Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(41,378)]
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([17, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.z (of order \(18\), 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: | \(144\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | −0.984808 | + | 0.173648i | −1.71625 | + | 0.233448i | 0.939693 | − | 0.342020i | −0.0848896 | − | 0.0712308i | 1.64964 | − | 0.527925i | −1.74235 | − | 1.99104i | −0.866025 | + | 0.500000i | 2.89100 | − | 0.801309i | 0.0959690 | + | 0.0554077i |
41.2 | −0.984808 | + | 0.173648i | −1.56441 | − | 0.743380i | 0.939693 | − | 0.342020i | −0.941136 | − | 0.789707i | 1.66973 | + | 0.460429i | 2.62387 | + | 0.339554i | −0.866025 | + | 0.500000i | 1.89477 | + | 2.32590i | 1.06397 | + | 0.614283i |
41.3 | −0.984808 | + | 0.173648i | −1.34465 | + | 1.09174i | 0.939693 | − | 0.342020i | −2.60134 | − | 2.18278i | 1.13465 | − | 1.30865i | −0.615646 | + | 2.57313i | −0.866025 | + | 0.500000i | 0.616190 | − | 2.93604i | 2.94085 | + | 1.69790i |
41.4 | −0.984808 | + | 0.173648i | −0.826027 | − | 1.52239i | 0.939693 | − | 0.342020i | 1.99843 | + | 1.67688i | 1.07784 | + | 1.35583i | 0.187552 | + | 2.63910i | −0.866025 | + | 0.500000i | −1.63536 | + | 2.51507i | −2.25926 | − | 1.30438i |
41.5 | −0.984808 | + | 0.173648i | −0.754516 | + | 1.55907i | 0.939693 | − | 0.342020i | 0.486827 | + | 0.408497i | 0.472323 | − | 1.66641i | −0.719601 | − | 2.54601i | −0.866025 | + | 0.500000i | −1.86141 | − | 2.35269i | −0.550366 | − | 0.317754i |
41.6 | −0.984808 | + | 0.173648i | −0.0283469 | − | 1.73182i | 0.939693 | − | 0.342020i | −2.71537 | − | 2.27847i | 0.328643 | + | 1.70059i | 0.173696 | − | 2.64004i | −0.866025 | + | 0.500000i | −2.99839 | + | 0.0981834i | 3.06977 | + | 1.77233i |
41.7 | −0.984808 | + | 0.173648i | 0.0283469 | + | 1.73182i | 0.939693 | − | 0.342020i | 2.71537 | + | 2.27847i | −0.328643 | − | 1.70059i | −1.56393 | + | 2.13404i | −0.866025 | + | 0.500000i | −2.99839 | + | 0.0981834i | −3.06977 | − | 1.77233i |
41.8 | −0.984808 | + | 0.173648i | 0.754516 | − | 1.55907i | 0.939693 | − | 0.342020i | −0.486827 | − | 0.408497i | −0.472323 | + | 1.66641i | −2.18779 | + | 1.48781i | −0.866025 | + | 0.500000i | −1.86141 | − | 2.35269i | 0.550366 | + | 0.317754i |
41.9 | −0.984808 | + | 0.173648i | 0.826027 | + | 1.52239i | 0.939693 | − | 0.342020i | −1.99843 | − | 1.67688i | −1.07784 | − | 1.35583i | 1.84005 | − | 1.90111i | −0.866025 | + | 0.500000i | −1.63536 | + | 2.51507i | 2.25926 | + | 1.30438i |
41.10 | −0.984808 | + | 0.173648i | 1.34465 | − | 1.09174i | 0.939693 | − | 0.342020i | 2.60134 | + | 2.18278i | −1.13465 | + | 1.30865i | 1.18236 | − | 2.36686i | −0.866025 | + | 0.500000i | 0.616190 | − | 2.93604i | −2.94085 | − | 1.69790i |
41.11 | −0.984808 | + | 0.173648i | 1.56441 | + | 0.743380i | 0.939693 | − | 0.342020i | 0.941136 | + | 0.789707i | −1.66973 | − | 0.460429i | 2.22826 | + | 1.42648i | −0.866025 | + | 0.500000i | 1.89477 | + | 2.32590i | −1.06397 | − | 0.614283i |
