Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2016,2,Mod(1025,2016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2016.1025");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2016.bt (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: | \(16.0978410475\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1025.1 | 0 | 0 | 0 | −1.67440 | − | 2.90015i | 0 | 2.56210 | − | 0.660025i | 0 | 0 | 0 | ||||||||||||||
1025.2 | 0 | 0 | 0 | −1.67440 | − | 2.90015i | 0 | −2.56210 | + | 0.660025i | 0 | 0 | 0 | ||||||||||||||
1025.3 | 0 | 0 | 0 | −1.34622 | − | 2.33172i | 0 | −1.77975 | + | 1.95768i | 0 | 0 | 0 | ||||||||||||||
1025.4 | 0 | 0 | 0 | −1.34622 | − | 2.33172i | 0 | 1.77975 | − | 1.95768i | 0 | 0 | 0 | ||||||||||||||
1025.5 | 0 | 0 | 0 | −0.808379 | − | 1.40015i | 0 | 1.50468 | + | 2.17622i | 0 | 0 | 0 | ||||||||||||||
1025.6 | 0 | 0 | 0 | −0.808379 | − | 1.40015i | 0 | −1.50468 | − | 2.17622i | 0 | 0 | 0 | ||||||||||||||
1025.7 | 0 | 0 | 0 | −0.480194 | − | 0.831721i | 0 | 0.0637645 | + | 2.64498i | 0 | 0 | 0 | ||||||||||||||
1025.8 | 0 | 0 | 0 | −0.480194 | − | 0.831721i | 0 | −0.0637645 | − | 2.64498i | 0 | 0 | 0 | ||||||||||||||
1025.9 | 0 | 0 | 0 | 0.480194 | + | 0.831721i | 0 | −0.0637645 | − | 2.64498i | 0 | 0 | 0 | ||||||||||||||
1025.10 | 0 | 0 | 0 | 0.480194 | + | 0.831721i | 0 | 0.0637645 | + | 2.64498i | 0 | 0 | 0 | ||||||||||||||
1025.11 | 0 | 0 | 0 | 0.808379 | + | 1.40015i | 0 | −1.50468 | − | 2.17622i | 0 | 0 | 0 | ||||||||||||||
1025.12 | 0 | 0 | 0 | 0.808379 | + | 1.40015i | 0 | 1.50468 | + | 2.17622i | 0 | 0 | 0 | ||||||||||||||
1025.13 | 0 | 0 | 0 | 1.34622 | + | 2.33172i | 0 | 1.77975 | − | 1.95768i | 0 | 0 | 0 | ||||||||||||||
1025.14 | 0 | 0 | 0 | 1.34622 | + | 2.33172i | 0 | −1.77975 | + | 1.95768i | 0 | 0 | 0 | ||||||||||||||
1025.15 | 0 | 0 | 0 | 1.67440 | + | 2.90015i | 0 | −2.56210 | + | 0.660025i | 0 | 0 | 0 | ||||||||||||||
1025.16 | 0 | 0 | 0 | 1.67440 | + | 2.90015i | 0 | 2.56210 | − | 0.660025i | 0 | 0 | 0 | ||||||||||||||
1601.1 | 0 | 0 | 0 | −1.67440 | + | 2.90015i | 0 | 2.56210 | + | 0.660025i | 0 | 0 | 0 | ||||||||||||||
1601.2 | 0 | 0 | 0 | −1.67440 | + | 2.90015i | 0 | −2.56210 | − | 0.660025i | 0 | 0 | 0 | ||||||||||||||
1601.3 | 0 | 0 | 0 | −1.34622 | + | 2.33172i | 0 | −1.77975 | − | 1.95768i | 0 | 0 | 0 | ||||||||||||||
1601.4 | 0 | 0 | 0 | −1.34622 | + | 2.33172i | 0 | 1.77975 | + | 1.95768i | 0 | 0 | 0 | ||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
12.b | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
28.f | even | 6 | 1 | inner |
84.j | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2016.2.bt.b | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
4.b | odd | 2 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
7.d | odd | 6 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
12.b | even | 2 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
21.g | even | 6 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
28.f | even | 6 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
84.j | odd | 6 | 1 | inner | 2016.2.bt.b | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2016.2.bt.b | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
2016.2.bt.b | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
2016.2.bt.b | ✓ | 32 | 4.b | odd | 2 | 1 | inner |
2016.2.bt.b | ✓ | 32 | 7.d | odd | 6 | 1 | inner |
2016.2.bt.b | ✓ | 32 | 12.b | even | 2 | 1 | inner |
2016.2.bt.b | ✓ | 32 | 21.g | even | 6 | 1 | inner |
2016.2.bt.b | ✓ | 32 | 28.f | even | 6 | 1 | inner |
2016.2.bt.b | ✓ | 32 | 84.j | odd | 6 | 1 | inner |