Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [855,2,Mod(221,855)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("855.221");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 855 = 3^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 855.x (of order \(6\), degree \(2\), 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: | \(6.82720937282\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(80\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
221.1 | −1.39068 | + | 2.40873i | −1.34399 | − | 1.09257i | −2.86800 | − | 4.96752i | − | 1.00000i | 4.50076 | − | 1.71789i | −1.67782 | − | 2.90607i | 10.3912 | 0.612597 | + | 2.93679i | 2.40873 | + | 1.39068i | |||
221.2 | −1.37032 | + | 2.37346i | 0.119237 | + | 1.72794i | −2.75553 | − | 4.77271i | − | 1.00000i | −4.26459 | − | 2.08482i | 2.01949 | + | 3.49786i | 9.62251 | −2.97156 | + | 0.412070i | 2.37346 | + | 1.37032i | |||
221.3 | −1.32825 | + | 2.30059i | 1.72632 | − | 0.140834i | −2.52849 | − | 4.37947i | 1.00000i | −1.96897 | + | 4.15861i | 1.99572 | + | 3.45669i | 8.12083 | 2.96033 | − | 0.486248i | −2.30059 | − | 1.32825i | ||||
221.4 | −1.31206 | + | 2.27256i | −1.72143 | − | 0.191545i | −2.44301 | − | 4.23142i | 1.00000i | 2.69392 | − | 3.66072i | 0.104302 | + | 0.180656i | 7.57328 | 2.92662 | + | 0.659461i | −2.27256 | − | 1.31206i | ||||
221.5 | −1.29201 | + | 2.23782i | 1.50336 | + | 0.860173i | −2.33857 | − | 4.05052i | − | 1.00000i | −3.86727 | + | 2.25291i | −1.25247 | − | 2.16933i | 6.91775 | 1.52020 | + | 2.58631i | 2.23782 | + | 1.29201i | |||
221.6 | −1.26189 | + | 2.18565i | −1.25173 | + | 1.19715i | −2.18472 | − | 3.78405i | 1.00000i | −1.03702 | − | 4.24652i | 0.694271 | + | 1.20251i | 5.97996 | 0.133647 | − | 2.99702i | −2.18565 | − | 1.26189i | ||||
221.7 | −1.25767 | + | 2.17835i | 1.17608 | − | 1.27155i | −2.16347 | − | 3.74723i | − | 1.00000i | 1.29077 | + | 4.16110i | 0.214792 | + | 0.372030i | 5.85302 | −0.233688 | − | 2.99088i | 2.17835 | + | 1.25767i | |||
221.8 | −1.23584 | + | 2.14054i | 1.72781 | + | 0.121162i | −2.05462 | − | 3.55871i | 1.00000i | −2.39465 | + | 3.54871i | −1.45823 | − | 2.52573i | 5.21338 | 2.97064 | + | 0.418690i | −2.14054 | − | 1.23584i | ||||
221.9 | −1.17311 | + | 2.03188i | 0.538875 | + | 1.64609i | −1.75236 | − | 3.03517i | 1.00000i | −3.97682 | − | 0.836111i | −1.95687 | − | 3.38939i | 3.53038 | −2.41923 | + | 1.77407i | −2.03188 | − | 1.17311i | ||||
221.10 | −1.07786 | + | 1.86691i | −0.828010 | − | 1.52131i | −1.32358 | − | 2.29250i | − | 1.00000i | 3.73264 | + | 0.0939451i | 0.975399 | + | 1.68944i | 1.39509 | −1.62880 | + | 2.51933i | 1.86691 | + | 1.07786i | |||
221.11 | −1.05211 | + | 1.82230i | −0.457890 | − | 1.67043i | −1.21385 | − | 2.10245i | 1.00000i | 3.52577 | + | 0.923056i | 0.178878 | + | 0.309826i | 0.899977 | −2.58067 | + | 1.52975i | −1.82230 | − | 1.05211i | ||||
221.12 | −1.04762 | + | 1.81453i | 0.863016 | + | 1.50173i | −1.19502 | − | 2.06983i | 1.00000i | −3.62906 | − | 0.00727657i | 0.784732 | + | 1.35920i | 0.817219 | −1.51041 | + | 2.59204i | −1.81453 | − | 1.04762i | ||||
221.13 | −1.02642 | + | 1.77782i | 0.386040 | − | 1.68848i | −1.10709 | − | 1.91754i | − | 1.00000i | 2.60557 | + | 2.41941i | −1.53541 | − | 2.65941i | 0.439683 | −2.70195 | − | 1.30364i | 1.77782 | + | 1.02642i | |||
221.14 | −1.01645 | + | 1.76054i | −1.58337 | + | 0.702094i | −1.06633 | − | 1.84694i | − | 1.00000i | 0.373349 | − | 3.50123i | −1.24801 | − | 2.16161i | 0.269685 | 2.01413 | − | 2.22335i | 1.76054 | + | 1.01645i | |||
221.15 | −1.00324 | + | 1.73766i | −1.59886 | − | 0.666057i | −1.01297 | − | 1.75452i | 1.00000i | 2.76142 | − | 2.11007i | −1.54448 | − | 2.67512i | 0.0520549 | 2.11274 | + | 2.12987i | −1.73766 | − | 1.00324i | ||||
221.16 | −0.949253 | + | 1.64416i | −0.882461 | + | 1.49039i | −0.802164 | − | 1.38939i | − | 1.00000i | −1.61275 | − | 2.86566i | 0.485880 | + | 0.841569i | −0.751187 | −1.44252 | − | 2.63042i | 1.64416 | + | 0.949253i | |||
221.17 | −0.927416 | + | 1.60633i | 1.26930 | − | 1.17850i | −0.720202 | − | 1.24743i | 1.00000i | 0.715896 | + | 3.13189i | −0.944202 | − | 1.63541i | −1.03796 | 0.222259 | − | 2.99176i | −1.60633 | − | 0.927416i | ||||
221.18 | −0.884257 | + | 1.53158i | −0.0232144 | + | 1.73190i | −0.563819 | − | 0.976563i | − | 1.00000i | −2.63200 | − | 1.56699i | 0.114765 | + | 0.198780i | −1.54278 | −2.99892 | − | 0.0804098i | 1.53158 | + | 0.884257i | |||
221.19 | −0.873054 | + | 1.51217i | 1.13991 | − | 1.30407i | −0.524448 | − | 0.908370i | 1.00000i | 0.976773 | + | 2.86227i | 0.508142 | + | 0.880128i | −1.66073 | −0.401193 | − | 2.97305i | −1.51217 | − | 0.873054i | ||||
221.20 | −0.852919 | + | 1.47730i | 1.63562 | − | 0.569861i | −0.454942 | − | 0.787983i | − | 1.00000i | −0.553197 | + | 2.90235i | 2.51393 | + | 4.35425i | −1.85956 | 2.35052 | − | 1.86415i | 1.47730 | + | 0.852919i | |||
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
171.k | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 855.2.x.a | ✓ | 160 |
9.d | odd | 6 | 1 | 855.2.bp.a | yes | 160 | |
19.d | odd | 6 | 1 | 855.2.bp.a | yes | 160 | |
171.k | even | 6 | 1 | inner | 855.2.x.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
855.2.x.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
855.2.x.a | ✓ | 160 | 171.k | even | 6 | 1 | inner |
855.2.bp.a | yes | 160 | 9.d | odd | 6 | 1 | |
855.2.bp.a | yes | 160 | 19.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(855, [\chi])\).