Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,2,Mod(8,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(60))
chi = DirichletCharacter(H, H._module([10, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 675.bd (of order \(60\), degree \(16\), 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: | \(5.38990213644\) |
Analytic rank: | \(0\) |
Dimension: | \(448\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{60})\) |
Twist minimal: | no (minimal twist has level 225) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{60}]$ |
$q$-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 | −1.42019 | − | 2.18691i | 0 | −1.95214 | + | 4.38458i | −0.865670 | − | 2.06170i | 0 | 0.0498024 | + | 0.185865i | 7.21013 | − | 1.14197i | 0 | −3.27933 | + | 4.82116i | ||||||
8.2 | −1.35622 | − | 2.08840i | 0 | −1.70861 | + | 3.83760i | −2.16393 | + | 0.563377i | 0 | 0.669023 | + | 2.49683i | 5.41275 | − | 0.857295i | 0 | 4.11134 | + | 3.75510i | ||||||
8.3 | −1.23818 | − | 1.90664i | 0 | −1.28869 | + | 2.89444i | 2.20972 | − | 0.342224i | 0 | 0.640500 | + | 2.39038i | 2.62344 | − | 0.415512i | 0 | −3.38854 | − | 3.78940i | ||||||
8.4 | −1.18213 | − | 1.82032i | 0 | −1.10267 | + | 2.47663i | 2.17921 | − | 0.501050i | 0 | −0.367194 | − | 1.37039i | 1.52424 | − | 0.241416i | 0 | −3.48818 | − | 3.37456i | ||||||
8.5 | −1.03790 | − | 1.59823i | 0 | −0.663624 | + | 1.49052i | −0.300573 | − | 2.21577i | 0 | −0.911535 | − | 3.40190i | −0.693439 | + | 0.109830i | 0 | −3.22935 | + | 2.78014i | ||||||
8.6 | −1.01926 | − | 1.56953i | 0 | −0.611049 | + | 1.37244i | −0.303082 | + | 2.21543i | 0 | −0.0934709 | − | 0.348838i | −0.919915 | + | 0.145700i | 0 | 3.78611 | − | 1.78242i | ||||||
8.7 | −0.754539 | − | 1.16189i | 0 | 0.0328191 | − | 0.0737129i | −0.331994 | + | 2.21128i | 0 | 0.541248 | + | 2.01996i | −2.84708 | + | 0.450933i | 0 | 2.81977 | − | 1.28276i | ||||||
8.8 | −0.752499 | − | 1.15875i | 0 | 0.0370346 | − | 0.0831810i | 1.32432 | − | 1.80172i | 0 | 1.05097 | + | 3.92229i | −2.85353 | + | 0.451954i | 0 | −3.08428 | − | 0.178760i | ||||||
8.9 | −0.745007 | − | 1.14721i | 0 | 0.0524168 | − | 0.117730i | −2.22341 | − | 0.237578i | 0 | −1.22522 | − | 4.57257i | −2.87621 | + | 0.455548i | 0 | 1.38391 | + | 2.72772i | ||||||
8.10 | −0.431818 | − | 0.664942i | 0 | 0.557793 | − | 1.25282i | 1.15892 | + | 1.91230i | 0 | 0.246375 | + | 0.919483i | −2.64010 | + | 0.418151i | 0 | 0.771129 | − | 1.59638i | ||||||
8.11 | −0.326894 | − | 0.503372i | 0 | 0.666949 | − | 1.49799i | −0.293865 | − | 2.21667i | 0 | −0.105719 | − | 0.394547i | −2.15769 | + | 0.341745i | 0 | −1.01975 | + | 0.872540i | ||||||
