Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [216,3,Mod(5,216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 9, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("216.5");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 216 = 2^{3} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 216.x (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: | \(5.88557371018\) |
Analytic rank: | \(0\) |
Dimension: | \(420\) |
Relative dimension: | \(70\) 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 | −1.99762 | − | 0.0976193i | −1.40653 | + | 2.64984i | 3.98094 | + | 0.390012i | −7.28649 | + | 2.65207i | 3.06839 | − | 5.15606i | −0.503077 | − | 2.85309i | −7.91432 | − | 1.16771i | −5.04332 | − | 7.45419i | 14.8145 | − | 4.58651i |
5.2 | −1.98021 | + | 0.280686i | −1.96788 | + | 2.26438i | 3.84243 | − | 1.11163i | 8.66971 | − | 3.15552i | 3.26123 | − | 5.03630i | −0.680238 | − | 3.85782i | −7.29678 | + | 3.27978i | −1.25487 | − | 8.91209i | −16.2821 | + | 8.68204i |
5.3 | −1.97743 | − | 0.299639i | 1.51508 | + | 2.58931i | 3.82043 | + | 1.18503i | 0.829293 | − | 0.301838i | −2.22009 | − | 5.57415i | 1.72439 | + | 9.77949i | −7.19954 | − | 3.48806i | −4.40908 | + | 7.84602i | −1.73031 | + | 0.348374i |
5.4 | −1.97718 | + | 0.301233i | 1.67983 | − | 2.48559i | 3.81852 | − | 1.19119i | −4.31690 | + | 1.57122i | −2.57260 | + | 5.42049i | 1.15077 | + | 6.52637i | −7.19109 | + | 3.50546i | −3.35631 | − | 8.35076i | 8.06201 | − | 4.40700i |
5.5 | −1.94771 | − | 0.454324i | −0.0785589 | − | 2.99897i | 3.58718 | + | 1.76979i | 0.792146 | − | 0.288318i | −1.20949 | + | 5.87683i | −1.04591 | − | 5.93165i | −6.18274 | − | 5.07678i | −8.98766 | + | 0.471192i | −1.67386 | + | 0.201669i |
5.6 | −1.93633 | + | 0.500639i | 2.99956 | − | 0.0513619i | 3.49872 | − | 1.93880i | −3.68720 | + | 1.34203i | −5.78241 | + | 1.60115i | −0.898934 | − | 5.09811i | −5.80403 | + | 5.50574i | 8.99472 | − | 0.308126i | 6.46775 | − | 4.44457i |
5.7 | −1.91201 | − | 0.586697i | −2.76561 | − | 1.16249i | 3.31157 | + | 2.24354i | 3.88145 | − | 1.41273i | 4.60586 | + | 3.84527i | 0.551431 | + | 3.12732i | −5.01548 | − | 6.23257i | 6.29724 | + | 6.42999i | −8.25022 | + | 0.423924i |
5.8 | −1.88656 | + | 0.664009i | −2.86047 | − | 0.904260i | 3.11818 | − | 2.50538i | −1.90855 | + | 0.694657i | 5.99688 | − | 0.193444i | −1.88770 | − | 10.7057i | −4.21904 | + | 6.79704i | 7.36463 | + | 5.17322i | 3.13934 | − | 2.57781i |
5.9 | −1.86507 | + | 0.722171i | −1.09942 | − | 2.79129i | 2.95694 | − | 2.69379i | 6.30966 | − | 2.29653i | 4.06627 | + | 4.41197i | 1.48683 | + | 8.43221i | −3.56950 | + | 7.15952i | −6.58256 | + | 6.13758i | −10.1094 | + | 8.83983i |
5.10 | −1.85293 | − | 0.752764i | 2.85359 | − | 0.925745i | 2.86669 | + | 2.78964i | 6.33667 | − | 2.30636i | −5.98437 | − | 0.432745i | −0.404112 | − | 2.29183i | −3.21184 | − | 7.32694i | 7.28599 | − | 5.28340i | −13.4775 | − | 0.496500i |
