Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,2,Mod(50,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("471.50");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 471.l (of order \(12\), degree \(4\), 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: | \(3.76095393520\) |
Analytic rank: | \(0\) |
Dimension: | \(200\) |
Relative dimension: | \(50\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
50.1 | −1.96449 | + | 1.96449i | 1.24725 | − | 1.20182i | − | 5.71848i | 0.309034 | + | 1.15333i | −0.0892417 | + | 4.81118i | 0.541767 | − | 0.541767i | 7.30492 | + | 7.30492i | 0.111255 | − | 2.99794i | −2.87281 | − | 1.65862i | |
50.2 | −1.93791 | + | 1.93791i | 0.655068 | + | 1.60340i | − | 5.51098i | −1.00937 | − | 3.76703i | −4.37670 | − | 1.83778i | −0.357569 | + | 0.357569i | 6.80395 | + | 6.80395i | −2.14177 | + | 2.10067i | 9.25622 | + | 5.34408i | |
50.3 | −1.82955 | + | 1.82955i | −1.72845 | − | 0.111624i | − | 4.69449i | 0.00122739 | + | 0.00458067i | 3.36650 | − | 2.95806i | 3.09988 | − | 3.09988i | 4.92970 | + | 4.92970i | 2.97508 | + | 0.385872i | −0.0106261 | − | 0.00613499i | |
50.4 | −1.73825 | + | 1.73825i | 1.29972 | + | 1.14486i | − | 4.04305i | 0.867835 | + | 3.23881i | −4.24932 | + | 0.269185i | −2.31299 | + | 2.31299i | 3.55133 | + | 3.55133i | 0.378569 | + | 2.97602i | −7.13838 | − | 4.12135i | |
50.5 | −1.70954 | + | 1.70954i | −1.10161 | + | 1.33658i | − | 3.84504i | 0.205161 | + | 0.765672i | −0.401690 | − | 4.16818i | −1.76931 | + | 1.76931i | 3.15416 | + | 3.15416i | −0.572902 | − | 2.94479i | −1.65968 | − | 0.958215i | |
50.6 | −1.59659 | + | 1.59659i | −1.63950 | − | 0.558616i | − | 3.09822i | 0.704955 | + | 2.63093i | 3.50949 | − | 1.72573i | −1.39752 | + | 1.39752i | 1.75342 | + | 1.75342i | 2.37590 | + | 1.83170i | −5.32605 | − | 3.07500i | |
50.7 | −1.59610 | + | 1.59610i | −0.929471 | − | 1.46153i | − | 3.09507i | −1.03956 | − | 3.87969i | 3.81628 | + | 0.849227i | 1.69458 | − | 1.69458i | 1.74784 | + | 1.74784i | −1.27217 | + | 2.71691i | 7.85161 | + | 4.53313i | |
50.8 | −1.52089 | + | 1.52089i | −0.258431 | − | 1.71266i | − | 2.62620i | 0.679058 | + | 2.53428i | 2.99781 | + | 2.21172i | 1.37133 | − | 1.37133i | 0.952382 | + | 0.952382i | −2.86643 | + | 0.885212i | −4.88713 | − | 2.82159i | |
50.9 | −1.47058 | + | 1.47058i | 1.73106 | + | 0.0586068i | − | 2.32524i | −0.337319 | − | 1.25889i | −2.63185 | + | 2.45948i | 0.411952 | − | 0.411952i | 0.478287 | + | 0.478287i | 2.99313 | + | 0.202904i | 2.34736 | + | 1.35525i | |
50.10 | −1.43787 | + | 1.43787i | 1.07451 | − | 1.35846i | − | 2.13496i | −0.618816 | − | 2.30945i | 0.408280 | + | 3.49831i | −2.17897 | + | 2.17897i | 0.194052 | + | 0.194052i | −0.690836 | − | 2.91937i | 4.21048 | + | 2.43092i | |
50.11 | −1.33901 | + | 1.33901i | −0.825865 | + | 1.52248i | − | 1.58591i | −0.378714 | − | 1.41338i | −0.932777 | − | 3.14447i | 1.10359 | − | 1.10359i | −0.554467 | − | 0.554467i | −1.63589 | − | 2.51473i | 2.39964 | + | 1.38543i | |
