Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2916,3,Mod(1457,2916)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2916, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2916.1457");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2916 = 2^{2} \cdot 3^{6} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 2916.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: | \(79.4552450875\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Twist minimal: | no (minimal twist has level 108) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1457.1 | 0 | 0 | 0 | − | 9.50324i | 0 | 2.36132 | 0 | 0 | 0 | |||||||||||||||||
1457.2 | 0 | 0 | 0 | − | 8.78835i | 0 | −4.31308 | 0 | 0 | 0 | |||||||||||||||||
1457.3 | 0 | 0 | 0 | − | 7.78294i | 0 | 3.61169 | 0 | 0 | 0 | |||||||||||||||||
1457.4 | 0 | 0 | 0 | − | 7.77099i | 0 | 13.6742 | 0 | 0 | 0 | |||||||||||||||||
1457.5 | 0 | 0 | 0 | − | 7.31297i | 0 | −9.37121 | 0 | 0 | 0 | |||||||||||||||||
1457.6 | 0 | 0 | 0 | − | 7.21405i | 0 | −4.39191 | 0 | 0 | 0 | |||||||||||||||||
1457.7 | 0 | 0 | 0 | − | 5.21679i | 0 | −11.6618 | 0 | 0 | 0 | |||||||||||||||||
1457.8 | 0 | 0 | 0 | − | 4.83775i | 0 | 9.71048 | 0 | 0 | 0 | |||||||||||||||||
1457.9 | 0 | 0 | 0 | − | 4.39539i | 0 | −3.57384 | 0 | 0 | 0 | |||||||||||||||||
1457.10 | 0 | 0 | 0 | − | 4.19339i | 0 | −11.8514 | 0 | 0 | 0 | |||||||||||||||||
1457.11 | 0 | 0 | 0 | − | 4.17955i | 0 | 5.19980 | 0 | 0 | 0 | |||||||||||||||||
1457.12 | 0 | 0 | 0 | − | 3.70301i | 0 | −0.419343 | 0 | 0 | 0 | |||||||||||||||||
1457.13 | 0 | 0 | 0 | − | 3.34301i | 0 | 5.01893 | 0 | 0 | 0 | |||||||||||||||||
1457.14 | 0 | 0 | 0 | − | 2.96937i | 0 | 0.934448 | 0 | 0 | 0 | |||||||||||||||||
1457.15 | 0 | 0 | 0 | − | 2.16582i | 0 | 6.23376 | 0 | 0 | 0 | |||||||||||||||||
1457.16 | 0 | 0 | 0 | − | 1.52502i | 0 | −3.44709 | 0 | 0 | 0 | |||||||||||||||||
1457.17 | 0 | 0 | 0 | − | 0.464464i | 0 | 10.8001 | 0 | 0 | 0 | |||||||||||||||||
1457.18 | 0 | 0 | 0 | − | 0.200781i | 0 | −8.51501 | 0 | 0 | 0 | |||||||||||||||||
1457.19 | 0 | 0 | 0 | 0.200781i | 0 | −8.51501 | 0 | 0 | 0 | ||||||||||||||||||
1457.20 | 0 | 0 | 0 | 0.464464i | 0 | 10.8001 | 0 | 0 | 0 | ||||||||||||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2916.3.c.b | 36 | |
3.b | odd | 2 | 1 | inner | 2916.3.c.b | 36 | |
27.e | even | 9 | 1 | 108.3.k.a | ✓ | 36 | |
27.e | even | 9 | 1 | 324.3.k.a | 36 | ||
27.f | odd | 18 | 1 | 108.3.k.a | ✓ | 36 | |
27.f | odd | 18 | 1 | 324.3.k.a | 36 | ||
108.j | odd | 18 | 1 | 432.3.bc.b | 36 | ||
108.l | even | 18 | 1 | 432.3.bc.b | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
108.3.k.a | ✓ | 36 | 27.e | even | 9 | 1 | |
108.3.k.a | ✓ | 36 | 27.f | odd | 18 | 1 | |
324.3.k.a | 36 | 27.e | even | 9 | 1 | ||
324.3.k.a | 36 | 27.f | odd | 18 | 1 | ||
432.3.bc.b | 36 | 108.j | odd | 18 | 1 | ||
432.3.bc.b | 36 | 108.l | even | 18 | 1 | ||
2916.3.c.b | 36 | 1.a | even | 1 | 1 | trivial | |
2916.3.c.b | 36 | 3.b | odd | 2 | 1 | inner |