7.c | even | 3 | 1 | |
378.2.w.b | yes | 72 | 27.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} + 3 T_{5}^{70} + 41 T_{5}^{69} - 105 T_{5}^{68} + 207 T_{5}^{67} + 6532 T_{5}^{66} + \cdots + 2678994081 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).