Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [891,2,Mod(8,891)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(90))
chi = DirichletCharacter(H, H._module([5, 27]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("891.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 891 = 3^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 891.bb (of order \(90\), degree \(24\), not 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: | \(7.11467082010\) |
Analytic rank: | \(0\) |
Dimension: | \(816\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{90})\) |
Twist minimal: | no (minimal twist has level 297) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{90}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −1.16150 | − | 2.38143i | 0 | −3.09081 | + | 3.95605i | 0.0429102 | + | 0.613644i | 0 | −1.71414 | + | 0.0598591i | 7.82768 | + | 1.66383i | 0 | 1.41151 | − | 0.814937i | ||||||
8.2 | −1.11997 | − | 2.29627i | 0 | −2.78723 | + | 3.56749i | 0.0405112 | + | 0.579337i | 0 | −1.72991 | + | 0.0604096i | 6.31552 | + | 1.34240i | 0 | 1.28495 | − | 0.741864i | ||||||
8.3 | −1.11045 | − | 2.27675i | 0 | −2.71918 | + | 3.48040i | −0.237504 | − | 3.39647i | 0 | 1.01734 | − | 0.0355264i | 5.98798 | + | 1.27279i | 0 | −7.46918 | + | 4.31233i | ||||||
8.4 | −1.10307 | − | 2.26163i | 0 | −2.66689 | + | 3.41346i | 0.284326 | + | 4.06604i | 0 | 4.04513 | − | 0.141259i | 5.73915 | + | 1.21989i | 0 | 8.88227 | − | 5.12818i | ||||||
8.5 | −0.936912 | − | 1.92095i | 0 | −1.58094 | + | 2.02351i | 0.0440697 | + | 0.630226i | 0 | 1.09943 | − | 0.0383930i | 1.18716 | + | 0.252338i | 0 | 1.16935 | − | 0.675122i | ||||||
8.6 | −0.847489 | − | 1.73761i | 0 | −1.06973 | + | 1.36919i | −0.164217 | − | 2.34841i | 0 | −0.729987 | + | 0.0254917i | −0.496339 | − | 0.105500i | 0 | −3.94146 | + | 2.27560i | ||||||
8.7 | −0.831591 | − | 1.70501i | 0 | −0.984207 | + | 1.25973i | −0.134621 | − | 1.92517i | 0 | −4.51387 | + | 0.157628i | −0.744786 | − | 0.158309i | 0 | −3.17049 | + | 1.83049i | ||||||
8.8 | −0.738685 | − | 1.51453i | 0 | −0.516818 | + | 0.661497i | 0.106711 | + | 1.52603i | 0 | 2.22195 | − | 0.0775923i | −1.91287 | − | 0.406592i | 0 | 2.23239 | − | 1.28887i | ||||||
8.9 | −0.626833 | − | 1.28520i | 0 | −0.0274932 | + | 0.0351897i | −0.0191541 | − | 0.273916i | 0 | −0.0722617 | + | 0.00252344i | −2.73488 | − | 0.581316i | 0 | −0.340030 | + | 0.196317i | ||||||
8.10 | −0.626532 | − | 1.28458i | 0 | −0.0262823 | + | 0.0336398i | 0.179932 | + | 2.57315i | 0 | −4.29989 | + | 0.150155i | −2.73631 | − | 0.581621i | 0 | 3.19268 | − | 1.84330i | ||||||
8.11 | −0.469140 | − | 0.961880i | 0 | 0.526202 | − | 0.673508i | 0.120802 | + | 1.72754i | 0 | 3.20463 | − | 0.111908i | −2.98830 | − | 0.635184i | 0 | 1.60502 | − | 0.926656i | ||||||
