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 | −0.530846 | − | 3.01058i | 0.766044 | + | 0.642788i | −1.91059 | + | 1.60317i | −0.530846 | + | 3.01058i | 0.304262 | − | 0.526998i | −0.500000 | − | 0.866025i | −5.96271 | + | 2.17025i | 2.34368 | − | 0.853031i |
23.2 | −0.939693 | − | 0.342020i | −0.235710 | − | 1.33678i | 0.766044 | + | 0.642788i | 2.14622 | − | 1.80090i | −0.235710 | + | 1.33678i | 2.28653 | − | 3.96039i | −0.500000 | − | 0.866025i | 1.08766 | − | 0.395875i | −2.63273 | + | 0.958237i |
23.3 | −0.939693 | − | 0.342020i | 0.146051 | + | 0.828296i | 0.766044 | + | 0.642788i | −1.03325 | + | 0.867001i | 0.146051 | − | 0.828296i | 0.273610 | − | 0.473907i | −0.500000 | − | 0.866025i | 2.15433 | − | 0.784114i | 1.26747 | − | 0.461322i |
23.4 | −0.939693 | − | 0.342020i | 0.197309 | + | 1.11900i | 0.766044 | + | 0.642788i | 2.56288 | − | 2.15051i | 0.197309 | − | 1.11900i | −0.480180 | + | 0.831696i | −0.500000 | − | 0.866025i | 1.60586 | − | 0.584483i | −3.14383 | + | 1.14426i |
23.5 | −0.939693 | − | 0.342020i | 0.596844 | + | 3.38487i | 0.766044 | + | 0.642788i | 0.532872 | − | 0.447132i | 0.596844 | − | 3.38487i | −0.678489 | + | 1.17518i | −0.500000 | − | 0.866025i | −8.28207 | + | 3.01443i | −0.653664 | + | 0.237914i |
111.1 | 0.766044 | + | 0.642788i | −2.46726 | + | 0.898008i | 0.173648 | + | 0.984808i | 0.624693 | − | 3.54281i | −2.46726 | − | 0.898008i | 2.28982 | + | 3.96608i | −0.500000 | + | 0.866025i | 2.98280 | − | 2.50287i | 2.75582 | − | 2.31240i |
111.2 | 0.766044 | + | 0.642788i | −2.46577 | + | 0.897465i | 0.173648 | + | 0.984808i | −0.00995446 | + | 0.0564545i | −2.46577 | − | 0.897465i | −1.80146 | − | 3.12022i | −0.500000 | + | 0.866025i | 2.97642 | − | 2.49751i | −0.0439138 | + | 0.0368481i |
111.3 | 0.766044 | + | 0.642788i | −0.317071 | + | 0.115404i | 0.173648 | + | 0.984808i | −0.456329 | + | 2.58797i | −0.317071 | − | 0.115404i | 0.0977794 | + | 0.169359i | −0.500000 | + | 0.866025i | −2.21092 | + | 1.85518i | −2.01309 | + | 1.68918i |
111.4 | 0.766044 | + | 0.642788i | 1.64633 | − | 0.599214i | 0.173648 | + | 0.984808i | 0.477535 | − | 2.70824i | 1.64633 | + | 0.599214i | −1.91741 | − | 3.32106i | −0.500000 | + | 0.866025i | 0.0532024 | − | 0.0446421i | 2.10664 | − | 1.76768i |
111.5 | 0.766044 | + | 0.642788i | 2.66407 | − | 0.969643i | 0.173648 | + | 0.984808i | −0.115000 | + | 0.652197i | 2.66407 | + | 0.969643i | 0.738881 | + | 1.27978i | −0.500000 | + | 0.866025i | 3.85894 | − | 3.23803i | −0.507320 | + | 0.425692i |
177.1 | 0.766044 | − | 0.642788i | −2.46726 | − | 0.898008i | 0.173648 | − | 0.984808i | 0.624693 | + | 3.54281i | −2.46726 | + | 0.898008i | 2.28982 | − | 3.96608i | −0.500000 | − | 0.866025i | 2.98280 | + | 2.50287i | 2.75582 | + | 2.31240i |
