Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(5,378)]
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([5, 15]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.bf (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −0.984808 | + | 0.173648i | −1.69494 | − | 0.356630i | 0.939693 | − | 0.342020i | 0.295406 | − | 1.67533i | 1.73112 | + | 0.0568892i | 2.46278 | − | 0.966803i | −0.866025 | + | 0.500000i | 2.74563 | + | 1.20893i | 1.70117i | ||
5.2 | −0.984808 | + | 0.173648i | −1.68531 | + | 0.399650i | 0.939693 | − | 0.342020i | −0.662712 | + | 3.75843i | 1.59031 | − | 0.686230i | −0.925433 | + | 2.47862i | −0.866025 | + | 0.500000i | 2.68056 | − | 1.34707i | − | 3.81641i | |
5.3 | −0.984808 | + | 0.173648i | −1.31364 | − | 1.12887i | 0.939693 | − | 0.342020i | −0.00506474 | + | 0.0287236i | 1.48971 | + | 0.883607i | −2.31258 | + | 1.28530i | −0.866025 | + | 0.500000i | 0.451313 | + | 2.96586i | − | 0.0291667i | |
5.4 | −0.984808 | + | 0.173648i | −0.906167 | + | 1.47610i | 0.939693 | − | 0.342020i | 0.392637 | − | 2.22675i | 0.636079 | − | 1.61103i | 1.14836 | + | 2.38354i | −0.866025 | + | 0.500000i | −1.35772 | − | 2.67518i | 2.26111i | ||
5.5 | −0.984808 | + | 0.173648i | −0.755599 | + | 1.55855i | 0.939693 | − | 0.342020i | −0.0946220 | + | 0.536628i | 0.473481 | − | 1.66608i | −2.19872 | − | 1.47161i | −0.866025 | + | 0.500000i | −1.85814 | − | 2.35527i | − | 0.544906i | |
5.6 | −0.984808 | + | 0.173648i | −0.542374 | − | 1.64494i | 0.939693 | − | 0.342020i | 0.381353 | − | 2.16276i | 0.819775 | + | 1.52577i | −1.53538 | − | 2.15467i | −0.866025 | + | 0.500000i | −2.41166 | + | 1.78435i | 2.19613i | ||
5.7 | −0.984808 | + | 0.173648i | −0.507755 | − | 1.65595i | 0.939693 | − | 0.342020i | −0.693451 | + | 3.93276i | 0.787595 | + | 1.54263i | 1.74878 | − | 1.98539i | −0.866025 | + | 0.500000i | −2.48437 | + | 1.68164i | − | 3.99342i | |
5.8 | −0.984808 | + | 0.173648i | 0.801094 | − | 1.53566i | 0.939693 | − | 0.342020i | 0.703607 | − | 3.99035i | −0.522259 | + | 1.65144i | 2.53004 | + | 0.773898i | −0.866025 | + | 0.500000i | −1.71650 | − | 2.46041i | 4.05191i | ||
5.9 | −0.984808 | + | 0.173648i | 0.891925 | + | 1.48475i | 0.939693 | − | 0.342020i | 0.217432 | − | 1.23312i | −1.13620 | − | 1.30731i | −2.23659 | + | 1.41339i | −0.866025 | + | 0.500000i | −1.40894 | + | 2.64856i | 1.25214i | ||
5.10 | −0.984808 | + | 0.173648i | 1.22171 | + | 1.22777i | 0.939693 | − | 0.342020i | 0.204833 | − | 1.16167i | −1.41635 | − | 0.996970i | 1.70692 | − | 2.02149i | −0.866025 | + | 0.500000i | −0.0148421 | + | 2.99996i | 1.17959i | ||
5.11 | −0.984808 | + | 0.173648i | 1.43254 | − | 0.973561i | 0.939693 | − | 0.342020i | −0.404896 | + | 2.29628i | −1.24172 | + | 1.20753i | −2.01803 | − | 1.71101i | −0.866025 | + | 0.500000i | 1.10436 | − | 2.78934i | − | 2.33170i | |