50.12 | −1.19074 | + | 1.19074i | 0.685591 | + | 1.59059i | − | 0.835708i | 0.604819 | + | 2.25721i | −2.71033 | − | 1.07761i | 0.837234 | − | 0.837234i | −1.38637 | − | 1.38637i | −2.05993 | + | 2.18098i | −3.40793 | − | 1.96757i | |
50.13 | −1.17584 | + | 1.17584i | −1.72956 | + | 0.0927798i | − | 0.765185i | −0.933382 | − | 3.48343i | 1.92459 | − | 2.14278i | −2.60917 | + | 2.60917i | −1.45194 | − | 1.45194i | 2.98278 | − | 0.320937i | 5.19345 | + | 2.99844i | |
50.14 | −1.00506 | + | 1.00506i | 0.156162 | − | 1.72500i | − | 0.0202720i | 0.729968 | + | 2.72428i | 1.57677 | + | 1.89067i | −2.45817 | + | 2.45817i | −1.98974 | − | 1.98974i | −2.95123 | − | 0.538757i | −3.47171 | − | 2.00439i | |
50.15 | −0.980409 | + | 0.980409i | 1.60565 | − | 0.649523i | 0.0775977i | 0.853262 | + | 3.18442i | −0.937397 | + | 2.21099i | 2.96274 | − | 2.96274i | −2.03689 | − | 2.03689i | 2.15624 | − | 2.08582i | −3.95858 | − | 2.28549i | ||
50.16 | −0.850678 | + | 0.850678i | −1.36219 | − | 1.06978i | 0.552695i | 0.0315701 | + | 0.117821i | 2.06883 | − | 0.248744i | 0.641782 | − | 0.641782i | −2.17152 | − | 2.17152i | 0.711127 | + | 2.91450i | −0.127084 | − | 0.0733718i | ||
50.17 | −0.668208 | + | 0.668208i | 1.73131 | + | 0.0507295i | 1.10700i | 0.363693 | + | 1.35732i | −1.19077 | + | 1.12298i | −2.84247 | + | 2.84247i | −2.07612 | − | 2.07612i | 2.99485 | + | 0.175657i | −1.14999 | − | 0.663950i | ||
50.18 | −0.648600 | + | 0.648600i | 0.734999 | − | 1.56837i | 1.15864i | −0.523398 | − | 1.95335i | 0.540523 | + | 1.49396i | 3.32511 | − | 3.32511i | −2.04869 | − | 2.04869i | −1.91955 | − | 2.30550i | 1.60642 | + | 0.927465i | ||
50.19 | −0.640983 | + | 0.640983i | 0.167689 | + | 1.72391i | 1.17828i | 0.280265 | + | 1.04596i | −1.21249 | − | 0.997514i | 0.268331 | − | 0.268331i | −2.03722 | − | 2.03722i | −2.94376 | + | 0.578164i | −0.850090 | − | 0.490800i | ||
50.20 | −0.513152 | + | 0.513152i | −1.60345 | + | 0.654944i | 1.47335i | −0.0432502 | − | 0.161412i | 0.486727 | − | 1.15890i | 2.54991 | − | 2.54991i | −1.78236 | − | 1.78236i | 2.14210 | − | 2.10034i | 0.105023 | + | 0.0606349i | ||
See next 80 embeddings (of 200 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
157.f | odd | 12 | 1 | inner |
471.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 471.2.l.b | ✓ | 200 |
3.b | odd | 2 | 1 | inner | 471.2.l.b | ✓ | 200 |
157.f | odd | 12 | 1 | inner | 471.2.l.b | ✓ | 200 |
471.l | even | 12 | 1 | inner | 471.2.l.b | ✓ | 200 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
471.2.l.b | ✓ | 200 | 1.a | even | 1 | 1 | trivial |
471.2.l.b | ✓ | 200 | 3.b | odd | 2 | 1 | inner |
471.2.l.b | ✓ | 200 | 157.f | odd | 12 | 1 | inner |
471.2.l.b | ✓ | 200 | 471.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{200} + 764 T_{2}^{196} + 277846 T_{2}^{192} + 64040328 T_{2}^{188} + 10508630733 T_{2}^{184} + \cdots + 92\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\).