Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1512,2,Mod(757,1512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1512.757");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1512.c (of order \(2\), degree \(1\), 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: | \(12.0733807856\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
757.1 | −1.40920 | − | 0.118957i | 0 | 1.97170 | + | 0.335269i | − | 3.46300i | 0 | 1.00000 | −2.73864 | − | 0.707009i | 0 | −0.411948 | + | 4.88007i | |||||||||
757.2 | −1.40920 | + | 0.118957i | 0 | 1.97170 | − | 0.335269i | 3.46300i | 0 | 1.00000 | −2.73864 | + | 0.707009i | 0 | −0.411948 | − | 4.88007i | ||||||||||
757.3 | −1.35918 | − | 0.390688i | 0 | 1.69473 | + | 1.06203i | 1.82986i | 0 | 1.00000 | −1.88851 | − | 2.10559i | 0 | 0.714906 | − | 2.48711i | ||||||||||
757.4 | −1.35918 | + | 0.390688i | 0 | 1.69473 | − | 1.06203i | − | 1.82986i | 0 | 1.00000 | −1.88851 | + | 2.10559i | 0 | 0.714906 | + | 2.48711i | |||||||||
757.5 | −1.13123 | − | 0.848721i | 0 | 0.559347 | + | 1.92019i | 0.940450i | 0 | 1.00000 | 0.996957 | − | 2.64690i | 0 | 0.798179 | − | 1.06386i | ||||||||||
757.6 | −1.13123 | + | 0.848721i | 0 | 0.559347 | − | 1.92019i | − | 0.940450i | 0 | 1.00000 | 0.996957 | + | 2.64690i | 0 | 0.798179 | + | 1.06386i | |||||||||
757.7 | −1.10240 | − | 0.885841i | 0 | 0.430570 | + | 1.95310i | − | 2.99001i | 0 | 1.00000 | 1.25548 | − | 2.53452i | 0 | −2.64867 | + | 3.29619i | |||||||||
757.8 | −1.10240 | + | 0.885841i | 0 | 0.430570 | − | 1.95310i | 2.99001i | 0 | 1.00000 | 1.25548 | + | 2.53452i | 0 | −2.64867 | − | 3.29619i | ||||||||||
757.9 | −0.497658 | − | 1.32376i | 0 | −1.50467 | + | 1.31756i | 1.25392i | 0 | 1.00000 | 2.49294 | + | 1.33613i | 0 | 1.65989 | − | 0.624024i | ||||||||||
757.10 | −0.497658 | + | 1.32376i | 0 | −1.50467 | − | 1.31756i | − | 1.25392i | 0 | 1.00000 | 2.49294 | − | 1.33613i | 0 | 1.65989 | + | 0.624024i | |||||||||
757.11 | −0.417332 | − | 1.35123i | 0 | −1.65167 | + | 1.12783i | − | 3.04340i | 0 | 1.00000 | 2.21325 | + | 1.76111i | 0 | −4.11235 | + | 1.27011i | |||||||||
757.12 | −0.417332 | + | 1.35123i | 0 | −1.65167 | − | 1.12783i | 3.04340i | 0 | 1.00000 | 2.21325 | − | 1.76111i | 0 | −4.11235 | − | 1.27011i | ||||||||||
757.13 | 0.417332 | − | 1.35123i | 0 | −1.65167 | − | 1.12783i | − | 3.04340i | 0 | 1.00000 | −2.21325 | + | 1.76111i | 0 | −4.11235 | − | 1.27011i | |||||||||
757.14 | 0.417332 | + | 1.35123i | 0 | −1.65167 | + | 1.12783i | 3.04340i | 0 | 1.00000 | −2.21325 | − | 1.76111i | 0 | −4.11235 | + | 1.27011i | ||||||||||
757.15 | 0.497658 | − | 1.32376i | 0 | −1.50467 | − | 1.31756i | 1.25392i | 0 | 1.00000 | −2.49294 | + | 1.33613i | 0 | 1.65989 | + | 0.624024i | ||||||||||
757.16 | 0.497658 | + | 1.32376i | 0 | −1.50467 | + | 1.31756i | − | 1.25392i | 0 | 1.00000 | −2.49294 | − | 1.33613i | 0 | 1.65989 | − | 0.624024i | |||||||||
757.17 | 1.10240 | − | 0.885841i | 0 | 0.430570 | − | 1.95310i | − | 2.99001i | 0 | 1.00000 | −1.25548 | − | 2.53452i | 0 | −2.64867 | − | 3.29619i | |||||||||
757.18 | 1.10240 | + | 0.885841i | 0 | 0.430570 | + | 1.95310i | 2.99001i | 0 | 1.00000 | −1.25548 | + | 2.53452i | 0 | −2.64867 | + | 3.29619i | ||||||||||
757.19 | 1.13123 | − | 0.848721i | 0 | 0.559347 | − | 1.92019i | 0.940450i | 0 | 1.00000 | −0.996957 | − | 2.64690i | 0 | 0.798179 | + | 1.06386i | ||||||||||
757.20 | 1.13123 | + | 0.848721i | 0 | 0.559347 | + | 1.92019i | − | 0.940450i | 0 | 1.00000 | −0.996957 | + | 2.64690i | 0 | 0.798179 | − | 1.06386i | |||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1512.2.c.g | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 1512.2.c.g | ✓ | 24 |
4.b | odd | 2 | 1 | 6048.2.c.f | 24 | ||
8.b | even | 2 | 1 | inner | 1512.2.c.g | ✓ | 24 |
8.d | odd | 2 | 1 | 6048.2.c.f | 24 | ||
12.b | even | 2 | 1 | 6048.2.c.f | 24 | ||
24.f | even | 2 | 1 | 6048.2.c.f | 24 | ||
24.h | odd | 2 | 1 | inner | 1512.2.c.g | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1512.2.c.g | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
1512.2.c.g | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
1512.2.c.g | ✓ | 24 | 8.b | even | 2 | 1 | inner |
1512.2.c.g | ✓ | 24 | 24.h | odd | 2 | 1 | inner |
6048.2.c.f | 24 | 4.b | odd | 2 | 1 | ||
6048.2.c.f | 24 | 8.d | odd | 2 | 1 | ||
6048.2.c.f | 24 | 12.b | even | 2 | 1 | ||
6048.2.c.f | 24 | 24.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):
\( T_{5}^{12} + 36T_{5}^{10} + 486T_{5}^{8} + 3036T_{5}^{6} + 8801T_{5}^{4} + 10952T_{5}^{2} + 4624 \) |
\( T_{17}^{12} - 100T_{17}^{10} + 3253T_{17}^{8} - 41296T_{17}^{6} + 197515T_{17}^{4} - 252108T_{17}^{2} + 63 \) |