Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(47,378)]
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([7, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.ba (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −0.984808 | − | 0.173648i | −1.69591 | + | 0.351999i | 0.939693 | + | 0.342020i | −2.70582 | − | 0.984837i | 1.73127 | − | 0.0521605i | 2.49997 | − | 0.866117i | −0.866025 | − | 0.500000i | 2.75219 | − | 1.19391i | 2.49370 | + | 1.43974i |
47.2 | −0.984808 | − | 0.173648i | −1.65689 | − | 0.504694i | 0.939693 | + | 0.342020i | 1.24189 | + | 0.452012i | 1.54408 | + | 0.784742i | −2.58194 | − | 0.577558i | −0.866025 | − | 0.500000i | 2.49057 | + | 1.67244i | −1.14453 | − | 0.660797i |
47.3 | −0.984808 | − | 0.173648i | −1.45666 | + | 0.937089i | 0.939693 | + | 0.342020i | 3.19574 | + | 1.16316i | 1.59726 | − | 0.669906i | 0.0714494 | + | 2.64479i | −0.866025 | − | 0.500000i | 1.24373 | − | 2.73004i | −2.94521 | − | 1.70042i |
47.4 | −0.984808 | − | 0.173648i | −1.17037 | − | 1.27680i | 0.939693 | + | 0.342020i | 1.68087 | + | 0.611788i | 0.930879 | + | 1.46064i | 1.98134 | − | 1.75337i | −0.866025 | − | 0.500000i | −0.260448 | + | 2.98867i | −1.54910 | − | 0.894374i |
47.5 | −0.984808 | − | 0.173648i | −0.235548 | − | 1.71596i | 0.939693 | + | 0.342020i | −1.87429 | − | 0.682186i | −0.0660042 | + | 1.73079i | −0.317208 | + | 2.62667i | −0.866025 | − | 0.500000i | −2.88903 | + | 0.808380i | 1.72736 | + | 0.997290i |
47.6 | −0.984808 | − | 0.173648i | 0.206907 | + | 1.71965i | 0.939693 | + | 0.342020i | −2.47224 | − | 0.899821i | 0.0948504 | − | 1.72945i | −1.11322 | + | 2.40015i | −0.866025 | − | 0.500000i | −2.91438 | + | 0.711614i | 2.27843 | + | 1.31545i |
47.7 | −0.984808 | − | 0.173648i | 0.363371 | + | 1.69351i | 0.939693 | + | 0.342020i | −0.772614 | − | 0.281208i | −0.0637767 | − | 1.73088i | −1.54265 | − | 2.14948i | −0.866025 | − | 0.500000i | −2.73592 | + | 1.23074i | 0.712045 | + | 0.411099i |
47.8 | −0.984808 | − | 0.173648i | 0.679714 | − | 1.59311i | 0.939693 | + | 0.342020i | 1.56056 | + | 0.567996i | −0.946028 | + | 1.45087i | −2.50129 | − | 0.862294i | −0.866025 | − | 0.500000i | −2.07598 | − | 2.16571i | −1.43822 | − | 0.830355i |
47.9 | −0.984808 | − | 0.173648i | 1.48209 | − | 0.896330i | 0.939693 | + | 0.342020i | −2.03467 | − | 0.740558i | −1.61522 | + | 0.625351i | 2.23500 | + | 1.41591i | −0.866025 | − | 0.500000i | 1.39318 | − | 2.65689i | 1.87516 | + | 1.08262i |
47.10 | −0.984808 | − | 0.173648i | 1.50439 | − | 0.858369i | 0.939693 | + | 0.342020i | 2.90863 | + | 1.05865i | −1.63059 | + | 0.584093i | 2.64574 | − | 0.00779754i | −0.866025 | − | 0.500000i | 1.52640 | − | 2.58265i | −2.68061 | − | 1.54765i |
47.11 | −0.984808 | − | 0.173648i | 1.63664 | + | 0.566931i | 0.939693 | + | 0.342020i | 1.63454 | + | 0.594924i | −1.51333 | − | 0.842517i | −2.03053 | + | 1.69616i | −0.866025 | − | 0.500000i | 2.35718 | + | 1.85572i | −1.50640 | − | 0.869721i |
