Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1560,2,Mod(49,1560)]
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, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1560.49");
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.dr (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: | \(44\) |
Relative dimension: | \(22\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | 0 | −0.866025 | + | 0.500000i | 0 | −2.22388 | + | 0.233190i | 0 | 2.24062 | − | 3.88087i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.2 | 0 | −0.866025 | + | 0.500000i | 0 | −1.70064 | − | 1.45184i | 0 | 1.91976 | − | 3.32513i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.3 | 0 | −0.866025 | + | 0.500000i | 0 | −2.16416 | − | 0.562520i | 0 | −1.46016 | + | 2.52908i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.4 | 0 | −0.866025 | + | 0.500000i | 0 | 0.455587 | + | 2.18916i | 0 | 1.28750 | − | 2.23001i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.5 | 0 | −0.866025 | + | 0.500000i | 0 | 1.63725 | − | 1.52296i | 0 | 1.13279 | − | 1.96206i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.6 | 0 | −0.866025 | + | 0.500000i | 0 | −1.75606 | + | 1.38429i | 0 | −1.15793 | + | 2.00559i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.7 | 0 | −0.866025 | + | 0.500000i | 0 | 2.23438 | − | 0.0868459i | 0 | 1.01600 | − | 1.75977i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.8 | 0 | −0.866025 | + | 0.500000i | 0 | −0.0598114 | + | 2.23527i | 0 | −0.402748 | + | 0.697580i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.9 | 0 | −0.866025 | + | 0.500000i | 0 | 1.24679 | − | 1.85621i | 0 | 0.168505 | − | 0.291860i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.10 | 0 | −0.866025 | + | 0.500000i | 0 | −0.260086 | − | 2.22089i | 0 | −1.86493 | + | 3.23015i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.11 | 0 | −0.866025 | + | 0.500000i | 0 | 2.09061 | + | 0.793323i | 0 | −2.37941 | + | 4.12126i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.12 | 0 | 0.866025 | − | 0.500000i | 0 | 2.20743 | − | 0.356749i | 0 | 2.27952 | − | 3.94824i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.13 | 0 | 0.866025 | − | 0.500000i | 0 | −1.08602 | + | 1.95462i | 0 | 1.85933 | − | 3.22046i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.14 | 0 | 0.866025 | − | 0.500000i | 0 | −1.82229 | + | 1.29586i | 0 | −1.66961 | + | 2.89185i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.15 | 0 | 0.866025 | − | 0.500000i | 0 | −1.49747 | − | 1.66059i | 0 | 1.56480 | − | 2.71031i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.16 | 0 | 0.866025 | − | 0.500000i | 0 | 0.987261 | − | 2.00632i | 0 | −0.931563 | + | 1.61351i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.17 | 0 | 0.866025 | − | 0.500000i | 0 | 0.644860 | − | 2.14106i | 0 | 0.855874 | − | 1.48242i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.18 | 0 | 0.866025 | − | 0.500000i | 0 | 1.50795 | + | 1.65109i | 0 | −0.673071 | + | 1.16579i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.19 | 0 | 0.866025 | − | 0.500000i | 0 | −0.868720 | + | 2.06042i | 0 | −0.489404 | + | 0.847672i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
49.20 | 0 | 0.866025 | − | 0.500000i | 0 | −2.23603 | − | 0.0122650i | 0 | 0.349261 | − | 0.604938i | 0 | 0.500000 | − | 0.866025i | 0 | ||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.l | 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.dr.a | ✓ | 44 |
5.b | even | 2 | 1 | 1560.2.dr.b | yes | 44 | |
13.e | even | 6 | 1 | 1560.2.dr.b | yes | 44 | |
65.l | even | 6 | 1 | inner | 1560.2.dr.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1560.2.dr.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
1560.2.dr.a | ✓ | 44 | 65.l | even | 6 | 1 | inner |
1560.2.dr.b | yes | 44 | 5.b | even | 2 | 1 | |
1560.2.dr.b | yes | 44 | 13.e | even | 6 | 1 |