Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,2,Mod(11,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(32))
chi = DirichletCharacter(H, H._module([16, 21, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 384.w (of order \(32\), degree \(16\), 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: | \(3.06625543762\) |
Analytic rank: | \(0\) |
Dimension: | \(992\) |
Relative dimension: | \(62\) over \(\Q(\zeta_{32})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{32}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.41413 | − | 0.0157861i | 1.41096 | + | 1.00458i | 1.99950 | + | 0.0446470i | −3.04456 | − | 1.62735i | −1.97942 | − | 1.44288i | −1.52522 | − | 0.303386i | −2.82684 | − | 0.0947007i | 0.981620 | + | 2.83486i | 4.27970 | + | 2.34934i |
11.2 | −1.41347 | + | 0.0458784i | −0.204040 | − | 1.71999i | 1.99579 | − | 0.129695i | −0.561681 | − | 0.300225i | 0.367315 | + | 2.42179i | −0.946225 | − | 0.188216i | −2.81504 | + | 0.274884i | −2.91674 | + | 0.701894i | 0.807693 | + | 0.398589i |
11.3 | −1.41282 | − | 0.0626704i | 1.71388 | − | 0.250263i | 1.99214 | + | 0.177085i | 1.34559 | + | 0.719232i | −2.43709 | + | 0.246168i | 2.49948 | + | 0.497176i | −2.80345 | − | 0.375038i | 2.87474 | − | 0.857839i | −1.85601 | − | 1.10048i |
11.4 | −1.39869 | − | 0.208968i | −0.339181 | + | 1.69852i | 1.91266 | + | 0.584564i | −0.321371 | − | 0.171776i | 0.829345 | − | 2.30482i | −1.36158 | − | 0.270834i | −2.55307 | − | 1.21731i | −2.76991 | − | 1.15221i | 0.413602 | + | 0.307418i |
11.5 | −1.37752 | + | 0.320040i | −1.61000 | − | 0.638681i | 1.79515 | − | 0.881727i | −0.665070 | − | 0.355487i | 2.42221 | + | 0.364535i | 0.505023 | + | 0.100455i | −2.19067 | + | 1.78912i | 2.18417 | + | 2.05655i | 1.02992 | + | 0.276843i |
11.6 | −1.30952 | + | 0.533994i | 1.15242 | + | 1.29303i | 1.42970 | − | 1.39855i | 3.25884 | + | 1.74189i | −2.19959 | − | 1.07787i | −3.48323 | − | 0.692857i | −1.12541 | + | 2.59489i | −0.343871 | + | 2.98023i | −5.19768 | − | 0.540839i |
11.7 | −1.30056 | − | 0.555477i | −1.67960 | + | 0.423001i | 1.38289 | + | 1.44486i | 3.35351 | + | 1.79249i | 2.41939 | + | 0.382846i | 3.20524 | + | 0.637561i | −0.995942 | − | 2.64728i | 2.64214 | − | 1.42095i | −3.36574 | − | 4.19402i |
11.8 | −1.29156 | − | 0.576082i | −1.73162 | + | 0.0387632i | 1.33626 | + | 1.48809i | −0.861594 | − | 0.460532i | 2.25882 | + | 0.947488i | −4.16278 | − | 0.828028i | −0.868599 | − | 2.69175i | 2.99699 | − | 0.134246i | 0.847497 | + | 1.09115i |
11.9 | −1.27198 | + | 0.618111i | −1.15591 | + | 1.28991i | 1.23588 | − | 1.57245i | 1.21911 | + | 0.651630i | 0.672992 | − | 2.35522i | 1.31145 | + | 0.260863i | −0.600063 | + | 2.76404i | −0.327733 | − | 2.98204i | −1.95347 | − | 0.0753137i |
11.10 | −1.24002 | + | 0.679964i | 1.09320 | − | 1.34347i | 1.07530 | − | 1.68634i | 0.708082 | + | 0.378478i | −0.442086 | + | 2.40927i | −4.90117 | − | 0.974904i | −0.186740 | + | 2.82226i | −0.609807 | − | 2.93737i | −1.13539 | + | 0.0121507i |
