Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,4,Mod(47,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.47");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 192.k (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: | \(11.3283667211\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 48) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | 0 | −5.19396 | − | 0.151014i | 0 | 4.66675 | + | 4.66675i | 0 | −0.405799 | 0 | 26.9544 | + | 1.56872i | 0 | ||||||||||||
47.2 | 0 | −5.07063 | − | 1.13522i | 0 | −11.2665 | − | 11.2665i | 0 | −30.2121 | 0 | 24.4225 | + | 11.5126i | 0 | ||||||||||||
47.3 | 0 | −4.68788 | + | 2.24138i | 0 | 2.69043 | + | 2.69043i | 0 | −10.6336 | 0 | 16.9524 | − | 21.0147i | 0 | ||||||||||||
47.4 | 0 | −4.43190 | − | 2.71261i | 0 | −3.17566 | − | 3.17566i | 0 | 32.3513 | 0 | 12.2835 | + | 24.0440i | 0 | ||||||||||||
47.5 | 0 | −3.86039 | + | 3.47813i | 0 | −13.5794 | − | 13.5794i | 0 | 19.7355 | 0 | 2.80518 | − | 26.8539i | 0 | ||||||||||||
47.6 | 0 | −3.76578 | − | 3.58035i | 0 | 4.71515 | + | 4.71515i | 0 | −4.67595 | 0 | 1.36225 | + | 26.9656i | 0 | ||||||||||||
47.7 | 0 | −3.47813 | + | 3.86039i | 0 | 13.5794 | + | 13.5794i | 0 | 19.7355 | 0 | −2.80518 | − | 26.8539i | 0 | ||||||||||||
47.8 | 0 | −2.28317 | − | 4.66767i | 0 | 11.5146 | + | 11.5146i | 0 | −0.829117 | 0 | −16.5743 | + | 21.3142i | 0 | ||||||||||||
47.9 | 0 | −2.24138 | + | 4.68788i | 0 | −2.69043 | − | 2.69043i | 0 | −10.6336 | 0 | −16.9524 | − | 21.0147i | 0 | ||||||||||||
47.10 | 0 | −0.563550 | − | 5.16550i | 0 | −13.1633 | − | 13.1633i | 0 | 13.2717 | 0 | −26.3648 | + | 5.82204i | 0 | ||||||||||||
47.11 | 0 | 0.0749974 | − | 5.19561i | 0 | −5.37662 | − | 5.37662i | 0 | −14.8575 | 0 | −26.9888 | − | 0.779314i | 0 | ||||||||||||
47.12 | 0 | 0.151014 | + | 5.19396i | 0 | −4.66675 | − | 4.66675i | 0 | −0.405799 | 0 | −26.9544 | + | 1.56872i | 0 | ||||||||||||
47.13 | 0 | 1.13522 | + | 5.07063i | 0 | 11.2665 | + | 11.2665i | 0 | −30.2121 | 0 | −24.4225 | + | 11.5126i | 0 | ||||||||||||
47.14 | 0 | 1.96089 | − | 4.81196i | 0 | 6.30133 | + | 6.30133i | 0 | −24.6728 | 0 | −19.3098 | − | 18.8714i | 0 | ||||||||||||
47.15 | 0 | 2.65327 | − | 4.46768i | 0 | 5.27809 | + | 5.27809i | 0 | 22.9284 | 0 | −12.9203 | − | 23.7079i | 0 | ||||||||||||
47.16 | 0 | 2.71261 | + | 4.43190i | 0 | 3.17566 | + | 3.17566i | 0 | 32.3513 | 0 | −12.2835 | + | 24.0440i | 0 | ||||||||||||
47.17 | 0 | 3.58035 | + | 3.76578i | 0 | −4.71515 | − | 4.71515i | 0 | −4.67595 | 0 | −1.36225 | + | 26.9656i | 0 | ||||||||||||
47.18 | 0 | 4.46768 | − | 2.65327i | 0 | −5.27809 | − | 5.27809i | 0 | 22.9284 | 0 | 12.9203 | − | 23.7079i | 0 | ||||||||||||
47.19 | 0 | 4.66767 | + | 2.28317i | 0 | −11.5146 | − | 11.5146i | 0 | −0.829117 | 0 | 16.5743 | + | 21.3142i | 0 | ||||||||||||
47.20 | 0 | 4.81196 | − | 1.96089i | 0 | −6.30133 | − | 6.30133i | 0 | −24.6728 | 0 | 19.3098 | − | 18.8714i | 0 | ||||||||||||
See all 44 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 | 192.4.k.a | 44 | |
3.b | odd | 2 | 1 | inner | 192.4.k.a | 44 | |
4.b | odd | 2 | 1 | 48.4.k.a | ✓ | 44 | |
8.b | even | 2 | 1 | 384.4.k.a | 44 | ||
8.d | odd | 2 | 1 | 384.4.k.b | 44 | ||
12.b | even | 2 | 1 | 48.4.k.a | ✓ | 44 | |
16.e | even | 4 | 1 | 48.4.k.a | ✓ | 44 | |
16.e | even | 4 | 1 | 384.4.k.b | 44 | ||
16.f | odd | 4 | 1 | inner | 192.4.k.a | 44 | |
16.f | odd | 4 | 1 | 384.4.k.a | 44 | ||
24.f | even | 2 | 1 | 384.4.k.b | 44 | ||
24.h | odd | 2 | 1 | 384.4.k.a | 44 | ||
48.i | odd | 4 | 1 | 48.4.k.a | ✓ | 44 | |
48.i | odd | 4 | 1 | 384.4.k.b | 44 | ||
48.k | even | 4 | 1 | inner | 192.4.k.a | 44 | |
48.k | even | 4 | 1 | 384.4.k.a | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.4.k.a | ✓ | 44 | 4.b | odd | 2 | 1 | |
48.4.k.a | ✓ | 44 | 12.b | even | 2 | 1 | |
48.4.k.a | ✓ | 44 | 16.e | even | 4 | 1 | |
48.4.k.a | ✓ | 44 | 48.i | odd | 4 | 1 | |
192.4.k.a | 44 | 1.a | even | 1 | 1 | trivial | |
192.4.k.a | 44 | 3.b | odd | 2 | 1 | inner | |
192.4.k.a | 44 | 16.f | odd | 4 | 1 | inner | |
192.4.k.a | 44 | 48.k | even | 4 | 1 | inner | |
384.4.k.a | 44 | 8.b | even | 2 | 1 | ||
384.4.k.a | 44 | 16.f | odd | 4 | 1 | ||
384.4.k.a | 44 | 24.h | odd | 2 | 1 | ||
384.4.k.a | 44 | 48.k | even | 4 | 1 | ||
384.4.k.b | 44 | 8.d | odd | 2 | 1 | ||
384.4.k.b | 44 | 16.e | even | 4 | 1 | ||
384.4.k.b | 44 | 24.f | even | 2 | 1 | ||
384.4.k.b | 44 | 48.i | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(192, [\chi])\).