Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1840,3,Mod(321,1840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, 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("1840.321");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1840 = 2^{4} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1840.k (of order \(2\), degree \(1\), not 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: | \(50.1363686423\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | no (minimal twist has level 920) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
321.1 | 0 | −5.39985 | 0 | − | 2.23607i | 0 | − | 8.04288i | 0 | 20.1584 | 0 | ||||||||||||||||
321.2 | 0 | −5.39985 | 0 | 2.23607i | 0 | 8.04288i | 0 | 20.1584 | 0 | ||||||||||||||||||
321.3 | 0 | −5.24002 | 0 | 2.23607i | 0 | − | 3.70696i | 0 | 18.4579 | 0 | |||||||||||||||||
321.4 | 0 | −5.24002 | 0 | − | 2.23607i | 0 | 3.70696i | 0 | 18.4579 | 0 | |||||||||||||||||
321.5 | 0 | −4.84326 | 0 | 2.23607i | 0 | − | 4.25633i | 0 | 14.4572 | 0 | |||||||||||||||||
321.6 | 0 | −4.84326 | 0 | − | 2.23607i | 0 | 4.25633i | 0 | 14.4572 | 0 | |||||||||||||||||
321.7 | 0 | −4.51681 | 0 | − | 2.23607i | 0 | 1.75483i | 0 | 11.4015 | 0 | |||||||||||||||||
321.8 | 0 | −4.51681 | 0 | 2.23607i | 0 | − | 1.75483i | 0 | 11.4015 | 0 | |||||||||||||||||
321.9 | 0 | −3.03006 | 0 | − | 2.23607i | 0 | − | 6.21370i | 0 | 0.181242 | 0 | ||||||||||||||||
321.10 | 0 | −3.03006 | 0 | 2.23607i | 0 | 6.21370i | 0 | 0.181242 | 0 | ||||||||||||||||||
321.11 | 0 | −2.78603 | 0 | − | 2.23607i | 0 | − | 9.63233i | 0 | −1.23801 | 0 | ||||||||||||||||
321.12 | 0 | −2.78603 | 0 | 2.23607i | 0 | 9.63233i | 0 | −1.23801 | 0 | ||||||||||||||||||
321.13 | 0 | −2.67503 | 0 | 2.23607i | 0 | − | 10.9996i | 0 | −1.84421 | 0 | |||||||||||||||||
321.14 | 0 | −2.67503 | 0 | − | 2.23607i | 0 | 10.9996i | 0 | −1.84421 | 0 | |||||||||||||||||
321.15 | 0 | −2.49795 | 0 | − | 2.23607i | 0 | 1.88865i | 0 | −2.76024 | 0 | |||||||||||||||||
321.16 | 0 | −2.49795 | 0 | 2.23607i | 0 | − | 1.88865i | 0 | −2.76024 | 0 | |||||||||||||||||
321.17 | 0 | −2.35475 | 0 | − | 2.23607i | 0 | 5.28829i | 0 | −3.45517 | 0 | |||||||||||||||||
321.18 | 0 | −2.35475 | 0 | 2.23607i | 0 | − | 5.28829i | 0 | −3.45517 | 0 | |||||||||||||||||
321.19 | 0 | −1.59835 | 0 | − | 2.23607i | 0 | − | 7.14235i | 0 | −6.44527 | 0 | ||||||||||||||||
321.20 | 0 | −1.59835 | 0 | 2.23607i | 0 | 7.14235i | 0 | −6.44527 | 0 | ||||||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1840.3.k.e | 48 | |
4.b | odd | 2 | 1 | 920.3.k.a | ✓ | 48 | |
23.b | odd | 2 | 1 | inner | 1840.3.k.e | 48 | |
92.b | even | 2 | 1 | 920.3.k.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
920.3.k.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
920.3.k.a | ✓ | 48 | 92.b | even | 2 | 1 | |
1840.3.k.e | 48 | 1.a | even | 1 | 1 | trivial | |
1840.3.k.e | 48 | 23.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 140 T_{3}^{22} + 8373 T_{3}^{20} - 32 T_{3}^{19} - 280772 T_{3}^{18} + 2664 T_{3}^{17} + \cdots - 2271855744 \) acting on \(S_{3}^{\mathrm{new}}(1840, [\chi])\).