Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4140,3,Mod(2161,4140)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("4140.2161");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 4140.d (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: | \(112.806829445\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2161.1 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 13.2033i | 0 | 0 | 0 | ||||||||||||||||
2161.2 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 9.55915i | 0 | 0 | 0 | ||||||||||||||||
2161.3 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 8.25712i | 0 | 0 | 0 | ||||||||||||||||
2161.4 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 7.68094i | 0 | 0 | 0 | ||||||||||||||||
2161.5 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 6.51414i | 0 | 0 | 0 | ||||||||||||||||
2161.6 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 4.50512i | 0 | 0 | 0 | ||||||||||||||||
2161.7 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 1.15266i | 0 | 0 | 0 | ||||||||||||||||
2161.8 | 0 | 0 | 0 | − | 2.23607i | 0 | − | 0.246464i | 0 | 0 | 0 | ||||||||||||||||
2161.9 | 0 | 0 | 0 | − | 2.23607i | 0 | 0.246464i | 0 | 0 | 0 | |||||||||||||||||
2161.10 | 0 | 0 | 0 | − | 2.23607i | 0 | 1.15266i | 0 | 0 | 0 | |||||||||||||||||
2161.11 | 0 | 0 | 0 | − | 2.23607i | 0 | 4.50512i | 0 | 0 | 0 | |||||||||||||||||
2161.12 | 0 | 0 | 0 | − | 2.23607i | 0 | 6.51414i | 0 | 0 | 0 | |||||||||||||||||
2161.13 | 0 | 0 | 0 | − | 2.23607i | 0 | 7.68094i | 0 | 0 | 0 | |||||||||||||||||
2161.14 | 0 | 0 | 0 | − | 2.23607i | 0 | 8.25712i | 0 | 0 | 0 | |||||||||||||||||
2161.15 | 0 | 0 | 0 | − | 2.23607i | 0 | 9.55915i | 0 | 0 | 0 | |||||||||||||||||
2161.16 | 0 | 0 | 0 | − | 2.23607i | 0 | 13.2033i | 0 | 0 | 0 | |||||||||||||||||
2161.17 | 0 | 0 | 0 | 2.23607i | 0 | − | 13.2033i | 0 | 0 | 0 | |||||||||||||||||
2161.18 | 0 | 0 | 0 | 2.23607i | 0 | − | 9.55915i | 0 | 0 | 0 | |||||||||||||||||
2161.19 | 0 | 0 | 0 | 2.23607i | 0 | − | 8.25712i | 0 | 0 | 0 | |||||||||||||||||
2161.20 | 0 | 0 | 0 | 2.23607i | 0 | − | 7.68094i | 0 | 0 | 0 | |||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
69.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4140.3.d.b | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 4140.3.d.b | ✓ | 32 |
23.b | odd | 2 | 1 | inner | 4140.3.d.b | ✓ | 32 |
69.c | even | 2 | 1 | inner | 4140.3.d.b | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4140.3.d.b | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
4140.3.d.b | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
4140.3.d.b | ✓ | 32 | 23.b | odd | 2 | 1 | inner |
4140.3.d.b | ✓ | 32 | 69.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 457 T_{7}^{14} + 79883 T_{7}^{12} + 6914455 T_{7}^{10} + 313947096 T_{7}^{8} + 7108236140 T_{7}^{6} + 64495964848 T_{7}^{4} + 77210147648 T_{7}^{2} + 4453693696 \)
acting on \(S_{3}^{\mathrm{new}}(4140, [\chi])\).