Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,2,Mod(51,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("380.51");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.be (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: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
51.1 | −1.41417 | − | 0.0114942i | 2.22536 | + | 0.809964i | 1.99974 | + | 0.0325095i | 0.173648 | + | 0.984808i | −3.13772 | − | 1.17100i | −1.02771 | − | 0.593350i | −2.82759 | − | 0.0689592i | 1.99805 | + | 1.67656i | −0.234248 | − | 1.39468i |
51.2 | −1.36060 | − | 0.385715i | 0.497127 | + | 0.180940i | 1.70245 | + | 1.04960i | 0.173648 | + | 0.984808i | −0.606599 | − | 0.437935i | −2.30321 | − | 1.32976i | −1.91150 | − | 2.08475i | −2.08374 | − | 1.74846i | 0.143590 | − | 1.40691i |
51.3 | −1.28569 | + | 0.589063i | −2.16384 | − | 0.787572i | 1.30601 | − | 1.51471i | 0.173648 | + | 0.984808i | 3.24596 | − | 0.262061i | −3.09267 | − | 1.78556i | −0.786867 | + | 2.71677i | 1.76379 | + | 1.48000i | −0.803372 | − | 1.16387i |
51.4 | −1.23339 | − | 0.691917i | −0.199361 | − | 0.0725616i | 1.04250 | + | 1.70681i | 0.173648 | + | 0.984808i | 0.195684 | + | 0.227438i | 3.46492 | + | 2.00047i | −0.104843 | − | 2.82648i | −2.26365 | − | 1.89943i | 0.467229 | − | 1.33480i |
51.5 | −1.04516 | − | 0.952704i | −2.85924 | − | 1.04068i | 0.184712 | + | 1.99145i | 0.173648 | + | 0.984808i | 1.99690 | + | 3.81168i | −0.0195893 | − | 0.0113099i | 1.70421 | − | 2.25736i | 4.79409 | + | 4.02272i | 0.756740 | − | 1.19472i |
51.6 | −0.845155 | + | 1.13389i | 0.191206 | + | 0.0695933i | −0.571427 | − | 1.91663i | 0.173648 | + | 0.984808i | −0.240510 | + | 0.157990i | −0.977956 | − | 0.564623i | 2.65620 | + | 0.971911i | −2.26642 | − | 1.90175i | −1.26343 | − | 0.635416i |
51.7 | −0.767584 | + | 1.18778i | 2.18403 | + | 0.794921i | −0.821629 | − | 1.82344i | 0.173648 | + | 0.984808i | −2.62061 | + | 1.98397i | 3.27406 | + | 1.89028i | 2.79651 | + | 0.423730i | 1.83994 | + | 1.54390i | −1.30302 | − | 0.549668i |
51.8 | −0.756740 | − | 1.19472i | 2.85924 | + | 1.04068i | −0.854689 | + | 1.80818i | 0.173648 | + | 0.984808i | −0.920386 | − | 4.20350i | 0.0195893 | + | 0.0113099i | 2.80703 | − | 0.347210i | 4.79409 | + | 4.02272i | 1.04516 | − | 0.952704i |
51.9 | −0.467229 | − | 1.33480i | 0.199361 | + | 0.0725616i | −1.56339 | + | 1.24732i | 0.173648 | + | 0.984808i | 0.00370794 | − | 0.300011i | −3.46492 | − | 2.00047i | 2.39538 | + | 1.50404i | −2.26365 | − | 1.89943i | 1.23339 | − | 0.691917i |
51.10 | −0.298614 | + | 1.38233i | −1.20014 | − | 0.436814i | −1.82166 | − | 0.825564i | 0.173648 | + | 0.984808i | 0.962198 | − | 1.52855i | −0.435003 | − | 0.251149i | 1.68517 | − | 2.27161i | −1.04861 | − | 0.879888i | −1.41318 | − | 0.0540382i |
51.11 | −0.143590 | − | 1.40691i | −0.497127 | − | 0.180940i | −1.95876 | + | 0.404034i | 0.173648 | + | 0.984808i | −0.183182 | + | 0.725392i | 2.30321 | + | 1.32976i | 0.849696 | + | 2.69778i | −2.08374 | − | 1.74846i | 1.36060 | − | 0.385715i |
