Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1503,2,Mod(1502,1503)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1503, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1503.1502");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1503 = 3^{2} \cdot 167 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1503.c (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: | \(12.0015154238\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1502.1 | − | 1.18160i | 0 | 0.603817 | −4.40626 | 0 | −3.48779 | − | 3.07668i | 0 | 5.20644i | ||||||||||||||||
1502.2 | 1.18160i | 0 | 0.603817 | −4.40626 | 0 | −3.48779 | 3.07668i | 0 | − | 5.20644i | |||||||||||||||||
1502.3 | − | 2.66906i | 0 | −5.12388 | −3.16492 | 0 | −0.263490 | 8.33782i | 0 | 8.44736i | |||||||||||||||||
1502.4 | 2.66906i | 0 | −5.12388 | −3.16492 | 0 | −0.263490 | − | 8.33782i | 0 | − | 8.44736i | ||||||||||||||||
1502.5 | − | 1.78744i | 0 | −1.19494 | 3.10541 | 0 | 3.91211 | − | 1.43899i | 0 | − | 5.55073i | |||||||||||||||
1502.6 | 1.78744i | 0 | −1.19494 | 3.10541 | 0 | 3.91211 | 1.43899i | 0 | 5.55073i | ||||||||||||||||||
1502.7 | − | 2.52227i | 0 | −4.36184 | 2.54201 | 0 | 2.74873 | 5.95720i | 0 | − | 6.41163i | ||||||||||||||||
1502.8 | 2.52227i | 0 | −4.36184 | 2.54201 | 0 | 2.74873 | − | 5.95720i | 0 | 6.41163i | |||||||||||||||||
1502.9 | − | 1.78653i | 0 | −1.19168 | 3.08210 | 0 | 4.41569 | − | 1.44408i | 0 | − | 5.50625i | |||||||||||||||
1502.10 | 1.78653i | 0 | −1.19168 | 3.08210 | 0 | 4.41569 | 1.44408i | 0 | 5.50625i | ||||||||||||||||||
1502.11 | − | 0.118553i | 0 | 1.98595 | 3.37331 | 0 | 2.05489 | − | 0.472544i | 0 | − | 0.399915i | |||||||||||||||
1502.12 | 0.118553i | 0 | 1.98595 | 3.37331 | 0 | 2.05489 | 0.472544i | 0 | 0.399915i | ||||||||||||||||||
1502.13 | − | 2.46644i | 0 | −4.08332 | 1.88017 | 0 | −1.78260 | 5.13837i | 0 | − | 4.63731i | ||||||||||||||||
1502.14 | 2.46644i | 0 | −4.08332 | 1.88017 | 0 | −1.78260 | − | 5.13837i | 0 | 4.63731i | |||||||||||||||||
1502.15 | − | 2.16730i | 0 | −2.69717 | −1.88079 | 0 | −3.73517 | 1.51098i | 0 | 4.07624i | |||||||||||||||||
1502.16 | 2.16730i | 0 | −2.69717 | −1.88079 | 0 | −3.73517 | − | 1.51098i | 0 | − | 4.07624i | ||||||||||||||||
1502.17 | − | 0.799957i | 0 | 1.36007 | −2.53587 | 0 | −2.19364 | − | 2.68791i | 0 | 2.02859i | ||||||||||||||||
1502.18 | 0.799957i | 0 | 1.36007 | −2.53587 | 0 | −2.19364 | 2.68791i | 0 | − | 2.02859i | |||||||||||||||||
1502.19 | − | 1.84274i | 0 | −1.39570 | −1.15735 | 0 | 0.906437 | − | 1.11357i | 0 | 2.13270i | ||||||||||||||||
1502.20 | 1.84274i | 0 | −1.39570 | −1.15735 | 0 | 0.906437 | 1.11357i | 0 | − | 2.13270i | |||||||||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
167.b | odd | 2 | 1 | inner |
501.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1503.2.c.a | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 1503.2.c.a | ✓ | 56 |
167.b | odd | 2 | 1 | inner | 1503.2.c.a | ✓ | 56 |
501.c | even | 2 | 1 | inner | 1503.2.c.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1503.2.c.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
1503.2.c.a | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
1503.2.c.a | ✓ | 56 | 167.b | odd | 2 | 1 | inner |
1503.2.c.a | ✓ | 56 | 501.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1503, [\chi])\).