Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [138,2,Mod(5,138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 138.f (of order \(22\), degree \(10\), 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: | \(1.10193554789\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −0.281733 | + | 0.959493i | −1.72467 | + | 0.159749i | −0.841254 | − | 0.540641i | 0.591943 | − | 4.11706i | 0.332618 | − | 1.69981i | 0.656690 | − | 0.299900i | 0.755750 | − | 0.654861i | 2.94896 | − | 0.551026i | 3.78352 | + | 1.72787i |
5.2 | −0.281733 | + | 0.959493i | −1.37386 | − | 1.05476i | −0.841254 | − | 0.540641i | −0.493501 | + | 3.43237i | 1.39909 | − | 1.02105i | −1.02427 | + | 0.467769i | 0.755750 | − | 0.654861i | 0.774979 | + | 2.89817i | −3.15430 | − | 1.44052i |
5.3 | −0.281733 | + | 0.959493i | 0.583374 | − | 1.63085i | −0.841254 | − | 0.540641i | 0.140990 | − | 0.980604i | 1.40043 | + | 1.01921i | 1.30276 | − | 0.594948i | 0.755750 | − | 0.654861i | −2.31935 | − | 1.90279i | 0.901161 | + | 0.411546i |
5.4 | −0.281733 | + | 0.959493i | 1.60896 | + | 0.641290i | −0.841254 | − | 0.540641i | −0.239432 | + | 1.66529i | −1.06861 | + | 1.36311i | −0.935173 | + | 0.427079i | 0.755750 | − | 0.654861i | 2.17749 | + | 2.06362i | −1.53038 | − | 0.698899i |
5.5 | 0.281733 | − | 0.959493i | −0.863742 | − | 1.50132i | −0.841254 | − | 0.540641i | 0.239432 | − | 1.66529i | −1.68385 | + | 0.405784i | −0.935173 | + | 0.427079i | −0.755750 | + | 0.654861i | −1.50790 | + | 2.59350i | −1.53038 | − | 0.698899i |
5.6 | 0.281733 | − | 0.959493i | 0.0873233 | + | 1.72985i | −0.841254 | − | 0.540641i | −0.591943 | + | 4.11706i | 1.68438 | + | 0.403568i | 0.656690 | − | 0.299900i | −0.755750 | + | 0.654861i | −2.98475 | + | 0.302112i | 3.78352 | + | 1.72787i |
5.7 | 0.281733 | − | 0.959493i | 1.23954 | + | 1.20977i | −0.841254 | − | 0.540641i | 0.493501 | − | 3.43237i | 1.50998 | − | 0.848500i | −1.02427 | + | 0.467769i | −0.755750 | + | 0.654861i | 0.0729229 | + | 2.99911i | −3.15430 | − | 1.44052i |
5.8 | 0.281733 | − | 0.959493i | 1.53123 | − | 0.809531i | −0.841254 | − | 0.540641i | −0.140990 | + | 0.980604i | −0.345342 | − | 1.69727i | 1.30276 | − | 0.594948i | −0.755750 | + | 0.654861i | 1.68932 | − | 2.47915i | 0.901161 | + | 0.411546i |
11.1 | −0.540641 | − | 0.841254i | −1.59667 | + | 0.671295i | −0.415415 | + | 0.909632i | 1.80115 | − | 0.528864i | 1.42796 | + | 0.980277i | 1.49075 | − | 1.29175i | 0.989821 | − | 0.142315i | 2.09873 | − | 2.14368i | −1.41868 | − | 1.22930i |
11.2 | −0.540641 | − | 0.841254i | 0.0700914 | + | 1.73063i | −0.415415 | + | 0.909632i | −3.21268 | + | 0.943327i | 1.41801 | − | 0.994615i | −1.80758 | + | 1.56628i | 0.989821 | − | 0.142315i | −2.99017 | + | 0.242605i | 2.53048 | + | 2.19268i |
11.3 | −0.540641 | − | 0.841254i | 1.29667 | − | 1.14833i | −0.415415 | + | 0.909632i | −1.43749 | + | 0.422085i | −1.66706 | − | 0.469995i | 3.34037 | − | 2.89445i | 0.989821 | − | 0.142315i | 0.362698 | − | 2.97799i | 1.13225 | + | 0.981098i |
