Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,3,Mod(161,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.161");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.n (of order \(2\), degree \(1\), 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: | \(26.1581053786\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
161.1 | 0 | −2.99223 | − | 0.215720i | 0 | −2.23607 | 0 | −8.57552 | 0 | 8.90693 | + | 1.29097i | 0 | ||||||||||||||
161.2 | 0 | −2.99223 | − | 0.215720i | 0 | 2.23607 | 0 | 8.57552 | 0 | 8.90693 | + | 1.29097i | 0 | ||||||||||||||
161.3 | 0 | −2.99223 | + | 0.215720i | 0 | −2.23607 | 0 | −8.57552 | 0 | 8.90693 | − | 1.29097i | 0 | ||||||||||||||
161.4 | 0 | −2.99223 | + | 0.215720i | 0 | 2.23607 | 0 | 8.57552 | 0 | 8.90693 | − | 1.29097i | 0 | ||||||||||||||
161.5 | 0 | −1.61015 | − | 2.53129i | 0 | −2.23607 | 0 | 8.16215 | 0 | −3.81483 | + | 8.15151i | 0 | ||||||||||||||
161.6 | 0 | −1.61015 | − | 2.53129i | 0 | 2.23607 | 0 | −8.16215 | 0 | −3.81483 | + | 8.15151i | 0 | ||||||||||||||
161.7 | 0 | −1.61015 | + | 2.53129i | 0 | −2.23607 | 0 | 8.16215 | 0 | −3.81483 | − | 8.15151i | 0 | ||||||||||||||
161.8 | 0 | −1.61015 | + | 2.53129i | 0 | 2.23607 | 0 | −8.16215 | 0 | −3.81483 | − | 8.15151i | 0 | ||||||||||||||
161.9 | 0 | −1.20580 | − | 2.74701i | 0 | −2.23607 | 0 | −4.45418 | 0 | −6.09210 | + | 6.62468i | 0 | ||||||||||||||
161.10 | 0 | −1.20580 | − | 2.74701i | 0 | 2.23607 | 0 | 4.45418 | 0 | −6.09210 | + | 6.62468i | 0 | ||||||||||||||
161.11 | 0 | −1.20580 | + | 2.74701i | 0 | −2.23607 | 0 | −4.45418 | 0 | −6.09210 | − | 6.62468i | 0 | ||||||||||||||
161.12 | 0 | −1.20580 | + | 2.74701i | 0 | 2.23607 | 0 | 4.45418 | 0 | −6.09210 | − | 6.62468i | 0 | ||||||||||||||
161.13 | 0 | 1.20580 | − | 2.74701i | 0 | −2.23607 | 0 | 4.45418 | 0 | −6.09210 | − | 6.62468i | 0 | ||||||||||||||
161.14 | 0 | 1.20580 | − | 2.74701i | 0 | 2.23607 | 0 | −4.45418 | 0 | −6.09210 | − | 6.62468i | 0 | ||||||||||||||
161.15 | 0 | 1.20580 | + | 2.74701i | 0 | −2.23607 | 0 | 4.45418 | 0 | −6.09210 | + | 6.62468i | 0 | ||||||||||||||
161.16 | 0 | 1.20580 | + | 2.74701i | 0 | 2.23607 | 0 | −4.45418 | 0 | −6.09210 | + | 6.62468i | 0 | ||||||||||||||
161.17 | 0 | 1.61015 | − | 2.53129i | 0 | −2.23607 | 0 | −8.16215 | 0 | −3.81483 | − | 8.15151i | 0 | ||||||||||||||
161.18 | 0 | 1.61015 | − | 2.53129i | 0 | 2.23607 | 0 | 8.16215 | 0 | −3.81483 | − | 8.15151i | 0 | ||||||||||||||
161.19 | 0 | 1.61015 | + | 2.53129i | 0 | −2.23607 | 0 | −8.16215 | 0 | −3.81483 | + | 8.15151i | 0 | ||||||||||||||
161.20 | 0 | 1.61015 | + | 2.53129i | 0 | 2.23607 | 0 | 8.16215 | 0 | −3.81483 | + | 8.15151i | 0 | ||||||||||||||
See all 24 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 |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
24.f | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.3.n.a | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
8.b | even | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
8.d | odd | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
12.b | even | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
24.f | even | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
24.h | odd | 2 | 1 | inner | 960.3.n.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
960.3.n.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
960.3.n.a | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
960.3.n.a | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
960.3.n.a | ✓ | 24 | 8.b | even | 2 | 1 | inner |
960.3.n.a | ✓ | 24 | 8.d | odd | 2 | 1 | inner |
960.3.n.a | ✓ | 24 | 12.b | even | 2 | 1 | inner |
960.3.n.a | ✓ | 24 | 24.f | even | 2 | 1 | inner |
960.3.n.a | ✓ | 24 | 24.h | odd | 2 | 1 | inner |