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: | \(24\) |
Relative dimension: | \(4\) 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 | −0.293621 | − | 1.66521i | 0.766044 | + | 0.642788i | −3.02184 | + | 2.53563i | 0.293621 | − | 1.66521i | −1.16294 | + | 2.01427i | 0.500000 | + | 0.866025i | 0.132366 | − | 0.0481775i | −3.70684 | + | 1.34918i |
23.2 | 0.939693 | + | 0.342020i | −0.232600 | − | 1.31914i | 0.766044 | + | 0.642788i | 0.527700 | − | 0.442792i | 0.232600 | − | 1.31914i | 0.106944 | − | 0.185233i | 0.500000 | + | 0.866025i | 1.13305 | − | 0.412397i | 0.647319 | − | 0.235605i |
23.3 | 0.939693 | + | 0.342020i | 0.291741 | + | 1.65454i | 0.766044 | + | 0.642788i | 1.53274 | − | 1.28612i | −0.291741 | + | 1.65454i | 1.45368 | − | 2.51785i | 0.500000 | + | 0.866025i | 0.166674 | − | 0.0606642i | 1.88018 | − | 0.684329i |
23.4 | 0.939693 | + | 0.342020i | 0.408129 | + | 2.31461i | 0.766044 | + | 0.642788i | −1.33673 | + | 1.12165i | −0.408129 | + | 2.31461i | −0.571336 | + | 0.989583i | 0.500000 | + | 0.866025i | −2.37179 | + | 0.863259i | −1.63974 | + | 0.596815i |
111.1 | −0.766044 | − | 0.642788i | −2.02265 | + | 0.736184i | 0.173648 | + | 0.984808i | −0.505679 | + | 2.86785i | 2.02265 | + | 0.736184i | 1.62478 | + | 2.81421i | 0.500000 | − | 0.866025i | 1.25101 | − | 1.04972i | 2.23079 | − | 1.87186i |
111.2 | −0.766044 | − | 0.642788i | −1.50994 | + | 0.549572i | 0.173648 | + | 0.984808i | 0.142998 | − | 0.810982i | 1.50994 | + | 0.549572i | −0.0412325 | − | 0.0714168i | 0.500000 | − | 0.866025i | −0.320257 | + | 0.268728i | −0.630832 | + | 0.529331i |
111.3 | −0.766044 | − | 0.642788i | 0.361908 | − | 0.131724i | 0.173648 | + | 0.984808i | −0.0315845 | + | 0.179124i | −0.361908 | − | 0.131724i | −2.25138 | − | 3.89951i | 0.500000 | − | 0.866025i | −2.18451 | + | 1.83302i | 0.139334 | − | 0.116915i |
111.4 | −0.766044 | − | 0.642788i | 2.23098 | − | 0.812012i | 0.173648 | + | 0.984808i | −0.126679 | + | 0.718430i | −2.23098 | − | 0.812012i | 1.60753 | + | 2.78432i | 0.500000 | − | 0.866025i | 2.01980 | − | 1.69481i | 0.558839 | − | 0.468922i |
177.1 | −0.766044 | + | 0.642788i | −2.02265 | − | 0.736184i | 0.173648 | − | 0.984808i | −0.505679 | − | 2.86785i | 2.02265 | − | 0.736184i | 1.62478 | − | 2.81421i | 0.500000 | + | 0.866025i | 1.25101 | + | 1.04972i | 2.23079 | + | 1.87186i |
177.2 | −0.766044 | + | 0.642788i | −1.50994 | − | 0.549572i | 0.173648 | − | 0.984808i | 0.142998 | + | 0.810982i | 1.50994 | − | 0.549572i | −0.0412325 | + | 0.0714168i | 0.500000 | + | 0.866025i | −0.320257 | − | 0.268728i | −0.630832 | − | 0.529331i |
177.3 | −0.766044 | + | 0.642788i | 0.361908 | + | 0.131724i | 0.173648 | − | 0.984808i | −0.0315845 | − | 0.179124i | −0.361908 | + | 0.131724i | −2.25138 | + | 3.89951i | 0.500000 | + | 0.866025i | −2.18451 | − | 1.83302i | 0.139334 | + | 0.116915i |
