Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [927,2,Mod(46,927)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(927, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("927.46");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 927 = 3^{2} \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 927.f (of order \(3\), degree \(2\), 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: | \(7.40213226737\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 | −1.35501 | − | 2.34695i | 0 | −2.67212 | + | 4.62824i | 1.61886 | − | 2.80395i | 0 | −0.805098 | + | 1.39447i | 9.06295 | 0 | −8.77431 | ||||||||||
46.2 | −1.24321 | − | 2.15331i | 0 | −2.09117 | + | 3.62201i | −1.45965 | + | 2.52819i | 0 | 1.35416 | − | 2.34547i | 5.42622 | 0 | 7.25864 | ||||||||||
46.3 | −0.980806 | − | 1.69881i | 0 | −0.923962 | + | 1.60035i | 0.348567 | − | 0.603736i | 0 | −1.12861 | + | 1.95480i | −0.298316 | 0 | −1.36751 | ||||||||||
46.4 | −0.959463 | − | 1.66184i | 0 | −0.841140 | + | 1.45690i | 1.43954 | − | 2.49335i | 0 | 2.39836 | − | 4.15408i | −0.609683 | 0 | −5.52473 | ||||||||||
46.5 | −0.711803 | − | 1.23288i | 0 | −0.0133274 | + | 0.0230838i | 0.396502 | − | 0.686762i | 0 | −0.249615 | + | 0.432345i | −2.80927 | 0 | −1.12893 | ||||||||||
46.6 | −0.609528 | − | 1.05573i | 0 | 0.256951 | − | 0.445052i | −1.48540 | + | 2.57279i | 0 | 0.0501933 | − | 0.0869373i | −3.06459 | 0 | 3.62157 | ||||||||||
46.7 | −0.315463 | − | 0.546398i | 0 | 0.800966 | − | 1.38731i | 1.85440 | − | 3.21192i | 0 | −1.44599 | + | 2.50453i | −2.27255 | 0 | −2.33998 | ||||||||||
46.8 | −0.0900175 | − | 0.155915i | 0 | 0.983794 | − | 1.70398i | −0.708871 | + | 1.22780i | 0 | 1.82660 | − | 3.16376i | −0.714305 | 0 | 0.255243 | ||||||||||
46.9 | 0.0900175 | + | 0.155915i | 0 | 0.983794 | − | 1.70398i | 0.708871 | − | 1.22780i | 0 | 1.82660 | − | 3.16376i | 0.714305 | 0 | 0.255243 | ||||||||||
46.10 | 0.315463 | + | 0.546398i | 0 | 0.800966 | − | 1.38731i | −1.85440 | + | 3.21192i | 0 | −1.44599 | + | 2.50453i | 2.27255 | 0 | −2.33998 | ||||||||||
46.11 | 0.609528 | + | 1.05573i | 0 | 0.256951 | − | 0.445052i | 1.48540 | − | 2.57279i | 0 | 0.0501933 | − | 0.0869373i | 3.06459 | 0 | 3.62157 | ||||||||||
46.12 | 0.711803 | + | 1.23288i | 0 | −0.0133274 | + | 0.0230838i | −0.396502 | + | 0.686762i | 0 | −0.249615 | + | 0.432345i | 2.80927 | 0 | −1.12893 | ||||||||||
46.13 | 0.959463 | + | 1.66184i | 0 | −0.841140 | + | 1.45690i | −1.43954 | + | 2.49335i | 0 | 2.39836 | − | 4.15408i | 0.609683 | 0 | −5.52473 | ||||||||||
46.14 | 0.980806 | + | 1.69881i | 0 | −0.923962 | + | 1.60035i | −0.348567 | + | 0.603736i | 0 | −1.12861 | + | 1.95480i | 0.298316 | 0 | −1.36751 | ||||||||||
46.15 | 1.24321 | + | 2.15331i | 0 | −2.09117 | + | 3.62201i | 1.45965 | − | 2.52819i | 0 | 1.35416 | − | 2.34547i | −5.42622 | 0 | 7.25864 | ||||||||||
46.16 | 1.35501 | + | 2.34695i | 0 | −2.67212 | + | 4.62824i | −1.61886 | + | 2.80395i | 0 | −0.805098 | + | 1.39447i | −9.06295 | 0 | −8.77431 | ||||||||||
262.1 | −1.35501 | + | 2.34695i | 0 | −2.67212 | − | 4.62824i | 1.61886 | + | 2.80395i | 0 | −0.805098 | − | 1.39447i | 9.06295 | 0 | −8.77431 | ||||||||||
262.2 | −1.24321 | + | 2.15331i | 0 | −2.09117 | − | 3.62201i | −1.45965 | − | 2.52819i | 0 | 1.35416 | + | 2.34547i | 5.42622 | 0 | 7.25864 | ||||||||||
262.3 | −0.980806 | + | 1.69881i | 0 | −0.923962 | − | 1.60035i | 0.348567 | + | 0.603736i | 0 | −1.12861 | − | 1.95480i | −0.298316 | 0 | −1.36751 | ||||||||||
262.4 | −0.959463 | + | 1.66184i | 0 | −0.841140 | − | 1.45690i | 1.43954 | + | 2.49335i | 0 | 2.39836 | + | 4.15408i | −0.609683 | 0 | −5.52473 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
103.c | even | 3 | 1 | inner |
309.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 927.2.f.f | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 927.2.f.f | ✓ | 32 |
103.c | even | 3 | 1 | inner | 927.2.f.f | ✓ | 32 |
309.h | odd | 6 | 1 | inner | 927.2.f.f | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
927.2.f.f | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
927.2.f.f | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
927.2.f.f | ✓ | 32 | 103.c | even | 3 | 1 | inner |
927.2.f.f | ✓ | 32 | 309.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 25 T_{2}^{30} + 376 T_{2}^{28} + 3691 T_{2}^{26} + 26818 T_{2}^{24} + 145258 T_{2}^{22} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\).