5.11 | −1.71853 | − | 1.02307i | −2.36755 | − | 1.84247i | 1.90667 | + | 3.51633i | −7.66878 | + | 2.79121i | 2.18374 | + | 5.58849i | 0.943246 | + | 5.34941i | 0.320773 | − | 7.99357i | 2.21063 | + | 8.72428i | 16.0346 | + | 3.04890i |
5.12 | −1.68433 | + | 1.07844i | −2.97464 | + | 0.389273i | 1.67392 | − | 3.63290i | −3.75258 | + | 1.36583i | 4.59046 | − | 3.86364i | 2.21734 | + | 12.5752i | 1.09844 | + | 7.92423i | 8.69693 | − | 2.31589i | 4.84761 | − | 6.34745i |
5.13 | −1.65199 | + | 1.12735i | 0.0725877 | + | 2.99912i | 1.45817 | − | 3.72475i | 2.01543 | − | 0.733558i | −3.50097 | − | 4.87270i | 0.236981 | + | 1.34399i | 1.79021 | + | 7.79713i | −8.98946 | + | 0.435399i | −2.50251 | + | 3.48393i |
5.14 | −1.64616 | + | 1.13585i | 2.96853 | + | 0.433379i | 1.41970 | − | 3.73958i | 6.44507 | − | 2.34581i | −5.37894 | + | 2.65839i | 1.04940 | + | 5.95146i | 1.91054 | + | 7.76852i | 8.62436 | + | 2.57300i | −7.94514 | + | 11.1822i |
5.15 | −1.62902 | − | 1.16029i | −2.58836 | + | 1.51670i | 1.30744 | + | 3.78029i | 0.174835 | − | 0.0636348i | 5.97632 | + | 0.532510i | −0.481503 | − | 2.73074i | 2.25640 | − | 7.67520i | 4.39922 | − | 7.85155i | −0.358646 | − | 0.0991975i |
5.16 | −1.59804 | − | 1.20262i | 2.77706 | + | 1.13487i | 1.10743 | + | 3.84364i | −6.71804 | + | 2.44517i | −3.07303 | − | 5.15330i | −0.389417 | − | 2.20849i | 2.85270 | − | 7.47409i | 6.42414 | + | 6.30321i | 13.6763 | + | 4.17175i |
5.17 | −1.54566 | − | 1.26923i | 0.649225 | + | 2.92891i | 0.778100 | + | 3.92359i | 2.81183 | − | 1.02342i | 2.71399 | − | 5.35110i | −1.90676 | − | 10.8138i | 3.77727 | − | 7.05211i | −8.15701 | + | 3.80304i | −5.64509 | − | 1.98701i |
5.18 | −1.42825 | + | 1.40003i | 1.48437 | − | 2.60704i | 0.0798093 | − | 3.99920i | 4.33790 | − | 1.57887i | 1.52990 | + | 5.80167i | −1.85705 | − | 10.5318i | 5.48503 | + | 5.82361i | −4.59332 | − | 7.73960i | −3.98515 | + | 8.32823i |
5.19 | −1.37398 | + | 1.45333i | 1.48412 | + | 2.60718i | −0.224349 | − | 3.99370i | −5.29643 | + | 1.92774i | −5.82825 | − | 1.42529i | −0.799013 | − | 4.53143i | 6.11243 | + | 5.16122i | −4.59476 | + | 7.73875i | 4.47555 | − | 10.3462i |
5.20 | −1.34054 | − | 1.48423i | 1.66458 | − | 2.49583i | −0.405897 | + | 3.97935i | −1.96446 | + | 0.715007i | −5.93583 | + | 0.875140i | 2.11808 | + | 12.0122i | 6.45041 | − | 4.73204i | −3.45835 | − | 8.30902i | 3.69468 | + | 1.95723i |
See next 80 embeddings (of 420 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
27.f | odd | 18 | 1 | inner |
216.x | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 216.3.x.a | ✓ | 420 |
8.b | even | 2 | 1 | inner | 216.3.x.a | ✓ | 420 |
27.f | odd | 18 | 1 | inner | 216.3.x.a | ✓ | 420 |
216.x | odd | 18 | 1 | inner | 216.3.x.a | ✓ | 420 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
216.3.x.a | ✓ | 420 | 1.a | even | 1 | 1 | trivial |
216.3.x.a | ✓ | 420 | 8.b | even | 2 | 1 | inner |
216.3.x.a | ✓ | 420 | 27.f | odd | 18 | 1 | inner |
216.3.x.a | ✓ | 420 | 216.x | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(216, [\chi])\).