8.12 | −0.270032 | − | 0.415813i | 0 | 0.713490 | − | 1.60253i | 2.01262 | + | 0.974349i | 0 | −0.995692 | − | 3.71597i | −1.83841 | + | 0.291175i | 0 | −0.138325 | − | 1.09998i | ||||||
8.13 | −0.0328244 | − | 0.0505452i | 0 | 0.811996 | − | 1.82377i | −2.19134 | + | 0.444987i | 0 | 0.835175 | + | 3.11691i | −0.237889 | + | 0.0376779i | 0 | 0.0944215 | + | 0.0961555i | ||||||
8.14 | 0.154007 | + | 0.237149i | 0 | 0.780951 | − | 1.75405i | 1.48145 | − | 1.67490i | 0 | 1.05334 | + | 3.93112i | 1.09482 | − | 0.173402i | 0 | 0.625356 | + | 0.0933787i | ||||||
8.15 | 0.191087 | + | 0.294249i | 0 | 0.763405 | − | 1.71464i | 1.24308 | + | 1.85869i | 0 | −0.223000 | − | 0.832248i | 1.34347 | − | 0.212785i | 0 | −0.309381 | + | 0.720948i | ||||||
8.16 | 0.299974 | + | 0.461920i | 0 | 0.690088 | − | 1.54996i | −1.56435 | + | 1.59775i | 0 | −0.838741 | − | 3.13022i | 2.01096 | − | 0.318504i | 0 | −1.20730 | − | 0.243322i | ||||||
8.17 | 0.336419 | + | 0.518040i | 0 | 0.658286 | − | 1.47853i | 1.21725 | − | 1.87571i | 0 | −0.707664 | − | 2.64104i | 2.20757 | − | 0.349645i | 0 | 1.38120 | 0.000442522i | |||||||
8.18 | 0.386810 | + | 0.595635i | 0 | 0.608314 | − | 1.36630i | −2.18497 | − | 0.475279i | 0 | 0.140157 | + | 0.523074i | 2.45205 | − | 0.388367i | 0 | −0.562077 | − | 1.48529i | ||||||
8.19 | 0.617042 | + | 0.950162i | 0 | 0.291407 | − | 0.654510i | 2.22710 | − | 0.200038i | 0 | 0.126426 | + | 0.471827i | 3.03968 | − | 0.481438i | 0 | 1.56428 | + | 1.99268i | ||||||
8.20 | 0.822576 | + | 1.26666i | 0 | −0.114312 | + | 0.256749i | −1.20491 | + | 1.88366i | 0 | 0.0589271 | + | 0.219919i | 2.56419 | − | 0.406129i | 0 | −3.37708 | + | 0.0232479i | ||||||
See next 80 embeddings (of 448 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
25.f | odd | 20 | 1 | inner |
225.w | even | 60 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 675.2.bd.a | 448 | |
3.b | odd | 2 | 1 | 225.2.w.a | ✓ | 448 | |
9.c | even | 3 | 1 | 225.2.w.a | ✓ | 448 | |
9.d | odd | 6 | 1 | inner | 675.2.bd.a | 448 | |
25.f | odd | 20 | 1 | inner | 675.2.bd.a | 448 | |
75.l | even | 20 | 1 | 225.2.w.a | ✓ | 448 | |
225.w | even | 60 | 1 | inner | 675.2.bd.a | 448 | |
225.x | odd | 60 | 1 | 225.2.w.a | ✓ | 448 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
225.2.w.a | ✓ | 448 | 3.b | odd | 2 | 1 | |
225.2.w.a | ✓ | 448 | 9.c | even | 3 | 1 | |
225.2.w.a | ✓ | 448 | 75.l | even | 20 | 1 | |
225.2.w.a | ✓ | 448 | 225.x | odd | 60 | 1 | |
675.2.bd.a | 448 | 1.a | even | 1 | 1 | trivial | |
675.2.bd.a | 448 | 9.d | odd | 6 | 1 | inner | |
675.2.bd.a | 448 | 25.f | odd | 20 | 1 | inner | |
675.2.bd.a | 448 | 225.w | even | 60 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(675, [\chi])\).