Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(239,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.239");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.bh (of order \(6\), 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: | \(7.28235666434\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
239.1 | 0 | −1.62428 | + | 0.601422i | 0 | −3.50986 | − | 2.02642i | 0 | 0.196665i | 0 | 2.27658 | − | 1.95376i | 0 | ||||||||||||
239.2 | 0 | −1.53419 | + | 0.803908i | 0 | 2.72078 | + | 1.57084i | 0 | − | 4.27804i | 0 | 1.70747 | − | 2.46669i | 0 | |||||||||||
239.3 | 0 | −0.579078 | + | 1.63238i | 0 | 2.37836 | + | 1.37315i | 0 | 4.29408i | 0 | −2.32934 | − | 1.89055i | 0 | ||||||||||||
239.4 | 0 | −0.462645 | + | 1.66912i | 0 | 0.299633 | + | 0.172993i | 0 | − | 0.167877i | 0 | −2.57192 | − | 1.54442i | 0 | |||||||||||
239.5 | 0 | −0.291294 | − | 1.70738i | 0 | 3.50986 | + | 2.02642i | 0 | 0.196665i | 0 | −2.83030 | + | 0.994699i | 0 | ||||||||||||
239.6 | 0 | −0.0708896 | − | 1.73060i | 0 | −2.72078 | − | 1.57084i | 0 | − | 4.27804i | 0 | −2.98995 | + | 0.245363i | 0 | |||||||||||
239.7 | 0 | 0.560953 | + | 1.63870i | 0 | −1.84530 | − | 1.06538i | 0 | − | 2.59062i | 0 | −2.37066 | + | 1.83847i | 0 | |||||||||||
239.8 | 0 | 1.12414 | − | 1.31769i | 0 | −2.37836 | − | 1.37315i | 0 | 4.29408i | 0 | −0.472598 | − | 2.96254i | 0 | ||||||||||||
239.9 | 0 | 1.21418 | − | 1.23522i | 0 | −0.299633 | − | 0.172993i | 0 | − | 0.167877i | 0 | −0.0515467 | − | 2.99956i | 0 | |||||||||||
239.10 | 0 | 1.34700 | + | 1.08885i | 0 | 1.83758 | + | 1.06093i | 0 | 2.54579i | 0 | 0.628815 | + | 2.93336i | 0 | ||||||||||||
239.11 | 0 | 1.61647 | + | 0.622111i | 0 | −1.83758 | − | 1.06093i | 0 | 2.54579i | 0 | 2.22596 | + | 2.01125i | 0 | ||||||||||||
239.12 | 0 | 1.69963 | − | 0.333549i | 0 | 1.84530 | + | 1.06538i | 0 | − | 2.59062i | 0 | 2.77749 | − | 1.13382i | 0 | |||||||||||
767.1 | 0 | −1.62428 | − | 0.601422i | 0 | −3.50986 | + | 2.02642i | 0 | − | 0.196665i | 0 | 2.27658 | + | 1.95376i | 0 | |||||||||||
767.2 | 0 | −1.53419 | − | 0.803908i | 0 | 2.72078 | − | 1.57084i | 0 | 4.27804i | 0 | 1.70747 | + | 2.46669i | 0 | ||||||||||||
767.3 | 0 | −0.579078 | − | 1.63238i | 0 | 2.37836 | − | 1.37315i | 0 | − | 4.29408i | 0 | −2.32934 | + | 1.89055i | 0 | |||||||||||
767.4 | 0 | −0.462645 | − | 1.66912i | 0 | 0.299633 | − | 0.172993i | 0 | 0.167877i | 0 | −2.57192 | + | 1.54442i | 0 | ||||||||||||
767.5 | 0 | −0.291294 | + | 1.70738i | 0 | 3.50986 | − | 2.02642i | 0 | − | 0.196665i | 0 | −2.83030 | − | 0.994699i | 0 | |||||||||||
767.6 | 0 | −0.0708896 | + | 1.73060i | 0 | −2.72078 | + | 1.57084i | 0 | 4.27804i | 0 | −2.98995 | − | 0.245363i | 0 | ||||||||||||
767.7 | 0 | 0.560953 | − | 1.63870i | 0 | −1.84530 | + | 1.06538i | 0 | 2.59062i | 0 | −2.37066 | − | 1.83847i | 0 | ||||||||||||
767.8 | 0 | 1.12414 | + | 1.31769i | 0 | −2.37836 | + | 1.37315i | 0 | − | 4.29408i | 0 | −0.472598 | + | 2.96254i | 0 | |||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
76.g | odd | 6 | 1 | inner |
228.m | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.2.bh.g | yes | 24 |
3.b | odd | 2 | 1 | inner | 912.2.bh.g | yes | 24 |
4.b | odd | 2 | 1 | 912.2.bh.e | ✓ | 24 | |
12.b | even | 2 | 1 | 912.2.bh.e | ✓ | 24 | |
19.c | even | 3 | 1 | 912.2.bh.e | ✓ | 24 | |
57.h | odd | 6 | 1 | 912.2.bh.e | ✓ | 24 | |
76.g | odd | 6 | 1 | inner | 912.2.bh.g | yes | 24 |
228.m | even | 6 | 1 | inner | 912.2.bh.g | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.2.bh.e | ✓ | 24 | 4.b | odd | 2 | 1 | |
912.2.bh.e | ✓ | 24 | 12.b | even | 2 | 1 | |
912.2.bh.e | ✓ | 24 | 19.c | even | 3 | 1 | |
912.2.bh.e | ✓ | 24 | 57.h | odd | 6 | 1 | |
912.2.bh.g | yes | 24 | 1.a | even | 1 | 1 | trivial |
912.2.bh.g | yes | 24 | 3.b | odd | 2 | 1 | inner |
912.2.bh.g | yes | 24 | 76.g | odd | 6 | 1 | inner |
912.2.bh.g | yes | 24 | 228.m | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):
\( T_{5}^{24} - 43 T_{5}^{22} + 1157 T_{5}^{20} - 19244 T_{5}^{18} + 233812 T_{5}^{16} - 2026568 T_{5}^{14} + \cdots + 8952064 \) |
\( T_{7}^{12} + 50T_{7}^{10} + 869T_{7}^{8} + 6108T_{7}^{6} + 15084T_{7}^{4} + 988T_{7}^{2} + 16 \) |
\( T_{43}^{12} - 21 T_{43}^{11} + 4 T_{43}^{10} + 3003 T_{43}^{9} - 5312 T_{43}^{8} - 381579 T_{43}^{7} + \cdots + 2834497600 \) |