Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(82,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.82");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.bc (of order \(12\), degree \(4\), 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: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
82.1 | −2.60685 | + | 0.698503i | 0.258819 | − | 0.965926i | 4.57570 | − | 2.64178i | 0 | 2.69881i | −0.543835 | − | 2.58926i | −6.26618 | + | 6.26618i | −0.866025 | − | 0.500000i | 0 | ||||||
82.2 | −2.01147 | + | 0.538972i | −0.258819 | + | 0.965926i | 2.02348 | − | 1.16825i | 0 | − | 2.08243i | −1.99060 | + | 1.74285i | −0.495509 | + | 0.495509i | −0.866025 | − | 0.500000i | 0 | |||||
82.3 | −0.595377 | + | 0.159531i | −0.258819 | + | 0.965926i | −1.40303 | + | 0.810038i | 0 | − | 0.616380i | 2.22322 | − | 1.43432i | 1.57780 | − | 1.57780i | −0.866025 | − | 0.500000i | 0 | |||||
82.4 | 0.595377 | − | 0.159531i | 0.258819 | − | 0.965926i | −1.40303 | + | 0.810038i | 0 | − | 0.616380i | −2.22322 | + | 1.43432i | −1.57780 | + | 1.57780i | −0.866025 | − | 0.500000i | 0 | |||||
82.5 | 2.01147 | − | 0.538972i | 0.258819 | − | 0.965926i | 2.02348 | − | 1.16825i | 0 | − | 2.08243i | 1.99060 | − | 1.74285i | 0.495509 | − | 0.495509i | −0.866025 | − | 0.500000i | 0 | |||||
82.6 | 2.60685 | − | 0.698503i | −0.258819 | + | 0.965926i | 4.57570 | − | 2.64178i | 0 | 2.69881i | 0.543835 | + | 2.58926i | 6.26618 | − | 6.26618i | −0.866025 | − | 0.500000i | 0 | ||||||
157.1 | −0.698503 | + | 2.60685i | −0.965926 | + | 0.258819i | −4.57570 | − | 2.64178i | 0 | − | 2.69881i | 2.58926 | + | 0.543835i | 6.26618 | − | 6.26618i | 0.866025 | − | 0.500000i | 0 | |||||
157.2 | −0.538972 | + | 2.01147i | 0.965926 | − | 0.258819i | −2.02348 | − | 1.16825i | 0 | 2.08243i | −1.74285 | + | 1.99060i | 0.495509 | − | 0.495509i | 0.866025 | − | 0.500000i | 0 | ||||||
157.3 | −0.159531 | + | 0.595377i | 0.965926 | − | 0.258819i | 1.40303 | + | 0.810038i | 0 | 0.616380i | 1.43432 | − | 2.22322i | −1.57780 | + | 1.57780i | 0.866025 | − | 0.500000i | 0 | ||||||
157.4 | 0.159531 | − | 0.595377i | −0.965926 | + | 0.258819i | 1.40303 | + | 0.810038i | 0 | 0.616380i | −1.43432 | + | 2.22322i | 1.57780 | − | 1.57780i | 0.866025 | − | 0.500000i | 0 | ||||||
157.5 | 0.538972 | − | 2.01147i | −0.965926 | + | 0.258819i | −2.02348 | − | 1.16825i | 0 | 2.08243i | 1.74285 | − | 1.99060i | −0.495509 | + | 0.495509i | 0.866025 | − | 0.500000i | 0 | ||||||
157.6 | 0.698503 | − | 2.60685i | 0.965926 | − | 0.258819i | −4.57570 | − | 2.64178i | 0 | − | 2.69881i | −2.58926 | − | 0.543835i | −6.26618 | + | 6.26618i | 0.866025 | − | 0.500000i | 0 | |||||
418.1 | −0.698503 | − | 2.60685i | −0.965926 | − | 0.258819i | −4.57570 | + | 2.64178i | 0 | 2.69881i | 2.58926 | − | 0.543835i | 6.26618 | + | 6.26618i | 0.866025 | + | 0.500000i | 0 | ||||||
418.2 | −0.538972 | − | 2.01147i | 0.965926 | + | 0.258819i | −2.02348 | + | 1.16825i | 0 | − | 2.08243i | −1.74285 | − | 1.99060i | 0.495509 | + | 0.495509i | 0.866025 | + | 0.500000i | 0 | |||||
418.3 | −0.159531 | − | 0.595377i | 0.965926 | + | 0.258819i | 1.40303 | − | 0.810038i | 0 | − | 0.616380i | 1.43432 | + | 2.22322i | −1.57780 | − | 1.57780i | 0.866025 | + | 0.500000i | 0 | |||||
418.4 | 0.159531 | + | 0.595377i | −0.965926 | − | 0.258819i | 1.40303 | − | 0.810038i | 0 | − | 0.616380i | −1.43432 | − | 2.22322i | 1.57780 | + | 1.57780i | 0.866025 | + | 0.500000i | 0 | |||||
418.5 | 0.538972 | + | 2.01147i | −0.965926 | − | 0.258819i | −2.02348 | + | 1.16825i | 0 | − | 2.08243i | 1.74285 | + | 1.99060i | −0.495509 | − | 0.495509i | 0.866025 | + | 0.500000i | 0 | |||||
418.6 | 0.698503 | + | 2.60685i | 0.965926 | + | 0.258819i | −4.57570 | + | 2.64178i | 0 | 2.69881i | −2.58926 | + | 0.543835i | −6.26618 | − | 6.26618i | 0.866025 | + | 0.500000i | 0 | ||||||
493.1 | −2.60685 | − | 0.698503i | 0.258819 | + | 0.965926i | 4.57570 | + | 2.64178i | 0 | − | 2.69881i | −0.543835 | + | 2.58926i | −6.26618 | − | 6.26618i | −0.866025 | + | 0.500000i | 0 | |||||
493.2 | −2.01147 | − | 0.538972i | −0.258819 | − | 0.965926i | 2.02348 | + | 1.16825i | 0 | 2.08243i | −1.99060 | − | 1.74285i | −0.495509 | − | 0.495509i | −0.866025 | + | 0.500000i | 0 | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
7.d | odd | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
35.k | even | 12 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.bc.d | ✓ | 24 |
5.b | even | 2 | 1 | inner | 525.2.bc.d | ✓ | 24 |
5.c | odd | 4 | 2 | inner | 525.2.bc.d | ✓ | 24 |
7.d | odd | 6 | 1 | inner | 525.2.bc.d | ✓ | 24 |
35.i | odd | 6 | 1 | inner | 525.2.bc.d | ✓ | 24 |
35.k | even | 12 | 2 | inner | 525.2.bc.d | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.bc.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
525.2.bc.d | ✓ | 24 | 5.b | even | 2 | 1 | inner |
525.2.bc.d | ✓ | 24 | 5.c | odd | 4 | 2 | inner |
525.2.bc.d | ✓ | 24 | 7.d | odd | 6 | 1 | inner |
525.2.bc.d | ✓ | 24 | 35.i | odd | 6 | 1 | inner |
525.2.bc.d | ✓ | 24 | 35.k | even | 12 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 72T_{2}^{20} + 4176T_{2}^{16} - 72288T_{2}^{12} + 1005696T_{2}^{8} - 145152T_{2}^{4} + 20736 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).