Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,4,Mod(25,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.h (of order \(3\), 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: | \(3.71712033036\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −5.09429 | 5.00043 | + | 1.41268i | 17.9518 | −3.42023 | + | 5.92401i | −25.4737 | − | 7.19661i | −0.241570 | − | 18.5187i | −50.6973 | 23.0087 | + | 14.1280i | 17.4236 | − | 30.1786i | ||||||
25.2 | −5.07998 | −4.95329 | + | 1.57000i | 17.8062 | −9.23499 | + | 15.9955i | 25.1626 | − | 7.97558i | −8.82618 | + | 16.2818i | −49.8155 | 22.0702 | − | 15.5533i | 46.9136 | − | 81.2567i | ||||||
25.3 | −4.87128 | −1.11598 | − | 5.07490i | 15.7294 | 3.10540 | − | 5.37871i | 5.43624 | + | 24.7213i | 17.8495 | − | 4.93906i | −37.6522 | −24.5092 | + | 11.3269i | −15.1273 | + | 26.2012i | ||||||
25.4 | −4.30573 | 0.221854 | + | 5.19141i | 10.5394 | 7.99829 | − | 13.8535i | −0.955246 | − | 22.3529i | 1.84582 | + | 18.4280i | −10.9338 | −26.9016 | + | 2.30348i | −34.4385 | + | 59.6493i | ||||||
25.5 | −3.37342 | 4.47377 | − | 2.64298i | 3.37993 | 4.87266 | − | 8.43970i | −15.0919 | + | 8.91589i | −16.1176 | + | 9.12268i | 15.5854 | 13.0293 | − | 23.6482i | −16.4375 | + | 28.4706i | ||||||
25.6 | −2.93457 | −3.89946 | − | 3.43427i | 0.611720 | 1.84855 | − | 3.20179i | 11.4432 | + | 10.0781i | −18.1273 | − | 3.79476i | 21.6814 | 3.41152 | + | 26.7836i | −5.42471 | + | 9.39588i | ||||||
25.7 | −2.65476 | −0.794408 | + | 5.13507i | −0.952261 | −3.67781 | + | 6.37015i | 2.10896 | − | 13.6324i | −1.32557 | − | 18.4728i | 23.7661 | −25.7378 | − | 8.15868i | 9.76368 | − | 16.9112i | ||||||
25.8 | −2.37882 | 3.09078 | − | 4.17697i | −2.34120 | −9.23374 | + | 15.9933i | −7.35242 | + | 9.93628i | 15.8733 | + | 9.54133i | 24.5999 | −7.89419 | − | 25.8202i | 21.9654 | − | 38.0453i | ||||||
25.9 | −1.80909 | −5.15261 | + | 0.671239i | −4.72719 | 1.04890 | − | 1.81674i | 9.32155 | − | 1.21433i | 18.3541 | + | 2.47507i | 23.0246 | 26.0989 | − | 6.91727i | −1.89755 | + | 3.28665i | ||||||
25.10 | −0.590775 | 4.35408 | + | 2.83584i | −7.65099 | −5.49223 | + | 9.51282i | −2.57228 | − | 1.67534i | −16.7104 | + | 7.98524i | 9.24621 | 10.9161 | + | 24.6949i | 3.24467 | − | 5.61993i | ||||||
25.11 | −0.438515 | 4.98047 | + | 1.48153i | −7.80770 | 8.04659 | − | 13.9371i | −2.18401 | − | 0.649674i | 16.9337 | − | 7.49990i | 6.93192 | 22.6101 | + | 14.7574i | −3.52855 | + | 6.11163i | ||||||
25.12 | 0.534259 | 1.18489 | − | 5.05925i | −7.71457 | 0.696621 | − | 1.20658i | 0.633036 | − | 2.70295i | −2.10659 | − | 18.4001i | −8.39564 | −24.1921 | − | 11.9893i | 0.372176 | − | 0.644627i | ||||||
25.13 | 0.983694 | −4.22206 | + | 3.02890i | −7.03235 | 9.35711 | − | 16.2070i | −4.15321 | + | 2.97951i | −18.4989 | − | 0.890133i | −14.7872 | 8.65150 | − | 25.5764i | 9.20454 | − | 15.9427i | ||||||
25.14 | 1.33560 | −3.51545 | − | 3.82643i | −6.21617 | −4.50235 | + | 7.79829i | −4.69524 | − | 5.11058i | −3.16069 | + | 18.2486i | −18.9871 | −2.28318 | + | 26.9033i | −6.01333 | + | 10.4154i | ||||||
25.15 | 1.44809 | 0.196108 | + | 5.19245i | −5.90304 | −2.21638 | + | 3.83887i | 0.283981 | + | 7.51913i | 9.71690 | + | 15.7665i | −20.1328 | −26.9231 | + | 2.03656i | −3.20951 | + | 5.55903i | ||||||
25.16 | 2.66292 | −4.74469 | + | 2.11846i | −0.908849 | −9.61903 | + | 16.6607i | −12.6348 | + | 5.64129i | −5.55741 | − | 17.6668i | −23.7236 | 18.0243 | − | 20.1029i | −25.6147 | + | 44.3660i | ||||||
25.17 | 3.35315 | 3.94919 | − | 3.37697i | 3.24362 | 4.35326 | − | 7.54007i | 13.2422 | − | 11.3235i | −2.88142 | + | 18.2947i | −15.9489 | 4.19213 | − | 26.6726i | 14.5971 | − | 25.2830i | ||||||
25.18 | 3.66978 | 5.19361 | + | 0.162683i | 5.46732 | −7.38708 | + | 12.7948i | 19.0594 | + | 0.597013i | 15.9165 | − | 9.46920i | −9.29438 | 26.9471 | + | 1.68983i | −27.1090 | + | 46.9542i | ||||||
25.19 | 3.86663 | −3.99989 | − | 3.31676i | 6.95086 | 6.67810 | − | 11.5668i | −15.4661 | − | 12.8247i | 15.1080 | − | 10.7120i | −4.05663 | 4.99821 | + | 26.5333i | 25.8218 | − | 44.7246i | ||||||
25.20 | 4.21476 | 2.84415 | + | 4.34866i | 9.76423 | 3.91201 | − | 6.77580i | 11.9874 | + | 18.3286i | −15.9002 | − | 9.49654i | 7.43579 | −10.8216 | + | 24.7365i | 16.4882 | − | 28.5584i | ||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 63.4.h.a | yes | 44 |
3.b | odd | 2 | 1 | 189.4.h.a | 44 | ||
7.c | even | 3 | 1 | 63.4.g.a | ✓ | 44 | |
9.c | even | 3 | 1 | 63.4.g.a | ✓ | 44 | |
9.d | odd | 6 | 1 | 189.4.g.a | 44 | ||
21.h | odd | 6 | 1 | 189.4.g.a | 44 | ||
63.h | even | 3 | 1 | inner | 63.4.h.a | yes | 44 |
63.j | odd | 6 | 1 | 189.4.h.a | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.4.g.a | ✓ | 44 | 7.c | even | 3 | 1 | |
63.4.g.a | ✓ | 44 | 9.c | even | 3 | 1 | |
63.4.h.a | yes | 44 | 1.a | even | 1 | 1 | trivial |
63.4.h.a | yes | 44 | 63.h | even | 3 | 1 | inner |
189.4.g.a | 44 | 9.d | odd | 6 | 1 | ||
189.4.g.a | 44 | 21.h | odd | 6 | 1 | ||
189.4.h.a | 44 | 3.b | odd | 2 | 1 | ||
189.4.h.a | 44 | 63.j | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(63, [\chi])\).