Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [418,2,Mod(23,418)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(418, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("418.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 418.j (of order \(9\), 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: | \(3.33774680449\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | 0.939693 | + | 0.342020i | −0.506748 | − | 2.87391i | 0.766044 | + | 0.642788i | −1.93343 | + | 1.62234i | 0.506748 | − | 2.87391i | 2.38935 | − | 4.13848i | 0.500000 | + | 0.866025i | −5.18349 | + | 1.88664i | −2.37170 | + | 0.863229i |
23.2 | 0.939693 | + | 0.342020i | −0.315286 | − | 1.78808i | 0.766044 | + | 0.642788i | 2.76685 | − | 2.32166i | 0.315286 | − | 1.78808i | −0.752408 | + | 1.30321i | 0.500000 | + | 0.866025i | −0.278741 | + | 0.101453i | 3.39404 | − | 1.23533i |
23.3 | 0.939693 | + | 0.342020i | 0.0304860 | + | 0.172895i | 0.766044 | + | 0.642788i | −0.190742 | + | 0.160051i | −0.0304860 | + | 0.172895i | 1.48347 | − | 2.56944i | 0.500000 | + | 0.866025i | 2.79011 | − | 1.01552i | −0.233979 | + | 0.0851616i |
23.4 | 0.939693 | + | 0.342020i | 0.102554 | + | 0.581612i | 0.766044 | + | 0.642788i | −1.23439 | + | 1.03577i | −0.102554 | + | 0.581612i | −1.96792 | + | 3.40854i | 0.500000 | + | 0.866025i | 2.49132 | − | 0.906767i | −1.51420 | + | 0.551123i |
23.5 | 0.939693 | + | 0.342020i | 0.515346 | + | 2.92267i | 0.766044 | + | 0.642788i | 2.88984 | − | 2.42486i | −0.515346 | + | 2.92267i | −1.44675 | + | 2.50585i | 0.500000 | + | 0.866025i | −5.45736 | + | 1.98632i | 3.54491 | − | 1.29024i |
111.1 | −0.766044 | − | 0.642788i | −3.00494 | + | 1.09371i | 0.173648 | + | 0.984808i | −0.0619925 | + | 0.351577i | 3.00494 | + | 1.09371i | −2.21848 | − | 3.84252i | 0.500000 | − | 0.866025i | 5.53533 | − | 4.64469i | 0.273478 | − | 0.229475i |
111.2 | −0.766044 | − | 0.642788i | −0.305857 | + | 0.111323i | 0.173648 | + | 0.984808i | 0.749682 | − | 4.25166i | 0.305857 | + | 0.111323i | −1.19826 | − | 2.07545i | 0.500000 | − | 0.866025i | −2.21698 | + | 1.86027i | −3.30720 | + | 2.77507i |
111.3 | −0.766044 | − | 0.642788i | 0.0916892 | − | 0.0333721i | 0.173648 | + | 0.984808i | 0.327930 | − | 1.85978i | −0.0916892 | − | 0.0333721i | 1.90575 | + | 3.30086i | 0.500000 | − | 0.866025i | −2.29084 | + | 1.92224i | −1.44665 | + | 1.21389i |
111.4 | −0.766044 | − | 0.642788i | 1.46132 | − | 0.531876i | 0.173648 | + | 0.984808i | −0.712779 | + | 4.04237i | −1.46132 | − | 0.531876i | −0.291586 | − | 0.505042i | 0.500000 | − | 0.866025i | −0.445574 | + | 0.373881i | 3.14441 | − | 2.63847i |
111.5 | −0.766044 | − | 0.642788i | 2.69748 | − | 0.981803i | 0.173648 | + | 0.984808i | 0.218104 | − | 1.23693i | −2.69748 | − | 0.981803i | −0.789822 | − | 1.36801i | 0.500000 | − | 0.866025i | 4.01433 | − | 3.36842i | −0.962162 | + | 0.807349i |
177.1 | −0.766044 | + | 0.642788i | −3.00494 | − | 1.09371i | 0.173648 | − | 0.984808i | −0.0619925 | − | 0.351577i | 3.00494 | − | 1.09371i | −2.21848 | + | 3.84252i | 0.500000 | + | 0.866025i | 5.53533 | + | 4.64469i | 0.273478 | + | 0.229475i |