177.2 | 0.766044 | − | 0.642788i | −2.46577 | − | 0.897465i | 0.173648 | − | 0.984808i | −0.00995446 | − | 0.0564545i | −2.46577 | + | 0.897465i | −1.80146 | + | 3.12022i | −0.500000 | − | 0.866025i | 2.97642 | + | 2.49751i | −0.0439138 | − | 0.0368481i |
177.3 | 0.766044 | − | 0.642788i | −0.317071 | − | 0.115404i | 0.173648 | − | 0.984808i | −0.456329 | − | 2.58797i | −0.317071 | + | 0.115404i | 0.0977794 | − | 0.169359i | −0.500000 | − | 0.866025i | −2.21092 | − | 1.85518i | −2.01309 | − | 1.68918i |
177.4 | 0.766044 | − | 0.642788i | 1.64633 | + | 0.599214i | 0.173648 | − | 0.984808i | 0.477535 | + | 2.70824i | 1.64633 | − | 0.599214i | −1.91741 | + | 3.32106i | −0.500000 | − | 0.866025i | 0.0532024 | + | 0.0446421i | 2.10664 | + | 1.76768i |
177.5 | 0.766044 | − | 0.642788i | 2.66407 | + | 0.969643i | 0.173648 | − | 0.984808i | −0.115000 | − | 0.652197i | 2.66407 | − | 0.969643i | 0.738881 | − | 1.27978i | −0.500000 | − | 0.866025i | 3.85894 | + | 3.23803i | −0.507320 | − | 0.425692i |
199.1 | 0.173648 | + | 0.984808i | −2.38321 | + | 1.99975i | −0.939693 | + | 0.342020i | −2.75425 | − | 1.00246i | −2.38321 | − | 1.99975i | −2.12923 | + | 3.68794i | −0.500000 | − | 0.866025i | 1.15974 | − | 6.57722i | 0.508964 | − | 2.88648i |
199.2 | 0.173648 | + | 0.984808i | −1.16564 | + | 0.978092i | −0.939693 | + | 0.342020i | 2.75856 | + | 1.00403i | −1.16564 | − | 0.978092i | 0.232407 | − | 0.402540i | −0.500000 | − | 0.866025i | −0.118881 | + | 0.674209i | −0.509761 | + | 2.89100i |
199.3 | 0.173648 | + | 0.984808i | 0.854877 | − | 0.717327i | −0.939693 | + | 0.342020i | −1.23363 | − | 0.449003i | 0.854877 | + | 0.717327i | −2.13725 | + | 3.70183i | −0.500000 | − | 0.866025i | −0.304688 | + | 1.72797i | 0.227965 | − | 1.29285i |
199.4 | 0.173648 | + | 0.984808i | 1.05344 | − | 0.883939i | −0.939693 | + | 0.342020i | 2.40688 | + | 0.876033i | 1.05344 | + | 0.883939i | 2.52622 | − | 4.37554i | −0.500000 | − | 0.866025i | −0.192562 | + | 1.09208i | −0.444774 | + | 2.52244i |
199.5 | 0.173648 | + | 0.984808i | 2.40658 | − | 2.01936i | −0.939693 | + | 0.342020i | −3.99665 | − | 1.45466i | 2.40658 | + | 2.01936i | 0.394516 | − | 0.683322i | −0.500000 | − | 0.866025i | 1.19287 | − | 6.76511i | 0.738551 | − | 4.18853i |
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.d | ✓ | 30 |
19.e | even | 9 | 1 | inner | 418.2.j.d | ✓ | 30 |
19.e | even | 9 | 1 | 7942.2.a.ca | 15 | ||
19.f | odd | 18 | 1 | 7942.2.a.by | 15 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
418.2.j.d | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
418.2.j.d | ✓ | 30 | 19.e | even | 9 | 1 | inner |
7942.2.a.by | 15 | 19.f | odd | 18 | 1 | ||
7942.2.a.ca | 15 | 19.e | even | 9 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + 7 T_{3}^{27} - 9 T_{3}^{26} - 63 T_{3}^{25} + 952 T_{3}^{24} + 735 T_{3}^{23} + \cdots + 12902464 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).