47.12 | −0.984808 | − | 0.173648i | 1.66909 | + | 0.462748i | 0.939693 | + | 0.342020i | −2.36261 | − | 0.859919i | −1.56338 | − | 0.745552i | 0.311323 | − | 2.62737i | −0.866025 | − | 0.500000i | 2.57173 | + | 1.54474i | 2.17739 | + | 1.25712i |
47.13 | 0.984808 | + | 0.173648i | −1.69863 | − | 0.338627i | 0.939693 | + | 0.342020i | −0.525051 | − | 0.191103i | −1.61402 | − | 0.628446i | −0.0974999 | − | 2.64395i | 0.866025 | + | 0.500000i | 2.77066 | + | 1.15040i | −0.483890 | − | 0.279374i |
47.14 | 0.984808 | + | 0.173648i | −1.63576 | − | 0.569458i | 0.939693 | + | 0.342020i | −3.08742 | − | 1.12373i | −1.51203 | − | 0.844853i | −0.0850968 | + | 2.64438i | 0.866025 | + | 0.500000i | 2.35144 | + | 1.86299i | −2.84538 | − | 1.64278i |
47.15 | 0.984808 | + | 0.173648i | −1.49340 | − | 0.877362i | 0.939693 | + | 0.342020i | 3.97042 | + | 1.44511i | −1.31836 | − | 1.12336i | −1.73420 | + | 1.99814i | 0.866025 | + | 0.500000i | 1.46047 | + | 2.62050i | 3.65916 | + | 2.11262i |
47.16 | 0.984808 | + | 0.173648i | −1.40287 | + | 1.01585i | 0.939693 | + | 0.342020i | 2.68457 | + | 0.977104i | −1.55796 | + | 0.756808i | 1.78067 | − | 1.95684i | 0.866025 | + | 0.500000i | 0.936109 | − | 2.85021i | 2.47411 | + | 1.42843i |
47.17 | 0.984808 | + | 0.173648i | −0.611133 | + | 1.62065i | 0.939693 | + | 0.342020i | −4.17040 | − | 1.51790i | −0.883272 | + | 1.48991i | −1.63543 | − | 2.07976i | 0.866025 | + | 0.500000i | −2.25303 | − | 1.98087i | −3.84346 | − | 2.21902i |
47.18 | 0.984808 | + | 0.173648i | −0.505130 | + | 1.65676i | 0.939693 | + | 0.342020i | −0.342190 | − | 0.124547i | −0.785149 | + | 1.54387i | 2.40698 | + | 1.09838i | 0.866025 | + | 0.500000i | −2.48969 | − | 1.67375i | −0.315364 | − | 0.182076i |
47.19 | 0.984808 | + | 0.173648i | −0.0904059 | − | 1.72969i | 0.939693 | + | 0.342020i | 1.22429 | + | 0.445607i | 0.211325 | − | 1.71911i | 2.52854 | + | 0.778769i | 0.866025 | + | 0.500000i | −2.98365 | + | 0.312748i | 1.12832 | + | 0.651433i |
47.20 | 0.984808 | + | 0.173648i | 0.692095 | + | 1.58777i | 0.939693 | + | 0.342020i | 0.235782 | + | 0.0858176i | 0.405868 | + | 1.68383i | −2.16551 | + | 1.52006i | 0.866025 | + | 0.500000i | −2.04201 | + | 2.19777i | 0.217298 | + | 0.125457i |
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
189.bd | 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.ba.a | ✓ | 144 |
7.d | odd | 6 | 1 | 378.2.bf.a | yes | 144 | |
27.f | odd | 18 | 1 | 378.2.bf.a | yes | 144 | |
189.bd | even | 18 | 1 | inner | 378.2.ba.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.ba.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
378.2.ba.a | ✓ | 144 | 189.bd | even | 18 | 1 | inner |
378.2.bf.a | yes | 144 | 7.d | odd | 6 | 1 | |
378.2.bf.a | yes | 144 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).