Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,2,Mod(97,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 5, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 768.n (of order \(8\), degree \(4\), 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: | \(6.13251087523\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{8})\) |
Twist minimal: | no (minimal twist has level 96) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 | 0 | −0.923880 | + | 0.382683i | 0 | −0.750897 | + | 1.81283i | 0 | 0.638460 | + | 0.638460i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.2 | 0 | −0.923880 | + | 0.382683i | 0 | 0.00259461 | − | 0.00626394i | 0 | −2.41880 | − | 2.41880i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.3 | 0 | −0.923880 | + | 0.382683i | 0 | 0.155637 | − | 0.375742i | 0 | 0.709092 | + | 0.709092i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.4 | 0 | −0.923880 | + | 0.382683i | 0 | 1.35803 | − | 3.27858i | 0 | 2.48546 | + | 2.48546i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.5 | 0 | 0.923880 | − | 0.382683i | 0 | −1.36206 | + | 3.28830i | 0 | −2.73097 | − | 2.73097i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.6 | 0 | 0.923880 | − | 0.382683i | 0 | −0.705805 | + | 1.70396i | 0 | 3.24150 | + | 3.24150i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.7 | 0 | 0.923880 | − | 0.382683i | 0 | −0.184062 | + | 0.444366i | 0 | −0.134531 | − | 0.134531i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
97.8 | 0 | 0.923880 | − | 0.382683i | 0 | 1.48656 | − | 3.58888i | 0 | 1.03821 | + | 1.03821i | 0 | 0.707107 | − | 0.707107i | 0 | ||||||||||
289.1 | 0 | −0.382683 | + | 0.923880i | 0 | −3.68816 | + | 1.52768i | 0 | −1.63704 | + | 1.63704i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.2 | 0 | −0.382683 | + | 0.923880i | 0 | −0.825824 | + | 0.342068i | 0 | 1.17750 | − | 1.17750i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.3 | 0 | −0.382683 | + | 0.923880i | 0 | 1.20409 | − | 0.498752i | 0 | 2.59422 | − | 2.59422i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.4 | 0 | −0.382683 | + | 0.923880i | 0 | 1.46213 | − | 0.605634i | 0 | −3.54889 | + | 3.54889i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.5 | 0 | 0.382683 | − | 0.923880i | 0 | −2.14986 | + | 0.890503i | 0 | −1.10001 | + | 1.10001i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.6 | 0 | 0.382683 | − | 0.923880i | 0 | −1.60930 | + | 0.666593i | 0 | −0.589445 | + | 0.589445i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.7 | 0 | 0.382683 | − | 0.923880i | 0 | 2.51374 | − | 1.04122i | 0 | 2.01027 | − | 2.01027i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
289.8 | 0 | 0.382683 | − | 0.923880i | 0 | 3.09318 | − | 1.28124i | 0 | −1.73503 | + | 1.73503i | 0 | −0.707107 | − | 0.707107i | 0 | ||||||||||
481.1 | 0 | −0.382683 | − | 0.923880i | 0 | −3.68816 | − | 1.52768i | 0 | −1.63704 | − | 1.63704i | 0 | −0.707107 | + | 0.707107i | 0 | ||||||||||
481.2 | 0 | −0.382683 | − | 0.923880i | 0 | −0.825824 | − | 0.342068i | 0 | 1.17750 | + | 1.17750i | 0 | −0.707107 | + | 0.707107i | 0 | ||||||||||
481.3 | 0 | −0.382683 | − | 0.923880i | 0 | 1.20409 | + | 0.498752i | 0 | 2.59422 | + | 2.59422i | 0 | −0.707107 | + | 0.707107i | 0 | ||||||||||
481.4 | 0 | −0.382683 | − | 0.923880i | 0 | 1.46213 | + | 0.605634i | 0 | −3.54889 | − | 3.54889i | 0 | −0.707107 | + | 0.707107i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 768.2.n.a | 32 | |
4.b | odd | 2 | 1 | 768.2.n.b | 32 | ||
8.b | even | 2 | 1 | 96.2.n.a | ✓ | 32 | |
8.d | odd | 2 | 1 | 384.2.n.a | 32 | ||
24.f | even | 2 | 1 | 1152.2.v.c | 32 | ||
24.h | odd | 2 | 1 | 288.2.v.d | 32 | ||
32.g | even | 8 | 1 | 96.2.n.a | ✓ | 32 | |
32.g | even | 8 | 1 | inner | 768.2.n.a | 32 | |
32.h | odd | 8 | 1 | 384.2.n.a | 32 | ||
32.h | odd | 8 | 1 | 768.2.n.b | 32 | ||
96.o | even | 8 | 1 | 1152.2.v.c | 32 | ||
96.p | odd | 8 | 1 | 288.2.v.d | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
96.2.n.a | ✓ | 32 | 8.b | even | 2 | 1 | |
96.2.n.a | ✓ | 32 | 32.g | even | 8 | 1 | |
288.2.v.d | 32 | 24.h | odd | 2 | 1 | ||
288.2.v.d | 32 | 96.p | odd | 8 | 1 | ||
384.2.n.a | 32 | 8.d | odd | 2 | 1 | ||
384.2.n.a | 32 | 32.h | odd | 8 | 1 | ||
768.2.n.a | 32 | 1.a | even | 1 | 1 | trivial | |
768.2.n.a | 32 | 32.g | even | 8 | 1 | inner | |
768.2.n.b | 32 | 4.b | odd | 2 | 1 | ||
768.2.n.b | 32 | 32.h | odd | 8 | 1 | ||
1152.2.v.c | 32 | 24.f | even | 2 | 1 | ||
1152.2.v.c | 32 | 96.o | even | 8 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} - 16 T_{7}^{29} + 960 T_{7}^{28} - 352 T_{7}^{27} + 128 T_{7}^{26} - 7328 T_{7}^{25} + \cdots + 1195499776 \) acting on \(S_{2}^{\mathrm{new}}(768, [\chi])\).