11.11 | −1.15643 | − | 0.814046i | −0.824531 | − | 1.52320i | 0.674657 | + | 1.88277i | −0.611958 | − | 0.327098i | −0.286446 | + | 2.43268i | 2.75895 | + | 0.548789i | 0.752472 | − | 2.72650i | −1.64030 | + | 2.51186i | 0.441413 | + | 0.876428i |
11.12 | −1.12499 | + | 0.856964i | 0.434523 | + | 1.67666i | 0.531225 | − | 1.92816i | −2.13158 | − | 1.13935i | −1.92567 | − | 1.51386i | 1.86189 | + | 0.370353i | 1.05474 | + | 2.62441i | −2.62238 | + | 1.45710i | 3.37440 | − | 0.544921i |
11.13 | −1.11732 | − | 0.866940i | 1.27976 | − | 1.16714i | 0.496829 | + | 1.93731i | −2.95304 | − | 1.57843i | −2.44175 | + | 0.194595i | −1.29330 | − | 0.257253i | 1.12441 | − | 2.59532i | 0.275578 | − | 2.98732i | 1.93109 | + | 4.32373i |
11.14 | −1.11401 | − | 0.871200i | 0.593110 | + | 1.62734i | 0.482021 | + | 1.94104i | 3.29266 | + | 1.75996i | 0.757006 | − | 2.32958i | −0.370976 | − | 0.0737916i | 1.15406 | − | 2.58227i | −2.29644 | + | 1.93038i | −2.13477 | − | 4.82918i |
11.15 | −1.10845 | + | 0.878258i | −0.117171 | − | 1.72808i | 0.457325 | − | 1.94701i | 3.15748 | + | 1.68771i | 1.64758 | + | 1.81259i | 4.43843 | + | 0.882860i | 1.20306 | + | 2.55982i | −2.97254 | + | 0.404962i | −4.98216 | + | 0.902342i |
11.16 | −1.07112 | + | 0.923423i | 1.49221 | − | 0.879386i | 0.294582 | − | 1.97819i | −1.88241 | − | 1.00617i | −0.786282 | + | 2.31986i | 2.66028 | + | 0.529163i | 1.51117 | + | 2.39089i | 1.45336 | − | 2.62445i | 2.94540 | − | 0.660535i |
11.17 | −1.02490 | − | 0.974463i | 1.27873 | + | 1.16826i | 0.100842 | + | 1.99746i | −1.18082 | − | 0.631161i | −0.172143 | − | 2.44343i | 4.72705 | + | 0.940268i | 1.84309 | − | 2.14546i | 0.270316 | + | 2.98780i | 0.595179 | + | 1.79754i |
11.18 | −0.917171 | + | 1.07647i | −0.843817 | − | 1.51260i | −0.317594 | − | 1.97462i | −3.45356 | − | 1.84597i | 2.40220 | + | 0.478969i | −1.46677 | − | 0.291758i | 2.41692 | + | 1.46919i | −1.57594 | + | 2.55272i | 5.15464 | − | 2.02460i |
11.19 | −0.845116 | − | 1.13392i | −0.765283 | + | 1.55382i | −0.571557 | + | 1.91659i | −0.974296 | − | 0.520772i | 2.40866 | − | 0.445384i | −1.27617 | − | 0.253846i | 2.65630 | − | 0.971642i | −1.82869 | − | 2.37822i | 0.232879 | + | 1.54489i |
11.20 | −0.813493 | + | 1.15682i | −1.73071 | − | 0.0681515i | −0.676460 | − | 1.88213i | 1.66341 | + | 0.889109i | 1.48676 | − | 1.94668i | −2.12405 | − | 0.422500i | 2.72758 | + | 0.748555i | 2.99071 | + | 0.235901i | −2.38171 | + | 1.20098i |
See next 80 embeddings (of 992 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
128.l | odd | 32 | 1 | inner |
384.w | even | 32 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 384.2.w.a | ✓ | 992 |
3.b | odd | 2 | 1 | inner | 384.2.w.a | ✓ | 992 |
128.l | odd | 32 | 1 | inner | 384.2.w.a | ✓ | 992 |
384.w | even | 32 | 1 | inner | 384.2.w.a | ✓ | 992 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
384.2.w.a | ✓ | 992 | 1.a | even | 1 | 1 | trivial |
384.2.w.a | ✓ | 992 | 3.b | odd | 2 | 1 | inner |
384.2.w.a | ✓ | 992 | 128.l | odd | 32 | 1 | inner |
384.2.w.a | ✓ | 992 | 384.w | even | 32 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(384, [\chi])\).