Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [500,2,Mod(307,500)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("500.307");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 500 = 2^{2} \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 500.e (of order \(4\), degree \(2\), 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: | \(3.99252010106\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
307.1 | −1.39022 | − | 0.259395i | 0.519784 | + | 0.519784i | 1.86543 | + | 0.721232i | 0 | −0.587785 | − | 0.857444i | −2.15096 | + | 2.15096i | −2.40627 | − | 1.48655i | − | 2.45965i | 0 | |||||
307.2 | −1.38743 | + | 0.273914i | 0.572461 | + | 0.572461i | 1.84994 | − | 0.760074i | 0 | −0.951057 | − | 0.637447i | 1.95303 | − | 1.95303i | −2.35848 | + | 1.56128i | − | 2.34458i | 0 | |||||
307.3 | −1.34942 | − | 0.423159i | −1.02677 | − | 1.02677i | 1.64187 | + | 1.14204i | 0 | 0.951057 | + | 1.82003i | −1.08889 | + | 1.08889i | −1.73231 | − | 2.23587i | − | 0.891491i | 0 | |||||
307.4 | −1.14764 | − | 0.826394i | −1.82972 | − | 1.82972i | 0.634146 | + | 1.89680i | 0 | 0.587785 | + | 3.61192i | 0.611043 | − | 0.611043i | 0.839736 | − | 2.70090i | 3.69572i | 0 | ||||||
307.5 | −0.826394 | − | 1.14764i | 1.82972 | + | 1.82972i | −0.634146 | + | 1.89680i | 0 | 0.587785 | − | 3.61192i | −0.611043 | + | 0.611043i | 2.70090 | − | 0.839736i | 3.69572i | 0 | ||||||
307.6 | −0.423159 | − | 1.34942i | 1.02677 | + | 1.02677i | −1.64187 | + | 1.14204i | 0 | 0.951057 | − | 1.82003i | 1.08889 | − | 1.08889i | 2.23587 | + | 1.73231i | − | 0.891491i | 0 | |||||
307.7 | −0.273914 | + | 1.38743i | 0.572461 | + | 0.572461i | −1.84994 | − | 0.760074i | 0 | −0.951057 | + | 0.637447i | 1.95303 | − | 1.95303i | 1.56128 | − | 2.35848i | − | 2.34458i | 0 | |||||
307.8 | −0.259395 | − | 1.39022i | −0.519784 | − | 0.519784i | −1.86543 | + | 0.721232i | 0 | −0.587785 | + | 0.857444i | 2.15096 | − | 2.15096i | 1.48655 | + | 2.40627i | − | 2.45965i | 0 | |||||
307.9 | 0.259395 | + | 1.39022i | 0.519784 | + | 0.519784i | −1.86543 | + | 0.721232i | 0 | −0.587785 | + | 0.857444i | −2.15096 | + | 2.15096i | −1.48655 | − | 2.40627i | − | 2.45965i | 0 | |||||
307.10 | 0.273914 | − | 1.38743i | −0.572461 | − | 0.572461i | −1.84994 | − | 0.760074i | 0 | −0.951057 | + | 0.637447i | −1.95303 | + | 1.95303i | −1.56128 | + | 2.35848i | − | 2.34458i | 0 | |||||
307.11 | 0.423159 | + | 1.34942i | −1.02677 | − | 1.02677i | −1.64187 | + | 1.14204i | 0 | 0.951057 | − | 1.82003i | −1.08889 | + | 1.08889i | −2.23587 | − | 1.73231i | − | 0.891491i | 0 | |||||
307.12 | 0.826394 | + | 1.14764i | −1.82972 | − | 1.82972i | −0.634146 | + | 1.89680i | 0 | 0.587785 | − | 3.61192i | 0.611043 | − | 0.611043i | −2.70090 | + | 0.839736i | 3.69572i | 0 | ||||||
307.13 | 1.14764 | + | 0.826394i | 1.82972 | + | 1.82972i | 0.634146 | + | 1.89680i | 0 | 0.587785 | + | 3.61192i | −0.611043 | + | 0.611043i | −0.839736 | + | 2.70090i | 3.69572i | 0 | ||||||
307.14 | 1.34942 | + | 0.423159i | 1.02677 | + | 1.02677i | 1.64187 | + | 1.14204i | 0 | 0.951057 | + | 1.82003i | 1.08889 | − | 1.08889i | 1.73231 | + | 2.23587i | − | 0.891491i | 0 | |||||
307.15 | 1.38743 | − | 0.273914i | −0.572461 | − | 0.572461i | 1.84994 | − | 0.760074i | 0 | −0.951057 | − | 0.637447i | −1.95303 | + | 1.95303i | 2.35848 | − | 1.56128i | − | 2.34458i | 0 | |||||
307.16 | 1.39022 | + | 0.259395i | −0.519784 | − | 0.519784i | 1.86543 | + | 0.721232i | 0 | −0.587785 | − | 0.857444i | 2.15096 | − | 2.15096i | 2.40627 | + | 1.48655i | − | 2.45965i | 0 | |||||
443.1 | −1.39022 | + | 0.259395i | 0.519784 | − | 0.519784i | 1.86543 | − | 0.721232i | 0 | −0.587785 | + | 0.857444i | −2.15096 | − | 2.15096i | −2.40627 | + | 1.48655i | 2.45965i | 0 | ||||||
443.2 | −1.38743 | − | 0.273914i | 0.572461 | − | 0.572461i | 1.84994 | + | 0.760074i | 0 | −0.951057 | + | 0.637447i | 1.95303 | + | 1.95303i | −2.35848 | − | 1.56128i | 2.34458i | 0 | ||||||
443.3 | −1.34942 | + | 0.423159i | −1.02677 | + | 1.02677i | 1.64187 | − | 1.14204i | 0 | 0.951057 | − | 1.82003i | −1.08889 | − | 1.08889i | −1.73231 | + | 2.23587i | 0.891491i | 0 | ||||||
443.4 | −1.14764 | + | 0.826394i | −1.82972 | + | 1.82972i | 0.634146 | − | 1.89680i | 0 | 0.587785 | − | 3.61192i | 0.611043 | + | 0.611043i | 0.839736 | + | 2.70090i | − | 3.69572i | 0 | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
20.d | odd | 2 | 1 | inner |
20.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 500.2.e.c | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 500.2.e.c | ✓ | 32 |
5.b | even | 2 | 1 | inner | 500.2.e.c | ✓ | 32 |
5.c | odd | 4 | 2 | inner | 500.2.e.c | ✓ | 32 |
20.d | odd | 2 | 1 | inner | 500.2.e.c | ✓ | 32 |
20.e | even | 4 | 2 | inner | 500.2.e.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
500.2.e.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
500.2.e.c | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
500.2.e.c | ✓ | 32 | 5.b | even | 2 | 1 | inner |
500.2.e.c | ✓ | 32 | 5.c | odd | 4 | 2 | inner |
500.2.e.c | ✓ | 32 | 20.d | odd | 2 | 1 | inner |
500.2.e.c | ✓ | 32 | 20.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 50T_{3}^{12} + 235T_{3}^{8} + 150T_{3}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\).