177.4 | −0.766044 | + | 0.642788i | 2.23098 | + | 0.812012i | 0.173648 | − | 0.984808i | −0.126679 | − | 0.718430i | −2.23098 | + | 0.812012i | 1.60753 | − | 2.78432i | 0.500000 | + | 0.866025i | 2.01980 | + | 1.69481i | 0.558839 | + | 0.468922i |
199.1 | −0.173648 | − | 0.984808i | −1.23590 | + | 1.03704i | −0.939693 | + | 0.342020i | −0.846117 | − | 0.307961i | 1.23590 | + | 1.03704i | 1.44710 | − | 2.50644i | 0.500000 | + | 0.866025i | −0.0689564 | + | 0.391071i | −0.156356 | + | 0.886739i |
199.2 | −0.173648 | − | 0.984808i | −0.404607 | + | 0.339506i | −0.939693 | + | 0.342020i | 2.97940 | + | 1.08441i | 0.404607 | + | 0.339506i | 0.0238226 | − | 0.0412620i | 0.500000 | + | 0.866025i | −0.472502 | + | 2.67969i | 0.550571 | − | 3.12244i |
199.3 | −0.173648 | − | 0.984808i | −0.0299784 | + | 0.0251548i | −0.939693 | + | 0.342020i | −2.55820 | − | 0.931110i | 0.0299784 | + | 0.0251548i | −0.483905 | + | 0.838147i | 0.500000 | + | 0.866025i | −0.520679 | + | 2.95292i | −0.472737 | + | 2.68103i |
199.4 | −0.173648 | − | 0.984808i | 2.43653 | − | 2.04449i | −0.939693 | + | 0.342020i | 3.24400 | + | 1.18072i | −2.43653 | − | 2.04449i | −1.75306 | + | 3.03639i | 0.500000 | + | 0.866025i | 1.23578 | − | 7.00848i | 0.599467 | − | 3.39975i |
309.1 | 0.939693 | − | 0.342020i | −0.293621 | + | 1.66521i | 0.766044 | − | 0.642788i | −3.02184 | − | 2.53563i | 0.293621 | + | 1.66521i | −1.16294 | − | 2.01427i | 0.500000 | − | 0.866025i | 0.132366 | + | 0.0481775i | −3.70684 | − | 1.34918i |
309.2 | 0.939693 | − | 0.342020i | −0.232600 | + | 1.31914i | 0.766044 | − | 0.642788i | 0.527700 | + | 0.442792i | 0.232600 | + | 1.31914i | 0.106944 | + | 0.185233i | 0.500000 | − | 0.866025i | 1.13305 | + | 0.412397i | 0.647319 | + | 0.235605i |
309.3 | 0.939693 | − | 0.342020i | 0.291741 | − | 1.65454i | 0.766044 | − | 0.642788i | 1.53274 | + | 1.28612i | −0.291741 | − | 1.65454i | 1.45368 | + | 2.51785i | 0.500000 | − | 0.866025i | 0.166674 | + | 0.0606642i | 1.88018 | + | 0.684329i |
309.4 | 0.939693 | − | 0.342020i | 0.408129 | − | 2.31461i | 0.766044 | − | 0.642788i | −1.33673 | − | 1.12165i | −0.408129 | − | 2.31461i | −0.571336 | − | 0.989583i | 0.500000 | − | 0.866025i | −2.37179 | − | 0.863259i | −1.63974 | − | 0.596815i |
See all 24 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.a | ✓ | 24 |
19.e | even | 9 | 1 | inner | 418.2.j.a | ✓ | 24 |
19.e | even | 9 | 1 | 7942.2.a.bt | 12 | ||
19.f | odd | 18 | 1 | 7942.2.a.bx | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
418.2.j.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
418.2.j.a | ✓ | 24 | 19.e | even | 9 | 1 | inner |
7942.2.a.bt | 12 | 19.e | even | 9 | 1 | ||
7942.2.a.bx | 12 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 13 T_{3}^{21} + 27 T_{3}^{20} + 99 T_{3}^{19} + 346 T_{3}^{18} + 333 T_{3}^{17} - 54 T_{3}^{16} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).