Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1560,2,Mod(289,1560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1560, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1560.289");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1560.dx (of order \(6\), 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: | \(12.4566627153\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 | 0 | −0.866025 | + | 0.500000i | 0 | −0.679446 | − | 2.13034i | 0 | −4.24457 | − | 2.45060i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.2 | 0 | −0.866025 | + | 0.500000i | 0 | −1.07256 | + | 1.96204i | 0 | 1.99530 | + | 1.15199i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.3 | 0 | −0.866025 | + | 0.500000i | 0 | −0.806317 | − | 2.08563i | 0 | 1.29895 | + | 0.749951i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.4 | 0 | −0.866025 | + | 0.500000i | 0 | 0.529325 | + | 2.17251i | 0 | −0.826715 | − | 0.477304i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.5 | 0 | −0.866025 | + | 0.500000i | 0 | 2.10902 | + | 0.742983i | 0 | 0.532024 | + | 0.307164i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.6 | 0 | −0.866025 | + | 0.500000i | 0 | −2.22917 | − | 0.175549i | 0 | −0.497119 | − | 0.287012i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.7 | 0 | −0.866025 | + | 0.500000i | 0 | 0.157435 | + | 2.23052i | 0 | −2.28206 | − | 1.31755i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.8 | 0 | −0.866025 | + | 0.500000i | 0 | 2.02056 | − | 0.957785i | 0 | −2.81002 | − | 1.62237i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.9 | 0 | −0.866025 | + | 0.500000i | 0 | 1.20368 | − | 1.88445i | 0 | 3.30209 | + | 1.90646i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.10 | 0 | −0.866025 | + | 0.500000i | 0 | −2.23253 | + | 0.125694i | 0 | 4.39815 | + | 2.53927i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.11 | 0 | 0.866025 | − | 0.500000i | 0 | −2.23253 | − | 0.125694i | 0 | −4.39815 | − | 2.53927i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.12 | 0 | 0.866025 | − | 0.500000i | 0 | 1.20368 | + | 1.88445i | 0 | −3.30209 | − | 1.90646i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.13 | 0 | 0.866025 | − | 0.500000i | 0 | 2.02056 | + | 0.957785i | 0 | 2.81002 | + | 1.62237i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.14 | 0 | 0.866025 | − | 0.500000i | 0 | 0.157435 | − | 2.23052i | 0 | 2.28206 | + | 1.31755i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.15 | 0 | 0.866025 | − | 0.500000i | 0 | −2.22917 | + | 0.175549i | 0 | 0.497119 | + | 0.287012i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.16 | 0 | 0.866025 | − | 0.500000i | 0 | 2.10902 | − | 0.742983i | 0 | −0.532024 | − | 0.307164i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.17 | 0 | 0.866025 | − | 0.500000i | 0 | 0.529325 | − | 2.17251i | 0 | 0.826715 | + | 0.477304i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.18 | 0 | 0.866025 | − | 0.500000i | 0 | −0.806317 | + | 2.08563i | 0 | −1.29895 | − | 0.749951i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.19 | 0 | 0.866025 | − | 0.500000i | 0 | −1.07256 | − | 1.96204i | 0 | −1.99530 | − | 1.15199i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
289.20 | 0 | 0.866025 | − | 0.500000i | 0 | −0.679446 | + | 2.13034i | 0 | 4.24457 | + | 2.45060i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
65.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1560.2.dx.b | ✓ | 40 |
5.b | even | 2 | 1 | inner | 1560.2.dx.b | ✓ | 40 |
13.c | even | 3 | 1 | inner | 1560.2.dx.b | ✓ | 40 |
65.n | even | 6 | 1 | inner | 1560.2.dx.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1560.2.dx.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
1560.2.dx.b | ✓ | 40 | 5.b | even | 2 | 1 | inner |
1560.2.dx.b | ✓ | 40 | 13.c | even | 3 | 1 | inner |
1560.2.dx.b | ✓ | 40 | 65.n | even | 6 | 1 | inner |