51.12 | 0.234248 | − | 1.39468i | −2.22536 | − | 0.809964i | −1.89026 | − | 0.653401i | 0.173648 | + | 0.984808i | −1.65093 | + | 2.91393i | 1.02771 | + | 0.593350i | −1.35407 | + | 2.48324i | 1.99805 | + | 1.67656i | 1.41417 | − | 0.0114942i |
51.13 | 0.262366 | + | 1.38966i | 2.24675 | + | 0.817752i | −1.86233 | + | 0.729199i | 0.173648 | + | 0.984808i | −0.546929 | + | 3.33678i | −0.386990 | − | 0.223429i | −1.50195 | − | 2.39669i | 2.08105 | + | 1.74621i | −1.32299 | + | 0.499692i |
51.14 | 0.579400 | + | 1.29008i | −2.37127 | − | 0.863072i | −1.32859 | + | 1.49494i | 0.173648 | + | 0.984808i | −0.260486 | − | 3.55918i | 3.42776 | + | 1.97902i | −2.69837 | − | 0.847816i | 2.57989 | + | 2.16479i | −1.16986 | + | 0.794617i |
51.15 | 0.803372 | − | 1.16387i | 2.16384 | + | 0.787572i | −0.709186 | − | 1.87004i | 0.173648 | + | 0.984808i | 2.65500 | − | 1.88571i | 3.09267 | + | 1.78556i | −2.74623 | − | 0.676939i | 1.76379 | + | 1.48000i | 1.28569 | + | 0.589063i |
51.16 | 1.16986 | + | 0.794617i | 2.37127 | + | 0.863072i | 0.737168 | + | 1.85919i | 0.173648 | + | 0.984808i | 2.08825 | + | 2.89393i | −3.42776 | − | 1.97902i | −0.614956 | + | 2.76077i | 2.57989 | + | 2.16479i | −0.579400 | + | 1.29008i |
51.17 | 1.26343 | − | 0.635416i | −0.191206 | − | 0.0695933i | 1.19249 | − | 1.60560i | 0.173648 | + | 0.984808i | −0.285795 | + | 0.0335694i | 0.977956 | + | 0.564623i | 0.486400 | − | 2.78629i | −2.26642 | − | 1.90175i | 0.845155 | + | 1.13389i |
51.18 | 1.30302 | − | 0.549668i | −2.18403 | − | 0.794921i | 1.39573 | − | 1.43246i | 0.173648 | + | 0.984808i | −3.28278 | + | 0.164690i | −3.27406 | − | 1.89028i | 1.03129 | − | 2.63371i | 1.83994 | + | 1.54390i | 0.767584 | + | 1.18778i |
51.19 | 1.32299 | + | 0.499692i | −2.24675 | − | 0.817752i | 1.50062 | + | 1.32218i | 0.173648 | + | 0.984808i | −2.56381 | − | 2.20456i | 0.386990 | + | 0.223429i | 1.32462 | + | 2.49908i | 2.08105 | + | 1.74621i | −0.262366 | + | 1.38966i |
51.20 | 1.41318 | − | 0.0540382i | 1.20014 | + | 0.436814i | 1.99416 | − | 0.152732i | 0.173648 | + | 0.984808i | 1.71962 | + | 0.552444i | 0.435003 | + | 0.251149i | 2.80985 | − | 0.323598i | −1.04861 | − | 0.879888i | 0.298614 | + | 1.38233i |
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.f | odd | 18 | 1 | inner |
76.k | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.be.a | ✓ | 120 |
4.b | odd | 2 | 1 | inner | 380.2.be.a | ✓ | 120 |
19.f | odd | 18 | 1 | inner | 380.2.be.a | ✓ | 120 |
76.k | even | 18 | 1 | inner | 380.2.be.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.be.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
380.2.be.a | ✓ | 120 | 4.b | odd | 2 | 1 | inner |
380.2.be.a | ✓ | 120 | 19.f | odd | 18 | 1 | inner |
380.2.be.a | ✓ | 120 | 76.k | even | 18 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} - 3 T_{3}^{118} + 48 T_{3}^{116} + 1856 T_{3}^{114} - 7803 T_{3}^{112} + 75999 T_{3}^{110} + \cdots + 7676563456 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).