11.4 | −0.540641 | − | 0.841254i | 1.70890 | + | 0.282234i | −0.415415 | + | 0.909632i | 2.84902 | − | 0.836549i | −0.686471 | − | 1.59021i | −3.02354 | + | 2.61991i | 0.989821 | − | 0.142315i | 2.84069 | + | 0.964621i | −2.24405 | − | 1.94448i |
11.5 | 0.540641 | + | 0.841254i | −1.71919 | − | 0.210651i | −0.415415 | + | 0.909632i | −2.84902 | + | 0.836549i | −0.752255 | − | 1.56016i | −3.02354 | + | 2.61991i | −0.989821 | + | 0.142315i | 2.91125 | + | 0.724301i | −2.24405 | − | 1.94448i |
11.6 | 0.540641 | + | 0.841254i | −0.920624 | − | 1.46712i | −0.415415 | + | 0.909632i | 1.43749 | − | 0.422085i | 0.736496 | − | 1.56766i | 3.34037 | − | 2.89445i | −0.989821 | + | 0.142315i | −1.30490 | + | 2.70134i | 1.13225 | + | 0.981098i |
11.7 | 0.540641 | + | 0.841254i | −0.554828 | + | 1.64078i | −0.415415 | + | 0.909632i | 3.21268 | − | 0.943327i | −1.68028 | + | 0.420323i | −1.80758 | + | 1.56628i | −0.989821 | + | 0.142315i | −2.38433 | − | 1.82070i | 2.53048 | + | 2.19268i |
11.8 | 0.540641 | + | 0.841254i | 1.34287 | + | 1.09394i | −0.415415 | + | 0.909632i | −1.80115 | + | 0.528864i | −0.194269 | + | 1.72112i | 1.49075 | − | 1.29175i | −0.989821 | + | 0.142315i | 0.606601 | + | 2.93803i | −1.41868 | − | 1.22930i |
17.1 | −0.909632 | − | 0.415415i | −1.72827 | − | 0.114314i | 0.654861 | + | 0.755750i | −1.19773 | + | 0.769732i | 1.52461 | + | 0.821935i | 3.68797 | + | 0.530250i | −0.281733 | − | 0.959493i | 2.97386 | + | 0.395131i | 1.40925 | − | 0.202619i |
17.2 | −0.909632 | − | 0.415415i | −0.580581 | + | 1.63185i | 0.654861 | + | 0.755750i | −1.63636 | + | 1.05163i | 1.20601 | − | 1.24320i | −3.52778 | − | 0.507219i | −0.281733 | − | 0.959493i | −2.32585 | − | 1.89484i | 1.92535 | − | 0.276823i |
17.3 | −0.909632 | − | 0.415415i | 0.322552 | − | 1.70175i | 0.654861 | + | 0.755750i | 1.12015 | − | 0.719874i | −1.00034 | + | 1.41398i | 1.49094 | + | 0.214365i | −0.281733 | − | 0.959493i | −2.79192 | − | 1.09781i | −1.31797 | + | 0.189495i |
17.4 | −0.909632 | − | 0.415415i | 1.47913 | + | 0.901205i | 0.654861 | + | 0.755750i | 1.71394 | − | 1.10148i | −0.971091 | − | 1.43422i | −1.65113 | − | 0.237396i | −0.281733 | − | 0.959493i | 1.37566 | + | 2.66600i | −2.01663 | + | 0.289948i |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 138.2.f.a | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 138.2.f.a | ✓ | 80 |
23.d | odd | 22 | 1 | inner | 138.2.f.a | ✓ | 80 |
69.g | even | 22 | 1 | inner | 138.2.f.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.2.f.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
138.2.f.a | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
138.2.f.a | ✓ | 80 | 23.d | odd | 22 | 1 | inner |
138.2.f.a | ✓ | 80 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(138, [\chi])\).