177.2 | −0.766044 | + | 0.642788i | −0.305857 | − | 0.111323i | 0.173648 | − | 0.984808i | 0.749682 | + | 4.25166i | 0.305857 | − | 0.111323i | −1.19826 | + | 2.07545i | 0.500000 | + | 0.866025i | −2.21698 | − | 1.86027i | −3.30720 | − | 2.77507i |
177.3 | −0.766044 | + | 0.642788i | 0.0916892 | + | 0.0333721i | 0.173648 | − | 0.984808i | 0.327930 | + | 1.85978i | −0.0916892 | + | 0.0333721i | 1.90575 | − | 3.30086i | 0.500000 | + | 0.866025i | −2.29084 | − | 1.92224i | −1.44665 | − | 1.21389i |
177.4 | −0.766044 | + | 0.642788i | 1.46132 | + | 0.531876i | 0.173648 | − | 0.984808i | −0.712779 | − | 4.04237i | −1.46132 | + | 0.531876i | −0.291586 | + | 0.505042i | 0.500000 | + | 0.866025i | −0.445574 | − | 0.373881i | 3.14441 | + | 2.63847i |
177.5 | −0.766044 | + | 0.642788i | 2.69748 | + | 0.981803i | 0.173648 | − | 0.984808i | 0.218104 | + | 1.23693i | −2.69748 | + | 0.981803i | −0.789822 | + | 1.36801i | 0.500000 | + | 0.866025i | 4.01433 | + | 3.36842i | −0.962162 | − | 0.807349i |
199.1 | −0.173648 | − | 0.984808i | −2.04235 | + | 1.71374i | −0.939693 | + | 0.342020i | −3.54102 | − | 1.28883i | 2.04235 | + | 1.71374i | 0.168894 | − | 0.292534i | 0.500000 | + | 0.866025i | 0.713363 | − | 4.04568i | −0.654355 | + | 3.71103i |
199.2 | −0.173648 | − | 0.984808i | −1.95711 | + | 1.64221i | −0.939693 | + | 0.342020i | 0.968941 | + | 0.352666i | 1.95711 | + | 1.64221i | −1.38692 | + | 2.40222i | 0.500000 | + | 0.866025i | 0.612477 | − | 3.47353i | 0.179053 | − | 1.01546i |
199.3 | −0.173648 | − | 0.984808i | 0.345094 | − | 0.289568i | −0.939693 | + | 0.342020i | 1.87337 | + | 0.681850i | −0.345094 | − | 0.289568i | −1.54380 | + | 2.67395i | 0.500000 | + | 0.866025i | −0.485704 | + | 2.75457i | 0.346185 | − | 1.96331i |
199.4 | −0.173648 | − | 0.984808i | 1.31855 | − | 1.10640i | −0.939693 | + | 0.342020i | −3.37653 | − | 1.22895i | −1.31855 | − | 1.10640i | −1.75103 | + | 3.03286i | 0.500000 | + | 0.866025i | −0.00647669 | + | 0.0367312i | −0.623957 | + | 3.53863i |
199.5 | −0.173648 | − | 0.984808i | 1.56977 | − | 1.31719i | −0.939693 | + | 0.342020i | 1.25616 | + | 0.457206i | −1.56977 | − | 1.31719i | 1.39952 | − | 2.42403i | 0.500000 | + | 0.866025i | 0.208231 | − | 1.18093i | 0.232130 | − | 1.31647i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 418.2.j.c | ✓ | 30 |
19.e | even | 9 | 1 | inner | 418.2.j.c | ✓ | 30 |
19.e | even | 9 | 1 | 7942.2.a.bz | 15 | ||
19.f | odd | 18 | 1 | 7942.2.a.cb | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
418.2.j.c | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
418.2.j.c | ✓ | 30 | 19.e | even | 9 | 1 | inner |
7942.2.a.bz | 15 | 19.e | even | 9 | 1 | ||
7942.2.a.cb | 15 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 7 T_{3}^{27} - 21 T_{3}^{26} - 81 T_{3}^{25} + 644 T_{3}^{24} - 129 T_{3}^{23} + 3042 T_{3}^{22} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).