Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,7,Mod(17,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.17");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 576.j (of order \(4\), degree \(2\), not 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: | \(132.511152165\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 144) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | 0 | 0 | −168.179 | − | 168.179i | 0 | 260.196i | 0 | 0 | 0 | ||||||||||||||||
17.2 | 0 | 0 | 0 | −158.404 | − | 158.404i | 0 | − | 133.106i | 0 | 0 | 0 | |||||||||||||||
17.3 | 0 | 0 | 0 | −143.326 | − | 143.326i | 0 | 548.969i | 0 | 0 | 0 | ||||||||||||||||
17.4 | 0 | 0 | 0 | −141.033 | − | 141.033i | 0 | − | 48.8223i | 0 | 0 | 0 | |||||||||||||||
17.5 | 0 | 0 | 0 | −140.082 | − | 140.082i | 0 | − | 123.966i | 0 | 0 | 0 | |||||||||||||||
17.6 | 0 | 0 | 0 | −128.928 | − | 128.928i | 0 | − | 317.300i | 0 | 0 | 0 | |||||||||||||||
17.7 | 0 | 0 | 0 | −125.068 | − | 125.068i | 0 | 645.212i | 0 | 0 | 0 | ||||||||||||||||
17.8 | 0 | 0 | 0 | −109.398 | − | 109.398i | 0 | − | 428.243i | 0 | 0 | 0 | |||||||||||||||
17.9 | 0 | 0 | 0 | −108.765 | − | 108.765i | 0 | − | 659.213i | 0 | 0 | 0 | |||||||||||||||
17.10 | 0 | 0 | 0 | −102.459 | − | 102.459i | 0 | 388.352i | 0 | 0 | 0 | ||||||||||||||||
17.11 | 0 | 0 | 0 | −100.508 | − | 100.508i | 0 | − | 48.6019i | 0 | 0 | 0 | |||||||||||||||
17.12 | 0 | 0 | 0 | −95.8031 | − | 95.8031i | 0 | 356.832i | 0 | 0 | 0 | ||||||||||||||||
17.13 | 0 | 0 | 0 | −78.6085 | − | 78.6085i | 0 | − | 255.650i | 0 | 0 | 0 | |||||||||||||||
17.14 | 0 | 0 | 0 | −71.4717 | − | 71.4717i | 0 | − | 315.658i | 0 | 0 | 0 | |||||||||||||||
17.15 | 0 | 0 | 0 | −68.0621 | − | 68.0621i | 0 | − | 538.920i | 0 | 0 | 0 | |||||||||||||||
17.16 | 0 | 0 | 0 | −67.0992 | − | 67.0992i | 0 | 212.431i | 0 | 0 | 0 | ||||||||||||||||
17.17 | 0 | 0 | 0 | −43.8575 | − | 43.8575i | 0 | 133.731i | 0 | 0 | 0 | ||||||||||||||||
17.18 | 0 | 0 | 0 | −41.7984 | − | 41.7984i | 0 | − | 240.685i | 0 | 0 | 0 | |||||||||||||||
17.19 | 0 | 0 | 0 | −33.4257 | − | 33.4257i | 0 | − | 175.919i | 0 | 0 | 0 | |||||||||||||||
17.20 | 0 | 0 | 0 | −15.2729 | − | 15.2729i | 0 | 498.002i | 0 | 0 | 0 | ||||||||||||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 576.7.j.a | 96 | |
3.b | odd | 2 | 1 | inner | 576.7.j.a | 96 | |
4.b | odd | 2 | 1 | 144.7.j.a | ✓ | 96 | |
12.b | even | 2 | 1 | 144.7.j.a | ✓ | 96 | |
16.e | even | 4 | 1 | inner | 576.7.j.a | 96 | |
16.f | odd | 4 | 1 | 144.7.j.a | ✓ | 96 | |
48.i | odd | 4 | 1 | inner | 576.7.j.a | 96 | |
48.k | even | 4 | 1 | 144.7.j.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
144.7.j.a | ✓ | 96 | 4.b | odd | 2 | 1 | |
144.7.j.a | ✓ | 96 | 12.b | even | 2 | 1 | |
144.7.j.a | ✓ | 96 | 16.f | odd | 4 | 1 | |
144.7.j.a | ✓ | 96 | 48.k | even | 4 | 1 | |
576.7.j.a | 96 | 1.a | even | 1 | 1 | trivial | |
576.7.j.a | 96 | 3.b | odd | 2 | 1 | inner | |
576.7.j.a | 96 | 16.e | even | 4 | 1 | inner | |
576.7.j.a | 96 | 48.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(576, [\chi])\).