Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [402,2,Mod(5,402)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(402, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("402.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 402 = 2 \cdot 3 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 402.j (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: | \(3.20998616126\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(12\) 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 | 0.654861 | − | 0.755750i | −1.71690 | + | 0.228591i | −0.142315 | − | 0.989821i | 0.242870 | − | 0.0713131i | −0.951573 | + | 1.44724i | −2.47201 | − | 2.14201i | −0.841254 | − | 0.540641i | 2.89549 | − | 0.784935i | 0.105151 | − | 0.230249i |
5.2 | 0.654861 | − | 0.755750i | −1.50108 | − | 0.864158i | −0.142315 | − | 0.989821i | 2.06705 | − | 0.606940i | −1.63608 | + | 0.568535i | 2.70814 | + | 2.34662i | −0.841254 | − | 0.540641i | 1.50646 | + | 2.59433i | 0.894933 | − | 1.95963i |
5.3 | 0.654861 | − | 0.755750i | −1.35285 | + | 1.08158i | −0.142315 | − | 0.989821i | 0.740480 | − | 0.217425i | −0.0685239 | + | 1.73069i | 1.96535 | + | 1.70298i | −0.841254 | − | 0.540641i | 0.660382 | − | 2.92641i | 0.320593 | − | 0.702000i |
5.4 | 0.654861 | − | 0.755750i | −1.14781 | − | 1.29712i | −0.142315 | − | 0.989821i | −2.61258 | + | 0.767123i | −1.73196 | + | 0.0180248i | 1.12824 | + | 0.977627i | −0.841254 | − | 0.540641i | −0.365051 | + | 2.97771i | −1.13112 | + | 2.47681i |
5.5 | 0.654861 | − | 0.755750i | −0.840609 | + | 1.51439i | −0.142315 | − | 0.989821i | −2.77179 | + | 0.813870i | 0.594018 | + | 1.62700i | −0.377481 | − | 0.327089i | −0.841254 | − | 0.540641i | −1.58675 | − | 2.54602i | −1.20005 | + | 2.62775i |
5.6 | 0.654861 | − | 0.755750i | −0.482917 | − | 1.66337i | −0.142315 | − | 0.989821i | 2.50946 | − | 0.736845i | −1.57333 | − | 0.724309i | −2.21975 | − | 1.92343i | −0.841254 | − | 0.540641i | −2.53358 | + | 1.60654i | 1.08648 | − | 2.37906i |
5.7 | 0.654861 | − | 0.755750i | 0.812042 | + | 1.52990i | −0.142315 | − | 0.989821i | 4.19490 | − | 1.23173i | 1.68799 | + | 0.388170i | 2.39898 | + | 2.07873i | −0.841254 | − | 0.540641i | −1.68118 | + | 2.48468i | 1.81619 | − | 3.97690i |
5.8 | 0.654861 | − | 0.755750i | 1.12510 | − | 1.31687i | −0.142315 | − | 0.989821i | −0.167968 | + | 0.0493200i | −0.258445 | − | 1.71266i | 3.46402 | + | 3.00159i | −0.841254 | − | 0.540641i | −0.468312 | − | 2.96322i | −0.0727223 | + | 0.159240i |
5.9 | 0.654861 | − | 0.755750i | 1.24596 | − | 1.20316i | −0.142315 | − | 0.989821i | 2.22460 | − | 0.653202i | −0.0933572 | − | 1.72953i | −0.639153 | − | 0.553829i | −0.841254 | − | 0.540641i | 0.104822 | − | 2.99817i | 0.963147 | − | 2.10900i |
5.10 | 0.654861 | − | 0.755750i | 1.30634 | + | 1.13731i | −0.142315 | − | 0.989821i | −3.41108 | + | 1.00158i | 1.71499 | − | 0.242485i | 3.24167 | + | 2.80892i | −0.841254 | − | 0.540641i | 0.413047 | + | 2.97143i | −1.47684 | + | 3.23382i |
