Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [294,4,Mod(293,294)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("294.293");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 294.d (of order \(2\), degree \(1\), 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: | \(17.3465615417\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
293.1 | − | 2.00000i | −0.390244 | + | 5.18148i | −4.00000 | 3.94125 | 10.3630 | + | 0.780489i | 0 | 8.00000i | −26.6954 | − | 4.04408i | − | 7.88249i | ||||||||||
293.2 | 2.00000i | −0.390244 | − | 5.18148i | −4.00000 | 3.94125 | 10.3630 | − | 0.780489i | 0 | − | 8.00000i | −26.6954 | + | 4.04408i | 7.88249i | |||||||||||
293.3 | − | 2.00000i | −5.18995 | + | 0.253820i | −4.00000 | 2.63152 | 0.507640 | + | 10.3799i | 0 | 8.00000i | 26.8712 | − | 2.63463i | − | 5.26303i | ||||||||||
293.4 | 2.00000i | −5.18995 | − | 0.253820i | −4.00000 | 2.63152 | 0.507640 | − | 10.3799i | 0 | − | 8.00000i | 26.8712 | + | 2.63463i | 5.26303i | |||||||||||
293.5 | − | 2.00000i | −4.92023 | + | 1.67071i | −4.00000 | 21.7262 | 3.34143 | + | 9.84047i | 0 | 8.00000i | 21.4174 | − | 16.4406i | − | 43.4525i | ||||||||||
293.6 | 2.00000i | −4.92023 | − | 1.67071i | −4.00000 | 21.7262 | 3.34143 | − | 9.84047i | 0 | − | 8.00000i | 21.4174 | + | 16.4406i | 43.4525i | |||||||||||
293.7 | − | 2.00000i | −4.16974 | − | 3.10053i | −4.00000 | −16.5401 | −6.20106 | + | 8.33948i | 0 | 8.00000i | 7.77344 | + | 25.8568i | 33.0801i | |||||||||||
293.8 | 2.00000i | −4.16974 | + | 3.10053i | −4.00000 | −16.5401 | −6.20106 | − | 8.33948i | 0 | − | 8.00000i | 7.77344 | − | 25.8568i | − | 33.0801i | ||||||||||
293.9 | − | 2.00000i | 2.88629 | + | 4.32080i | −4.00000 | −9.77682 | 8.64161 | − | 5.77258i | 0 | 8.00000i | −10.3387 | + | 24.9422i | 19.5536i | |||||||||||
293.10 | 2.00000i | 2.88629 | − | 4.32080i | −4.00000 | −9.77682 | 8.64161 | + | 5.77258i | 0 | − | 8.00000i | −10.3387 | − | 24.9422i | − | 19.5536i | ||||||||||
293.11 | − | 2.00000i | 4.47058 | − | 2.64839i | −4.00000 | −0.593965 | −5.29677 | − | 8.94115i | 0 | 8.00000i | 12.9721 | − | 23.6796i | 1.18793i | |||||||||||
293.12 | 2.00000i | 4.47058 | + | 2.64839i | −4.00000 | −0.593965 | −5.29677 | + | 8.94115i | 0 | − | 8.00000i | 12.9721 | + | 23.6796i | − | 1.18793i | ||||||||||
293.13 | − | 2.00000i | −4.47058 | + | 2.64839i | −4.00000 | 0.593965 | 5.29677 | + | 8.94115i | 0 | 8.00000i | 12.9721 | − | 23.6796i | − | 1.18793i | ||||||||||
293.14 | 2.00000i | −4.47058 | − | 2.64839i | −4.00000 | 0.593965 | 5.29677 | − | 8.94115i | 0 | − | 8.00000i | 12.9721 | + | 23.6796i | 1.18793i | |||||||||||
293.15 | − | 2.00000i | −2.88629 | − | 4.32080i | −4.00000 | 9.77682 | −8.64161 | + | 5.77258i | 0 | 8.00000i | −10.3387 | + | 24.9422i | − | 19.5536i | ||||||||||
293.16 | 2.00000i | −2.88629 | + | 4.32080i | −4.00000 | 9.77682 | −8.64161 | − | 5.77258i | 0 | − | 8.00000i | −10.3387 | − | 24.9422i | 19.5536i | |||||||||||
293.17 | − | 2.00000i | 4.16974 | + | 3.10053i | −4.00000 | 16.5401 | 6.20106 | − | 8.33948i | 0 | 8.00000i | 7.77344 | + | 25.8568i | − | 33.0801i | ||||||||||
293.18 | 2.00000i | 4.16974 | − | 3.10053i | −4.00000 | 16.5401 | 6.20106 | + | 8.33948i | 0 | − | 8.00000i | 7.77344 | − | 25.8568i | 33.0801i | |||||||||||
293.19 | − | 2.00000i | 4.92023 | − | 1.67071i | −4.00000 | −21.7262 | −3.34143 | − | 9.84047i | 0 | 8.00000i | 21.4174 | − | 16.4406i | 43.4525i | |||||||||||
293.20 | 2.00000i | 4.92023 | + | 1.67071i | −4.00000 | −21.7262 | −3.34143 | + | 9.84047i | 0 | − | 8.00000i | 21.4174 | + | 16.4406i | − | 43.4525i | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 294.4.d.b | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 294.4.d.b | ✓ | 24 |
7.b | odd | 2 | 1 | inner | 294.4.d.b | ✓ | 24 |
7.c | even | 3 | 2 | 294.4.f.c | 48 | ||
7.d | odd | 6 | 2 | 294.4.f.c | 48 | ||
21.c | even | 2 | 1 | inner | 294.4.d.b | ✓ | 24 |
21.g | even | 6 | 2 | 294.4.f.c | 48 | ||
21.h | odd | 6 | 2 | 294.4.f.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
294.4.d.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
294.4.d.b | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
294.4.d.b | ✓ | 24 | 7.b | odd | 2 | 1 | inner |
294.4.d.b | ✓ | 24 | 21.c | even | 2 | 1 | inner |
294.4.f.c | 48 | 7.c | even | 3 | 2 | ||
294.4.f.c | 48 | 7.d | odd | 6 | 2 | ||
294.4.f.c | 48 | 21.g | even | 6 | 2 | ||
294.4.f.c | 48 | 21.h | odd | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 864T_{5}^{10} + 219708T_{5}^{8} - 17012128T_{5}^{6} + 304745412T_{5}^{4} - 1433158464T_{5}^{2} + 468424832 \) acting on \(S_{4}^{\mathrm{new}}(294, [\chi])\).