41.12 | −0.984808 | + | 0.173648i | 1.71625 | − | 0.233448i | 0.939693 | − | 0.342020i | 0.0848896 | + | 0.0712308i | −1.64964 | + | 0.527925i | −2.61453 | + | 0.405264i | −0.866025 | + | 0.500000i | 2.89100 | − | 0.801309i | −0.0959690 | − | 0.0554077i |
41.13 | 0.984808 | − | 0.173648i | −1.64560 | − | 0.540384i | 0.939693 | − | 0.342020i | 0.881233 | + | 0.739442i | −1.71443 | − | 0.246420i | −0.336437 | + | 2.62427i | 0.866025 | − | 0.500000i | 2.41597 | + | 1.77851i | 0.996248 | + | 0.575184i |
41.14 | 0.984808 | − | 0.173648i | −1.47964 | + | 0.900370i | 0.939693 | − | 0.342020i | −1.23395 | − | 1.03541i | −1.30081 | + | 1.14363i | 1.54951 | − | 2.14453i | 0.866025 | − | 0.500000i | 1.37867 | − | 2.66445i | −1.39500 | − | 0.805406i |
41.15 | 0.984808 | − | 0.173648i | −1.41919 | + | 0.992922i | 0.939693 | − | 0.342020i | 3.28093 | + | 2.75302i | −1.22521 | + | 1.22428i | −2.27542 | − | 1.34999i | 0.866025 | − | 0.500000i | 1.02821 | − | 2.81829i | 3.70914 | + | 2.14147i |
41.16 | 0.984808 | − | 0.173648i | −1.33409 | − | 1.10463i | 0.939693 | − | 0.342020i | −1.22837 | − | 1.03073i | −1.50564 | − | 0.856190i | −2.38077 | − | 1.15409i | 0.866025 | − | 0.500000i | 0.559569 | + | 2.94735i | −1.38870 | − | 0.801764i |
41.17 | 0.984808 | − | 0.173648i | −0.419652 | − | 1.68044i | 0.939693 | − | 0.342020i | 1.95067 | + | 1.63681i | −0.705083 | − | 1.58204i | 2.43278 | − | 1.04000i | 0.866025 | − | 0.500000i | −2.64778 | + | 1.41040i | 2.20527 | + | 1.27321i |
41.18 | 0.984808 | − | 0.173648i | −0.0750633 | − | 1.73042i | 0.939693 | − | 0.342020i | −2.14704 | − | 1.80158i | −0.374408 | − | 1.69110i | 2.53928 | + | 0.743006i | 0.866025 | − | 0.500000i | −2.98873 | + | 0.259783i | −2.42726 | − | 1.40138i |
41.19 | 0.984808 | − | 0.173648i | 0.0750633 | + | 1.73042i | 0.939693 | − | 0.342020i | 2.14704 | + | 1.80158i | 0.374408 | + | 1.69110i | 2.42280 | + | 1.06304i | 0.866025 | − | 0.500000i | −2.98873 | + | 0.259783i | 2.42726 | + | 1.40138i |
41.20 | 0.984808 | − | 0.173648i | 0.419652 | + | 1.68044i | 0.939693 | − | 0.342020i | −1.95067 | − | 1.63681i | 0.705083 | + | 1.58204i | 1.19511 | + | 2.36045i | 0.866025 | − | 0.500000i | −2.64778 | + | 1.41040i | −2.20527 | − | 1.27321i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
27.f | odd | 18 | 1 | inner |
189.be | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.z.a | ✓ | 144 |
7.b | odd | 2 | 1 | inner | 378.2.z.a | ✓ | 144 |
27.f | odd | 18 | 1 | inner | 378.2.z.a | ✓ | 144 |
189.be | even | 18 | 1 | inner | 378.2.z.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.z.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
378.2.z.a | ✓ | 144 | 7.b | odd | 2 | 1 | inner |
378.2.z.a | ✓ | 144 | 27.f | odd | 18 | 1 | inner |
378.2.z.a | ✓ | 144 | 189.be | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).