Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1088,2,Mod(225,1088)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1088, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1088.225");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1088 = 2^{6} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1088.m (of order \(4\), 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: | \(8.68772373992\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
225.1 | 0 | −1.74370 | − | 1.74370i | 0 | −2.07081 | − | 2.07081i | 0 | 0.691235 | + | 0.691235i | 0 | 3.08101i | 0 | ||||||||||||
225.2 | 0 | −1.74370 | − | 1.74370i | 0 | 2.07081 | + | 2.07081i | 0 | −0.691235 | − | 0.691235i | 0 | 3.08101i | 0 | ||||||||||||
225.3 | 0 | −1.36000 | − | 1.36000i | 0 | −1.15824 | − | 1.15824i | 0 | 2.45561 | + | 2.45561i | 0 | 0.699218i | 0 | ||||||||||||
225.4 | 0 | −1.36000 | − | 1.36000i | 0 | 1.15824 | + | 1.15824i | 0 | −2.45561 | − | 2.45561i | 0 | 0.699218i | 0 | ||||||||||||
225.5 | 0 | −0.391124 | − | 0.391124i | 0 | 2.42619 | + | 2.42619i | 0 | 3.52212 | + | 3.52212i | 0 | − | 2.69404i | 0 | |||||||||||
225.6 | 0 | −0.391124 | − | 0.391124i | 0 | −2.42619 | − | 2.42619i | 0 | −3.52212 | − | 3.52212i | 0 | − | 2.69404i | 0 | |||||||||||
225.7 | 0 | 0.514752 | + | 0.514752i | 0 | −1.38937 | − | 1.38937i | 0 | 0.251879 | + | 0.251879i | 0 | − | 2.47006i | 0 | |||||||||||
225.8 | 0 | 0.514752 | + | 0.514752i | 0 | 1.38937 | + | 1.38937i | 0 | −0.251879 | − | 0.251879i | 0 | − | 2.47006i | 0 | |||||||||||
225.9 | 0 | 1.13543 | + | 1.13543i | 0 | −2.26352 | − | 2.26352i | 0 | 1.15552 | + | 1.15552i | 0 | − | 0.421600i | 0 | |||||||||||
225.10 | 0 | 1.13543 | + | 1.13543i | 0 | 2.26352 | + | 2.26352i | 0 | −1.15552 | − | 1.15552i | 0 | − | 0.421600i | 0 | |||||||||||
225.11 | 0 | 1.84465 | + | 1.84465i | 0 | 0.655716 | + | 0.655716i | 0 | 2.58615 | + | 2.58615i | 0 | 3.80548i | 0 | ||||||||||||
225.12 | 0 | 1.84465 | + | 1.84465i | 0 | −0.655716 | − | 0.655716i | 0 | −2.58615 | − | 2.58615i | 0 | 3.80548i | 0 | ||||||||||||
353.1 | 0 | −1.74370 | + | 1.74370i | 0 | −2.07081 | + | 2.07081i | 0 | 0.691235 | − | 0.691235i | 0 | − | 3.08101i | 0 | |||||||||||
353.2 | 0 | −1.74370 | + | 1.74370i | 0 | 2.07081 | − | 2.07081i | 0 | −0.691235 | + | 0.691235i | 0 | − | 3.08101i | 0 | |||||||||||
353.3 | 0 | −1.36000 | + | 1.36000i | 0 | −1.15824 | + | 1.15824i | 0 | 2.45561 | − | 2.45561i | 0 | − | 0.699218i | 0 | |||||||||||
353.4 | 0 | −1.36000 | + | 1.36000i | 0 | 1.15824 | − | 1.15824i | 0 | −2.45561 | + | 2.45561i | 0 | − | 0.699218i | 0 | |||||||||||
353.5 | 0 | −0.391124 | + | 0.391124i | 0 | 2.42619 | − | 2.42619i | 0 | 3.52212 | − | 3.52212i | 0 | 2.69404i | 0 | ||||||||||||
353.6 | 0 | −0.391124 | + | 0.391124i | 0 | −2.42619 | + | 2.42619i | 0 | −3.52212 | + | 3.52212i | 0 | 2.69404i | 0 | ||||||||||||
353.7 | 0 | 0.514752 | − | 0.514752i | 0 | −1.38937 | + | 1.38937i | 0 | 0.251879 | − | 0.251879i | 0 | 2.47006i | 0 | ||||||||||||
353.8 | 0 | 0.514752 | − | 0.514752i | 0 | 1.38937 | − | 1.38937i | 0 | −0.251879 | + | 0.251879i | 0 | 2.47006i | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
68.f | odd | 4 | 1 | inner |
136.i | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1088.2.m.i | ✓ | 24 |
4.b | odd | 2 | 1 | 1088.2.m.j | yes | 24 | |
8.b | even | 2 | 1 | 1088.2.m.j | yes | 24 | |
8.d | odd | 2 | 1 | inner | 1088.2.m.i | ✓ | 24 |
17.c | even | 4 | 1 | 1088.2.m.j | yes | 24 | |
68.f | odd | 4 | 1 | inner | 1088.2.m.i | ✓ | 24 |
136.i | even | 4 | 1 | inner | 1088.2.m.i | ✓ | 24 |
136.j | odd | 4 | 1 | 1088.2.m.j | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1088.2.m.i | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
1088.2.m.i | ✓ | 24 | 8.d | odd | 2 | 1 | inner |
1088.2.m.i | ✓ | 24 | 68.f | odd | 4 | 1 | inner |
1088.2.m.i | ✓ | 24 | 136.i | even | 4 | 1 | inner |
1088.2.m.j | yes | 24 | 4.b | odd | 2 | 1 | |
1088.2.m.j | yes | 24 | 8.b | even | 2 | 1 | |
1088.2.m.j | yes | 24 | 17.c | even | 4 | 1 | |
1088.2.m.j | yes | 24 | 136.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1088, [\chi])\):
\( T_{3}^{12} + 52T_{3}^{8} + 4T_{3}^{7} - 40T_{3}^{5} + 416T_{3}^{4} - 96T_{3}^{3} + 8T_{3}^{2} + 32T_{3} + 64 \) |
\( T_{5}^{24} + 340T_{5}^{20} + 39840T_{5}^{16} + 1851520T_{5}^{12} + 28493056T_{5}^{8} + 134931456T_{5}^{4} + 84934656 \) |