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 algebraic \(q\)-expansion of this newform has not been computed, 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 | −1.63593 | + | 0.568988i | 0 | 0.164566 | + | 0.0950122i | 0 | 0.883190i | 0 | 2.35251 | − | 1.86164i | 0 | ||||||||||||
239.2 | 0 | −1.63249 | − | 0.578768i | 0 | 3.06808 | + | 1.77136i | 0 | 0.587143i | 0 | 2.33006 | + | 1.88967i | 0 | ||||||||||||
239.3 | 0 | −1.31747 | − | 1.12440i | 0 | −3.06808 | − | 1.77136i | 0 | 0.587143i | 0 | 0.471472 | + | 2.96272i | 0 | ||||||||||||
239.4 | 0 | −1.08081 | + | 1.35346i | 0 | −1.81928 | − | 1.05036i | 0 | 3.85684i | 0 | −0.663700 | − | 2.92566i | 0 | ||||||||||||
239.5 | 0 | −0.631725 | + | 1.61274i | 0 | 1.81928 | + | 1.05036i | 0 | − | 3.85684i | 0 | −2.20185 | − | 2.03761i | 0 | |||||||||||
239.6 | 0 | −0.325205 | − | 1.70125i | 0 | −0.164566 | − | 0.0950122i | 0 | 0.883190i | 0 | −2.78848 | + | 1.10651i | 0 | ||||||||||||
239.7 | 0 | 0.325205 | + | 1.70125i | 0 | −0.164566 | − | 0.0950122i | 0 | − | 0.883190i | 0 | −2.78848 | + | 1.10651i | 0 | |||||||||||
239.8 | 0 | 0.631725 | − | 1.61274i | 0 | 1.81928 | + | 1.05036i | 0 | 3.85684i | 0 | −2.20185 | − | 2.03761i | 0 | ||||||||||||
239.9 | 0 | 1.08081 | − | 1.35346i | 0 | −1.81928 | − | 1.05036i | 0 | − | 3.85684i | 0 | −0.663700 | − | 2.92566i | 0 | |||||||||||
239.10 | 0 | 1.31747 | + | 1.12440i | 0 | −3.06808 | − | 1.77136i | 0 | − | 0.587143i | 0 | 0.471472 | + | 2.96272i | 0 | |||||||||||
239.11 | 0 | 1.63249 | + | 0.578768i | 0 | 3.06808 | + | 1.77136i | 0 | − | 0.587143i | 0 | 2.33006 | + | 1.88967i | 0 | |||||||||||
239.12 | 0 | 1.63593 | − | 0.568988i | 0 | 0.164566 | + | 0.0950122i | 0 | − | 0.883190i | 0 | 2.35251 | − | 1.86164i | 0 | |||||||||||
767.1 | 0 | −1.63593 | − | 0.568988i | 0 | 0.164566 | − | 0.0950122i | 0 | − | 0.883190i | 0 | 2.35251 | + | 1.86164i | 0 | |||||||||||
767.2 | 0 | −1.63249 | + | 0.578768i | 0 | 3.06808 | − | 1.77136i | 0 | − | 0.587143i | 0 | 2.33006 | − | 1.88967i | 0 | |||||||||||
767.3 | 0 | −1.31747 | + | 1.12440i | 0 | −3.06808 | + | 1.77136i | 0 | − | 0.587143i | 0 | 0.471472 | − | 2.96272i | 0 | |||||||||||
767.4 | 0 | −1.08081 | − | 1.35346i | 0 | −1.81928 | + | 1.05036i | 0 | − | 3.85684i | 0 | −0.663700 | + | 2.92566i | 0 | |||||||||||
767.5 | 0 | −0.631725 | − | 1.61274i | 0 | 1.81928 | − | 1.05036i | 0 | 3.85684i | 0 | −2.20185 | + | 2.03761i | 0 | ||||||||||||
767.6 | 0 | −0.325205 | + | 1.70125i | 0 | −0.164566 | + | 0.0950122i | 0 | − | 0.883190i | 0 | −2.78848 | − | 1.10651i | 0 | |||||||||||
767.7 | 0 | 0.325205 | − | 1.70125i | 0 | −0.164566 | + | 0.0950122i | 0 | 0.883190i | 0 | −2.78848 | − | 1.10651i | 0 | ||||||||||||
767.8 | 0 | 0.631725 | + | 1.61274i | 0 | 1.81928 | − | 1.05036i | 0 | − | 3.85684i | 0 | −2.20185 | + | 2.03761i | 0 | |||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
57.h | odd | 6 | 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.f | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 912.2.bh.f | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 912.2.bh.f | ✓ | 24 |
12.b | even | 2 | 1 | inner | 912.2.bh.f | ✓ | 24 |
19.c | even | 3 | 1 | inner | 912.2.bh.f | ✓ | 24 |
57.h | odd | 6 | 1 | inner | 912.2.bh.f | ✓ | 24 |
76.g | odd | 6 | 1 | inner | 912.2.bh.f | ✓ | 24 |
228.m | even | 6 | 1 | inner | 912.2.bh.f | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.2.bh.f | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
912.2.bh.f | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
912.2.bh.f | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
912.2.bh.f | ✓ | 24 | 12.b | even | 2 | 1 | inner |
912.2.bh.f | ✓ | 24 | 19.c | even | 3 | 1 | inner |
912.2.bh.f | ✓ | 24 | 57.h | odd | 6 | 1 | inner |
912.2.bh.f | ✓ | 24 | 76.g | odd | 6 | 1 | inner |
912.2.bh.f | ✓ | 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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):
\( T_{5}^{12} - 17T_{5}^{10} + 233T_{5}^{8} - 948T_{5}^{6} + 3102T_{5}^{4} - 112T_{5}^{2} + 4 \)
|
\( T_{7}^{6} + 16T_{7}^{4} + 17T_{7}^{2} + 4 \)
|
\( T_{43}^{12} - 149T_{43}^{10} + 16204T_{43}^{8} - 773503T_{43}^{6} + 27020284T_{43}^{4} - 359969925T_{43}^{2} + 3603000625 \)
|