8.12 | −0.361589 | − | 0.741367i | 0 | 0.812444 | − | 1.03988i | −0.228051 | − | 3.26129i | 0 | −3.64248 | + | 0.127198i | −2.67835 | − | 0.569301i | 0 | −2.33535 | + | 1.34832i | ||||||
8.13 | −0.337236 | − | 0.691437i | 0 | 0.866966 | − | 1.10967i | 0.278822 | + | 3.98735i | 0 | −2.50352 | + | 0.0874247i | −2.56460 | − | 0.545123i | 0 | 2.66297 | − | 1.53747i | ||||||
8.14 | −0.294358 | − | 0.603523i | 0 | 0.953729 | − | 1.22072i | −0.136587 | − | 1.95329i | 0 | 1.03639 | − | 0.0361914i | −2.33108 | − | 0.495487i | 0 | −1.13865 | + | 0.657401i | ||||||
8.15 | −0.131962 | − | 0.270562i | 0 | 1.17553 | − | 1.50461i | −0.115008 | − | 1.64470i | 0 | 4.68773 | − | 0.163699i | −1.15112 | − | 0.244677i | 0 | −0.429816 | + | 0.248154i | ||||||
8.16 | −0.0361822 | − | 0.0741844i | 0 | 1.22713 | − | 1.57065i | 0.116205 | + | 1.66180i | 0 | −4.36097 | + | 0.152288i | −0.322386 | − | 0.0685253i | 0 | 0.119076 | − | 0.0687483i | ||||||
8.17 | −0.0350715 | − | 0.0719072i | 0 | 1.22738 | − | 1.57098i | 0.103569 | + | 1.48111i | 0 | −0.412525 | + | 0.0144057i | −0.312522 | − | 0.0664287i | 0 | 0.102870 | − | 0.0593920i | ||||||
8.18 | −0.00652561 | − | 0.0133795i | 0 | 1.23119 | − | 1.57585i | −0.107633 | − | 1.53923i | 0 | −0.727962 | + | 0.0254210i | −0.0582398 | − | 0.0123792i | 0 | −0.0198917 | + | 0.0114845i | ||||||
8.19 | 0.234840 | + | 0.481493i | 0 | 1.05464 | − | 1.34987i | 0.140627 | + | 2.01106i | 0 | 2.77243 | − | 0.0968153i | 1.94563 | + | 0.413557i | 0 | −0.935288 | + | 0.539989i | ||||||
8.20 | 0.257823 | + | 0.528615i | 0 | 1.01836 | − | 1.30344i | −0.252670 | − | 3.61335i | 0 | 1.32393 | − | 0.0462327i | 2.10215 | + | 0.446826i | 0 | 1.84493 | − | 1.06517i | ||||||
See next 80 embeddings (of 816 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
27.f | odd | 18 | 1 | inner |
297.x | even | 90 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 891.2.bb.a | 816 | |
3.b | odd | 2 | 1 | 297.2.x.a | ✓ | 816 | |
11.d | odd | 10 | 1 | inner | 891.2.bb.a | 816 | |
27.e | even | 9 | 1 | 297.2.x.a | ✓ | 816 | |
27.f | odd | 18 | 1 | inner | 891.2.bb.a | 816 | |
33.f | even | 10 | 1 | 297.2.x.a | ✓ | 816 | |
297.w | odd | 90 | 1 | 297.2.x.a | ✓ | 816 | |
297.x | even | 90 | 1 | inner | 891.2.bb.a | 816 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
297.2.x.a | ✓ | 816 | 3.b | odd | 2 | 1 | |
297.2.x.a | ✓ | 816 | 27.e | even | 9 | 1 | |
297.2.x.a | ✓ | 816 | 33.f | even | 10 | 1 | |
297.2.x.a | ✓ | 816 | 297.w | odd | 90 | 1 | |
891.2.bb.a | 816 | 1.a | even | 1 | 1 | trivial | |
891.2.bb.a | 816 | 11.d | odd | 10 | 1 | inner | |
891.2.bb.a | 816 | 27.f | odd | 18 | 1 | inner | |
891.2.bb.a | 816 | 297.x | even | 90 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(891, [\chi])\).