Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(193,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 2, 12]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.193");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.bp (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: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(144\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
193.1 | 0 | −1.72645 | + | 0.139144i | 0 | −0.604458 | + | 0.220005i | 0 | −2.43449 | − | 1.03598i | 0 | 2.96128 | − | 0.480450i | 0 | ||||||||||
193.2 | 0 | −1.72611 | − | 0.143355i | 0 | 1.14483 | − | 0.416685i | 0 | −1.12940 | + | 2.39259i | 0 | 2.95890 | + | 0.494894i | 0 | ||||||||||
193.3 | 0 | −1.68394 | + | 0.405388i | 0 | 1.57900 | − | 0.574710i | 0 | 2.62692 | + | 0.315138i | 0 | 2.67132 | − | 1.36530i | 0 | ||||||||||
193.4 | 0 | −1.67981 | − | 0.422185i | 0 | −3.86396 | + | 1.40636i | 0 | 0.267801 | − | 2.63216i | 0 | 2.64352 | + | 1.41838i | 0 | ||||||||||
193.5 | 0 | −1.35655 | − | 1.07693i | 0 | 2.04264 | − | 0.743461i | 0 | 0.979055 | − | 2.45794i | 0 | 0.680440 | + | 2.92181i | 0 | ||||||||||
193.6 | 0 | −1.22275 | + | 1.22673i | 0 | −1.75075 | + | 0.637221i | 0 | 2.59849 | + | 0.497853i | 0 | −0.00975629 | − | 2.99998i | 0 | ||||||||||
193.7 | 0 | −0.985812 | + | 1.42414i | 0 | −1.13435 | + | 0.412871i | 0 | −2.60567 | − | 0.458792i | 0 | −1.05635 | − | 2.80787i | 0 | ||||||||||
193.8 | 0 | −0.844572 | − | 1.51218i | 0 | −0.569103 | + | 0.207137i | 0 | 2.00271 | + | 1.72892i | 0 | −1.57340 | + | 2.55430i | 0 | ||||||||||
193.9 | 0 | −0.699375 | + | 1.58457i | 0 | 3.09474 | − | 1.12639i | 0 | −0.555998 | + | 2.58667i | 0 | −2.02175 | − | 2.21642i | 0 | ||||||||||
193.10 | 0 | −0.636453 | − | 1.61088i | 0 | −2.33777 | + | 0.850878i | 0 | −2.09399 | + | 1.61716i | 0 | −2.18985 | + | 2.05050i | 0 | ||||||||||
193.11 | 0 | −0.513768 | + | 1.65410i | 0 | −0.633570 | + | 0.230601i | 0 | 1.95184 | − | 1.78614i | 0 | −2.47208 | − | 1.69965i | 0 | ||||||||||
193.12 | 0 | 0.128228 | − | 1.72730i | 0 | 3.80842 | − | 1.38615i | 0 | −2.63217 | − | 0.267690i | 0 | −2.96712 | − | 0.442975i | 0 | ||||||||||
193.13 | 0 | 0.156536 | − | 1.72496i | 0 | −1.43093 | + | 0.520817i | 0 | −0.946337 | − | 2.47072i | 0 | −2.95099 | − | 0.540038i | 0 | ||||||||||
193.14 | 0 | 0.378367 | + | 1.69022i | 0 | −3.32039 | + | 1.20852i | 0 | 0.693513 | + | 2.55324i | 0 | −2.71368 | + | 1.27905i | 0 | ||||||||||
193.15 | 0 | 0.557871 | − | 1.63975i | 0 | 2.46711 | − | 0.897956i | 0 | 1.73437 | + | 1.99799i | 0 | −2.37756 | − | 1.82954i | 0 | ||||||||||
193.16 | 0 | 0.726734 | + | 1.57221i | 0 | 1.04532 | − | 0.380466i | 0 | −2.56389 | + | 0.653064i | 0 | −1.94371 | + | 2.28516i | 0 | ||||||||||
193.17 | 0 | 0.963882 | + | 1.43907i | 0 | −0.981872 | + | 0.357372i | 0 | 0.459325 | − | 2.60557i | 0 | −1.14186 | + | 2.77419i | 0 | ||||||||||
193.18 | 0 | 1.12192 | − | 1.31958i | 0 | −3.46735 | + | 1.26201i | 0 | 2.63004 | − | 0.287949i | 0 | −0.482607 | − | 2.96093i | 0 | ||||||||||
193.19 | 0 | 1.17989 | + | 1.26802i | 0 | 3.75808 | − | 1.36783i | 0 | 2.59104 | − | 0.535278i | 0 | −0.215726 | + | 2.99223i | 0 | ||||||||||
193.20 | 0 | 1.44372 | − | 0.956902i | 0 | −2.40619 | + | 0.875781i | 0 | −1.62201 | + | 2.09024i | 0 | 1.16868 | − | 2.76300i | 0 | ||||||||||
See next 80 embeddings (of 144 total) |
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 | 756.2.bp.a | ✓ | 144 |
7.c | even | 3 | 1 | 756.2.bq.a | yes | 144 | |
27.e | even | 9 | 1 | 756.2.bq.a | yes | 144 | |
189.u | even | 9 | 1 | inner | 756.2.bp.a | ✓ | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
756.2.bp.a | ✓ | 144 | 1.a | even | 1 | 1 | trivial |
756.2.bp.a | ✓ | 144 | 189.u | even | 9 | 1 | inner |
756.2.bq.a | yes | 144 | 7.c | even | 3 | 1 | |
756.2.bq.a | yes | 144 | 27.e | even | 9 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).