5.11 | 0.654861 | − | 0.755750i | 1.32337 | − | 1.11744i | −0.142315 | − | 0.989821i | −3.81422 | + | 1.11996i | 0.0221185 | − | 1.73191i | −2.93792 | − | 2.54572i | −0.841254 | − | 0.540641i | 0.502640 | − | 2.95759i | −1.65138 | + | 3.61601i |
5.12 | 0.654861 | − | 0.755750i | 1.64476 | + | 0.542907i | −0.142315 | − | 0.989821i | 0.798282 | − | 0.234397i | 1.48739 | − | 0.887502i | −1.77174 | − | 1.53522i | −0.841254 | − | 0.540641i | 2.41050 | + | 1.78591i | 0.345618 | − | 0.756798i |
53.1 | −0.415415 | − | 0.909632i | −1.70193 | − | 0.321604i | −0.654861 | + | 0.755750i | −0.487223 | − | 3.38871i | 0.414466 | + | 1.68173i | 2.37486 | − | 1.08456i | 0.959493 | + | 0.281733i | 2.79314 | + | 1.09470i | −2.88008 | + | 1.85091i |
53.2 | −0.415415 | − | 0.909632i | −1.57028 | + | 0.730902i | −0.654861 | + | 0.755750i | −0.0640359 | − | 0.445379i | 1.31717 | + | 1.12475i | −1.48804 | + | 0.679564i | 0.959493 | + | 0.281733i | 1.93156 | − | 2.29544i | −0.378530 | + | 0.243266i |
53.3 | −0.415415 | − | 0.909632i | −1.09017 | + | 1.34593i | −0.654861 | + | 0.755750i | 0.621342 | + | 4.32153i | 1.67717 | + | 0.432535i | 0.522788 | − | 0.238749i | 0.959493 | + | 0.281733i | −0.623054 | − | 2.93459i | 3.67289 | − | 2.36042i |
53.4 | −0.415415 | − | 0.909632i | −1.06328 | − | 1.36728i | −0.654861 | + | 0.755750i | −0.0682677 | − | 0.474812i | −0.802016 | + | 1.53518i | −2.74677 | + | 1.25441i | 0.959493 | + | 0.281733i | −0.738882 | + | 2.90759i | −0.403545 | + | 0.259343i |
53.5 | −0.415415 | − | 0.909632i | −0.891192 | − | 1.48519i | −0.654861 | + | 0.755750i | 0.275987 | + | 1.91953i | −0.980758 | + | 1.42763i | 3.39983 | − | 1.55265i | 0.959493 | + | 0.281733i | −1.41155 | + | 2.64717i | 1.63142 | − | 1.04845i |
53.6 | −0.415415 | − | 0.909632i | 0.158062 | + | 1.72482i | −0.654861 | + | 0.755750i | −0.574881 | − | 3.99839i | 1.50329 | − | 0.860296i | −3.24959 | + | 1.48404i | 0.959493 | + | 0.281733i | −2.95003 | + | 0.545259i | −3.39825 | + | 2.18392i |
53.7 | −0.415415 | − | 0.909632i | 0.227861 | − | 1.71700i | −0.654861 | + | 0.755750i | 0.280377 | + | 1.95006i | −1.65649 | + | 0.505997i | 1.38208 | − | 0.631177i | 0.959493 | + | 0.281733i | −2.89616 | − | 0.782473i | 1.65737 | − | 1.06512i |
53.8 | −0.415415 | − | 0.909632i | 1.06320 | + | 1.36734i | −0.654861 | + | 0.755750i | 0.294281 | + | 2.04677i | 0.802103 | − | 1.53513i | −0.870333 | + | 0.397468i | 0.959493 | + | 0.281733i | −0.739213 | + | 2.90750i | 1.73956 | − | 1.11795i |
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
201.j | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 402.2.j.b | yes | 120 |
3.b | odd | 2 | 1 | 402.2.j.a | ✓ | 120 | |
67.f | odd | 22 | 1 | 402.2.j.a | ✓ | 120 | |
201.j | even | 22 | 1 | inner | 402.2.j.b | yes | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
402.2.j.a | ✓ | 120 | 3.b | odd | 2 | 1 | |
402.2.j.a | ✓ | 120 | 67.f | odd | 22 | 1 | |
402.2.j.b | yes | 120 | 1.a | even | 1 | 1 | trivial |
402.2.j.b | yes | 120 | 201.j | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{120} + 42 T_{5}^{118} + 12 T_{5}^{117} + 747 T_{5}^{116} + 536 T_{5}^{115} + \cdots + 15\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\).