Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,4,Mod(143,576)]
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([2, 1, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.143");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 576.l (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: | \(33.9851001633\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 144) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 | 0 | 0 | 0 | −15.2065 | + | 15.2065i | 0 | −24.4971 | 0 | 0 | 0 | ||||||||||||||||
143.2 | 0 | 0 | 0 | −11.8474 | + | 11.8474i | 0 | 12.7523 | 0 | 0 | 0 | ||||||||||||||||
143.3 | 0 | 0 | 0 | −11.7679 | + | 11.7679i | 0 | 11.9978 | 0 | 0 | 0 | ||||||||||||||||
143.4 | 0 | 0 | 0 | −11.2962 | + | 11.2962i | 0 | 19.1985 | 0 | 0 | 0 | ||||||||||||||||
143.5 | 0 | 0 | 0 | −9.40057 | + | 9.40057i | 0 | 3.57327 | 0 | 0 | 0 | ||||||||||||||||
143.6 | 0 | 0 | 0 | −7.28730 | + | 7.28730i | 0 | −29.9574 | 0 | 0 | 0 | ||||||||||||||||
143.7 | 0 | 0 | 0 | −6.31601 | + | 6.31601i | 0 | −16.2772 | 0 | 0 | 0 | ||||||||||||||||
143.8 | 0 | 0 | 0 | −6.30986 | + | 6.30986i | 0 | −27.2034 | 0 | 0 | 0 | ||||||||||||||||
143.9 | 0 | 0 | 0 | −3.62348 | + | 3.62348i | 0 | 33.7361 | 0 | 0 | 0 | ||||||||||||||||
143.10 | 0 | 0 | 0 | −3.43656 | + | 3.43656i | 0 | −8.14652 | 0 | 0 | 0 | ||||||||||||||||
143.11 | 0 | 0 | 0 | −3.22357 | + | 3.22357i | 0 | 13.1030 | 0 | 0 | 0 | ||||||||||||||||
143.12 | 0 | 0 | 0 | −2.40838 | + | 2.40838i | 0 | 11.7205 | 0 | 0 | 0 | ||||||||||||||||
143.13 | 0 | 0 | 0 | 2.40838 | − | 2.40838i | 0 | 11.7205 | 0 | 0 | 0 | ||||||||||||||||
143.14 | 0 | 0 | 0 | 3.22357 | − | 3.22357i | 0 | 13.1030 | 0 | 0 | 0 | ||||||||||||||||
143.15 | 0 | 0 | 0 | 3.43656 | − | 3.43656i | 0 | −8.14652 | 0 | 0 | 0 | ||||||||||||||||
143.16 | 0 | 0 | 0 | 3.62348 | − | 3.62348i | 0 | 33.7361 | 0 | 0 | 0 | ||||||||||||||||
143.17 | 0 | 0 | 0 | 6.30986 | − | 6.30986i | 0 | −27.2034 | 0 | 0 | 0 | ||||||||||||||||
143.18 | 0 | 0 | 0 | 6.31601 | − | 6.31601i | 0 | −16.2772 | 0 | 0 | 0 | ||||||||||||||||
143.19 | 0 | 0 | 0 | 7.28730 | − | 7.28730i | 0 | −29.9574 | 0 | 0 | 0 | ||||||||||||||||
143.20 | 0 | 0 | 0 | 9.40057 | − | 9.40057i | 0 | 3.57327 | 0 | 0 | 0 | ||||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 576.4.l.a | 48 | |
3.b | odd | 2 | 1 | inner | 576.4.l.a | 48 | |
4.b | odd | 2 | 1 | 144.4.l.a | ✓ | 48 | |
8.b | even | 2 | 1 | 1152.4.l.a | 48 | ||
8.d | odd | 2 | 1 | 1152.4.l.b | 48 | ||
12.b | even | 2 | 1 | 144.4.l.a | ✓ | 48 | |
16.e | even | 4 | 1 | 144.4.l.a | ✓ | 48 | |
16.e | even | 4 | 1 | 1152.4.l.b | 48 | ||
16.f | odd | 4 | 1 | inner | 576.4.l.a | 48 | |
16.f | odd | 4 | 1 | 1152.4.l.a | 48 | ||
24.f | even | 2 | 1 | 1152.4.l.b | 48 | ||
24.h | odd | 2 | 1 | 1152.4.l.a | 48 | ||
48.i | odd | 4 | 1 | 144.4.l.a | ✓ | 48 | |
48.i | odd | 4 | 1 | 1152.4.l.b | 48 | ||
48.k | even | 4 | 1 | inner | 576.4.l.a | 48 | |
48.k | even | 4 | 1 | 1152.4.l.a | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
144.4.l.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
144.4.l.a | ✓ | 48 | 12.b | even | 2 | 1 | |
144.4.l.a | ✓ | 48 | 16.e | even | 4 | 1 | |
144.4.l.a | ✓ | 48 | 48.i | odd | 4 | 1 | |
576.4.l.a | 48 | 1.a | even | 1 | 1 | trivial | |
576.4.l.a | 48 | 3.b | odd | 2 | 1 | inner | |
576.4.l.a | 48 | 16.f | odd | 4 | 1 | inner | |
576.4.l.a | 48 | 48.k | even | 4 | 1 | inner | |
1152.4.l.a | 48 | 8.b | even | 2 | 1 | ||
1152.4.l.a | 48 | 16.f | odd | 4 | 1 | ||
1152.4.l.a | 48 | 24.h | odd | 2 | 1 | ||
1152.4.l.a | 48 | 48.k | even | 4 | 1 | ||
1152.4.l.b | 48 | 8.d | odd | 2 | 1 | ||
1152.4.l.b | 48 | 16.e | even | 4 | 1 | ||
1152.4.l.b | 48 | 24.f | even | 2 | 1 | ||
1152.4.l.b | 48 | 48.i | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(576, [\chi])\).