Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1288,2,Mod(321,1288)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1288.321");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1288 = 2^{3} \cdot 7 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1288.f (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: | \(10.2847317803\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
321.1 | 0 | − | 3.35841i | 0 | −3.31812 | 0 | 2.64107 | + | 0.157353i | 0 | −8.27890 | 0 | |||||||||||||||
321.2 | 0 | 3.35841i | 0 | −3.31812 | 0 | 2.64107 | − | 0.157353i | 0 | −8.27890 | 0 | ||||||||||||||||
321.3 | 0 | − | 2.02563i | 0 | 4.00951 | 0 | 1.45896 | + | 2.20713i | 0 | −1.10319 | 0 | |||||||||||||||
321.4 | 0 | 2.02563i | 0 | 4.00951 | 0 | 1.45896 | − | 2.20713i | 0 | −1.10319 | 0 | ||||||||||||||||
321.5 | 0 | − | 0.294736i | 0 | 3.26890 | 0 | −1.37990 | + | 2.25740i | 0 | 2.91313 | 0 | |||||||||||||||
321.6 | 0 | 0.294736i | 0 | 3.26890 | 0 | −1.37990 | − | 2.25740i | 0 | 2.91313 | 0 | ||||||||||||||||
321.7 | 0 | − | 2.26017i | 0 | −0.642418 | 0 | 0.344791 | + | 2.62319i | 0 | −2.10835 | 0 | |||||||||||||||
321.8 | 0 | 2.26017i | 0 | −0.642418 | 0 | 0.344791 | − | 2.62319i | 0 | −2.10835 | 0 | ||||||||||||||||
321.9 | 0 | − | 2.29674i | 0 | −0.705527 | 0 | −2.39836 | + | 1.11708i | 0 | −2.27499 | 0 | |||||||||||||||
321.10 | 0 | 2.29674i | 0 | −0.705527 | 0 | −2.39836 | − | 1.11708i | 0 | −2.27499 | 0 | ||||||||||||||||
321.11 | 0 | − | 0.384309i | 0 | 1.14794 | 0 | 2.57314 | + | 0.615588i | 0 | 2.85231 | 0 | |||||||||||||||
321.12 | 0 | 0.384309i | 0 | 1.14794 | 0 | 2.57314 | − | 0.615588i | 0 | 2.85231 | 0 | ||||||||||||||||
321.13 | 0 | − | 0.384309i | 0 | −1.14794 | 0 | −2.57314 | − | 0.615588i | 0 | 2.85231 | 0 | |||||||||||||||
321.14 | 0 | 0.384309i | 0 | −1.14794 | 0 | −2.57314 | + | 0.615588i | 0 | 2.85231 | 0 | ||||||||||||||||
321.15 | 0 | − | 2.29674i | 0 | 0.705527 | 0 | 2.39836 | − | 1.11708i | 0 | −2.27499 | 0 | |||||||||||||||
321.16 | 0 | 2.29674i | 0 | 0.705527 | 0 | 2.39836 | + | 1.11708i | 0 | −2.27499 | 0 | ||||||||||||||||
321.17 | 0 | − | 2.26017i | 0 | 0.642418 | 0 | −0.344791 | − | 2.62319i | 0 | −2.10835 | 0 | |||||||||||||||
321.18 | 0 | 2.26017i | 0 | 0.642418 | 0 | −0.344791 | + | 2.62319i | 0 | −2.10835 | 0 | ||||||||||||||||
321.19 | 0 | − | 0.294736i | 0 | −3.26890 | 0 | 1.37990 | − | 2.25740i | 0 | 2.91313 | 0 | |||||||||||||||
321.20 | 0 | 0.294736i | 0 | −3.26890 | 0 | 1.37990 | + | 2.25740i | 0 | 2.91313 | 0 | ||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
161.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1288.2.f.a | ✓ | 24 |
4.b | odd | 2 | 1 | 2576.2.f.h | 24 | ||
7.b | odd | 2 | 1 | inner | 1288.2.f.a | ✓ | 24 |
23.b | odd | 2 | 1 | inner | 1288.2.f.a | ✓ | 24 |
28.d | even | 2 | 1 | 2576.2.f.h | 24 | ||
92.b | even | 2 | 1 | 2576.2.f.h | 24 | ||
161.c | even | 2 | 1 | inner | 1288.2.f.a | ✓ | 24 |
644.h | odd | 2 | 1 | 2576.2.f.h | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1288.2.f.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
1288.2.f.a | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
1288.2.f.a | ✓ | 24 | 23.b | odd | 2 | 1 | inner |
1288.2.f.a | ✓ | 24 | 161.c | even | 2 | 1 | inner |
2576.2.f.h | 24 | 4.b | odd | 2 | 1 | ||
2576.2.f.h | 24 | 28.d | even | 2 | 1 | ||
2576.2.f.h | 24 | 92.b | even | 2 | 1 | ||
2576.2.f.h | 24 | 644.h | odd | 2 | 1 |