Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1380,2,Mod(689,1380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 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("1380.689");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1380.n (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: | \(11.0193554789\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
689.1 | 0 | −1.71333 | − | 0.253996i | 0 | −0.923440 | − | 2.03648i | 0 | −2.63922 | 0 | 2.87097 | + | 0.870355i | 0 | ||||||||||||
689.2 | 0 | −1.71333 | − | 0.253996i | 0 | 0.923440 | + | 2.03648i | 0 | 2.63922 | 0 | 2.87097 | + | 0.870355i | 0 | ||||||||||||
689.3 | 0 | −1.71333 | + | 0.253996i | 0 | −0.923440 | + | 2.03648i | 0 | −2.63922 | 0 | 2.87097 | − | 0.870355i | 0 | ||||||||||||
689.4 | 0 | −1.71333 | + | 0.253996i | 0 | 0.923440 | − | 2.03648i | 0 | 2.63922 | 0 | 2.87097 | − | 0.870355i | 0 | ||||||||||||
689.5 | 0 | −1.63297 | − | 0.577419i | 0 | −2.15297 | + | 0.603919i | 0 | 1.63712 | 0 | 2.33317 | + | 1.88581i | 0 | ||||||||||||
689.6 | 0 | −1.63297 | − | 0.577419i | 0 | 2.15297 | − | 0.603919i | 0 | −1.63712 | 0 | 2.33317 | + | 1.88581i | 0 | ||||||||||||
689.7 | 0 | −1.63297 | + | 0.577419i | 0 | −2.15297 | − | 0.603919i | 0 | 1.63712 | 0 | 2.33317 | − | 1.88581i | 0 | ||||||||||||
689.8 | 0 | −1.63297 | + | 0.577419i | 0 | 2.15297 | + | 0.603919i | 0 | −1.63712 | 0 | 2.33317 | − | 1.88581i | 0 | ||||||||||||
689.9 | 0 | −1.35581 | − | 1.07786i | 0 | −1.70581 | − | 1.44575i | 0 | −5.00303 | 0 | 0.676420 | + | 2.92275i | 0 | ||||||||||||
689.10 | 0 | −1.35581 | − | 1.07786i | 0 | 1.70581 | + | 1.44575i | 0 | 5.00303 | 0 | 0.676420 | + | 2.92275i | 0 | ||||||||||||
689.11 | 0 | −1.35581 | + | 1.07786i | 0 | −1.70581 | + | 1.44575i | 0 | −5.00303 | 0 | 0.676420 | − | 2.92275i | 0 | ||||||||||||
689.12 | 0 | −1.35581 | + | 1.07786i | 0 | 1.70581 | − | 1.44575i | 0 | 5.00303 | 0 | 0.676420 | − | 2.92275i | 0 | ||||||||||||
689.13 | 0 | −0.888021 | − | 1.48708i | 0 | −1.29815 | + | 1.82066i | 0 | 0.470527 | 0 | −1.42284 | + | 2.64112i | 0 | ||||||||||||
689.14 | 0 | −0.888021 | − | 1.48708i | 0 | 1.29815 | − | 1.82066i | 0 | −0.470527 | 0 | −1.42284 | + | 2.64112i | 0 | ||||||||||||
689.15 | 0 | −0.888021 | + | 1.48708i | 0 | −1.29815 | − | 1.82066i | 0 | 0.470527 | 0 | −1.42284 | − | 2.64112i | 0 | ||||||||||||
689.16 | 0 | −0.888021 | + | 1.48708i | 0 | 1.29815 | + | 1.82066i | 0 | −0.470527 | 0 | −1.42284 | − | 2.64112i | 0 | ||||||||||||
689.17 | 0 | −0.779801 | − | 1.54658i | 0 | −0.268100 | − | 2.21994i | 0 | 2.49920 | 0 | −1.78382 | + | 2.41205i | 0 | ||||||||||||
689.18 | 0 | −0.779801 | − | 1.54658i | 0 | 0.268100 | + | 2.21994i | 0 | −2.49920 | 0 | −1.78382 | + | 2.41205i | 0 | ||||||||||||
689.19 | 0 | −0.779801 | + | 1.54658i | 0 | −0.268100 | + | 2.21994i | 0 | 2.49920 | 0 | −1.78382 | − | 2.41205i | 0 | ||||||||||||
689.20 | 0 | −0.779801 | + | 1.54658i | 0 | 0.268100 | − | 2.21994i | 0 | −2.49920 | 0 | −1.78382 | − | 2.41205i | 0 | ||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
69.c | even | 2 | 1 | inner |
115.c | odd | 2 | 1 | inner |
345.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1380.2.n.a | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
15.d | odd | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
23.b | odd | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
69.c | even | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
115.c | odd | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
345.h | even | 2 | 1 | inner | 1380.2.n.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1380.2.n.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
1380.2.n.a | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
1380.2.n.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
1380.2.n.a | ✓ | 48 | 15.d | odd | 2 | 1 | inner |
1380.2.n.a | ✓ | 48 | 23.b | odd | 2 | 1 | inner |
1380.2.n.a | ✓ | 48 | 69.c | even | 2 | 1 | inner |
1380.2.n.a | ✓ | 48 | 115.c | odd | 2 | 1 | inner |
1380.2.n.a | ✓ | 48 | 345.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1380, [\chi])\).