5.12 | −0.984808 | + | 0.173648i | 1.73169 | + | 0.0354625i | 0.939693 | − | 0.342020i | −0.334523 | + | 1.89717i | −1.71154 | + | 0.265781i | 2.15385 | + | 1.53653i | −0.866025 | + | 0.500000i | 2.99748 | + | 0.122820i | − | 1.92644i | |
5.13 | 0.984808 | − | 0.173648i | −1.73120 | − | 0.0542063i | 0.939693 | − | 0.342020i | −0.139489 | + | 0.791082i | −1.71431 | + | 0.247237i | 1.46739 | + | 2.20154i | 0.866025 | − | 0.500000i | 2.99412 | + | 0.187684i | 0.803286i | ||
5.14 | 0.984808 | − | 0.173648i | −1.68862 | + | 0.385431i | 0.939693 | − | 0.342020i | 0.136034 | − | 0.771489i | −1.59604 | + | 0.672801i | −2.64466 | + | 0.0761338i | 0.866025 | − | 0.500000i | 2.70289 | − | 1.30169i | − | 0.783391i | |
5.15 | 0.984808 | − | 0.173648i | −1.15375 | + | 1.29185i | 0.939693 | − | 0.342020i | 0.485901 | − | 2.75568i | −0.911891 | + | 1.47257i | 1.15554 | − | 2.38007i | 0.866025 | − | 0.500000i | −0.337739 | − | 2.98093i | − | 2.79820i | |
5.16 | 0.984808 | − | 0.173648i | −0.926008 | − | 1.46373i | 0.939693 | − | 0.342020i | 0.475930 | − | 2.69913i | −1.16611 | − | 1.28069i | 2.49768 | + | 0.872702i | 0.866025 | − | 0.500000i | −1.28502 | + | 2.71085i | − | 2.74077i | |
5.17 | 0.984808 | − | 0.173648i | −0.424397 | + | 1.67925i | 0.939693 | − | 0.342020i | −0.661063 | + | 3.74907i | −0.126350 | + | 1.72744i | −2.47370 | − | 0.938514i | 0.866025 | − | 0.500000i | −2.63977 | − | 1.42534i | 3.80691i | ||
5.18 | 0.984808 | − | 0.173648i | 0.228430 | − | 1.71692i | 0.939693 | − | 0.342020i | 0.0564431 | − | 0.320105i | −0.0731809 | − | 1.73050i | −0.543232 | − | 2.58938i | 0.866025 | − | 0.500000i | −2.89564 | − | 0.784392i | − | 0.325043i | |
5.19 | 0.984808 | − | 0.173648i | 0.261649 | + | 1.71217i | 0.939693 | − | 0.342020i | −0.00558222 | + | 0.0316584i | 0.554990 | + | 1.64073i | 2.62068 | − | 0.363383i | 0.866025 | − | 0.500000i | −2.86308 | + | 0.895977i | 0.0321467i | ||
5.20 | 0.984808 | − | 0.173648i | 0.963600 | − | 1.43926i | 0.939693 | − | 0.342020i | −0.689038 | + | 3.90773i | 0.699036 | − | 1.58472i | 0.360258 | + | 2.62111i | 0.866025 | − | 0.500000i | −1.14295 | − | 2.77375i | 3.96801i | ||
See next 80 embeddings (of 144 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
189.ba | 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.bf.a | yes | 144 |
7.d | odd | 6 | 1 | 378.2.ba.a | ✓ | 144 | |
27.f | odd | 18 | 1 | 378.2.ba.a | ✓ | 144 | |
189.ba | even | 18 | 1 | inner | 378.2.bf.a | yes | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.ba.a | ✓ | 144 | 7.d | odd | 6 | 1 | |
378.2.ba.a | ✓ | 144 | 27.f | odd | 18 | 1 | |
378.2.bf.a | yes | 144 | 1.a | even | 1 | 1 | trivial |
378.2.bf.a | yes | 144 | 189.ba | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).