Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [561,2,Mod(560,561)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(561, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("561.560");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 561 = 3 \cdot 11 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 561.h (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: | \(4.47960755339\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
560.1 | −2.62229 | −0.492467 | − | 1.66056i | 4.87641 | −1.87989 | 1.29139 | + | 4.35448i | −4.40416 | −7.54277 | −2.51495 | + | 1.63555i | 4.92962 | ||||||||||||
560.2 | −2.62229 | −0.492467 | + | 1.66056i | 4.87641 | −1.87989 | 1.29139 | − | 4.35448i | −4.40416 | −7.54277 | −2.51495 | − | 1.63555i | 4.92962 | ||||||||||||
560.3 | −2.62229 | 0.492467 | − | 1.66056i | 4.87641 | 1.87989 | −1.29139 | + | 4.35448i | 4.40416 | −7.54277 | −2.51495 | − | 1.63555i | −4.92962 | ||||||||||||
560.4 | −2.62229 | 0.492467 | + | 1.66056i | 4.87641 | 1.87989 | −1.29139 | − | 4.35448i | 4.40416 | −7.54277 | −2.51495 | + | 1.63555i | −4.92962 | ||||||||||||
560.5 | −2.41427 | −1.39719 | − | 1.02364i | 3.82869 | 1.10630 | 3.37320 | + | 2.47135i | −0.177376 | −4.41495 | 0.904305 | + | 2.86046i | −2.67090 | ||||||||||||
560.6 | −2.41427 | −1.39719 | + | 1.02364i | 3.82869 | 1.10630 | 3.37320 | − | 2.47135i | −0.177376 | −4.41495 | 0.904305 | − | 2.86046i | −2.67090 | ||||||||||||
560.7 | −2.41427 | 1.39719 | − | 1.02364i | 3.82869 | −1.10630 | −3.37320 | + | 2.47135i | 0.177376 | −4.41495 | 0.904305 | − | 2.86046i | 2.67090 | ||||||||||||
560.8 | −2.41427 | 1.39719 | + | 1.02364i | 3.82869 | −1.10630 | −3.37320 | − | 2.47135i | 0.177376 | −4.41495 | 0.904305 | + | 2.86046i | 2.67090 | ||||||||||||
560.9 | −1.96484 | −0.118967 | − | 1.72796i | 1.86059 | 3.79231 | 0.233751 | + | 3.39516i | −2.62787 | 0.273922 | −2.97169 | + | 0.411141i | −7.45127 | ||||||||||||
560.10 | −1.96484 | −0.118967 | + | 1.72796i | 1.86059 | 3.79231 | 0.233751 | − | 3.39516i | −2.62787 | 0.273922 | −2.97169 | − | 0.411141i | −7.45127 | ||||||||||||
560.11 | −1.96484 | 0.118967 | − | 1.72796i | 1.86059 | −3.79231 | −0.233751 | + | 3.39516i | 2.62787 | 0.273922 | −2.97169 | − | 0.411141i | 7.45127 | ||||||||||||
560.12 | −1.96484 | 0.118967 | + | 1.72796i | 1.86059 | −3.79231 | −0.233751 | − | 3.39516i | 2.62787 | 0.273922 | −2.97169 | + | 0.411141i | 7.45127 | ||||||||||||
560.13 | −1.75720 | −1.65768 | − | 0.502106i | 1.08774 | −2.02373 | 2.91286 | + | 0.882298i | −0.533738 | 1.60303 | 2.49578 | + | 1.66466i | 3.55610 | ||||||||||||
560.14 | −1.75720 | −1.65768 | + | 0.502106i | 1.08774 | −2.02373 | 2.91286 | − | 0.882298i | −0.533738 | 1.60303 | 2.49578 | − | 1.66466i | 3.55610 | ||||||||||||
560.15 | −1.75720 | 1.65768 | − | 0.502106i | 1.08774 | 2.02373 | −2.91286 | + | 0.882298i | 0.533738 | 1.60303 | 2.49578 | − | 1.66466i | −3.55610 | ||||||||||||
560.16 | −1.75720 | 1.65768 | + | 0.502106i | 1.08774 | 2.02373 | −2.91286 | − | 0.882298i | 0.533738 | 1.60303 | 2.49578 | + | 1.66466i | −3.55610 | ||||||||||||
560.17 | −1.63012 | −1.13840 | − | 1.30539i | 0.657285 | 1.34971 | 1.85573 | + | 2.12793i | 4.67831 | 2.18878 | −0.408070 | + | 2.97212i | −2.20018 | ||||||||||||
560.18 | −1.63012 | −1.13840 | + | 1.30539i | 0.657285 | 1.34971 | 1.85573 | − | 2.12793i | 4.67831 | 2.18878 | −0.408070 | − | 2.97212i | −2.20018 | ||||||||||||
560.19 | −1.63012 | 1.13840 | − | 1.30539i | 0.657285 | −1.34971 | −1.85573 | + | 2.12793i | −4.67831 | 2.18878 | −0.408070 | − | 2.97212i | 2.20018 | ||||||||||||
560.20 | −1.63012 | 1.13840 | + | 1.30539i | 0.657285 | −1.34971 | −1.85573 | − | 2.12793i | −4.67831 | 2.18878 | −0.408070 | + | 2.97212i | 2.20018 | ||||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
17.b | even | 2 | 1 | inner |
33.d | even | 2 | 1 | inner |
51.c | odd | 2 | 1 | inner |
187.b | odd | 2 | 1 | inner |
561.h | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 561.2.h.d | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
11.b | odd | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
17.b | even | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
33.d | even | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
51.c | odd | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
187.b | odd | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
561.h | even | 2 | 1 | inner | 561.2.h.d | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
561.2.h.d | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
561.2.h.d | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
561.2.h.d | ✓ | 56 | 11.b | odd | 2 | 1 | inner |
561.2.h.d | ✓ | 56 | 17.b | even | 2 | 1 | inner |
561.2.h.d | ✓ | 56 | 33.d | even | 2 | 1 | inner |
561.2.h.d | ✓ | 56 | 51.c | odd | 2 | 1 | inner |
561.2.h.d | ✓ | 56 | 187.b | odd | 2 | 1 | inner |
561.2.h.d | ✓ | 56 | 561.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 23T_{2}^{12} + 208T_{2}^{10} - 938T_{2}^{8} + 2196T_{2}^{6} - 2480T_{2}^{4} + 1064T_{2}^{2} - 148 \)
acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\).