Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1710,2,Mod(179,1710)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1710.179");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1710.q (of order \(6\), degree \(2\), 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: | \(13.6544187456\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
179.1 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.65982 | + | 1.49833i | 0 | − | 1.61258i | − | 1.00000i | 0 | 2.18661 | − | 0.467676i | ||||||||
179.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.467676 | + | 2.18661i | 0 | 1.61258i | − | 1.00000i | 0 | 1.49833 | − | 1.65982i | |||||||||
179.3 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.67574 | − | 1.48050i | 0 | 4.01357i | − | 1.00000i | 0 | 0.710988 | + | 2.12002i | |||||||||
179.4 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 2.12002 | + | 0.710988i | 0 | − | 4.01357i | − | 1.00000i | 0 | −1.48050 | − | 1.67574i | ||||||||
179.5 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −2.17236 | + | 0.529970i | 0 | 0.317852i | − | 1.00000i | 0 | 2.14630 | + | 0.627210i | |||||||||
179.6 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.627210 | + | 2.14630i | 0 | − | 0.317852i | − | 1.00000i | 0 | 0.529970 | − | 2.17236i | ||||||||
179.7 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −2.05514 | − | 0.881135i | 0 | − | 0.180093i | − | 1.00000i | 0 | 1.33924 | + | 1.79066i | ||||||||
179.8 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.79066 | + | 1.33924i | 0 | 0.180093i | − | 1.00000i | 0 | −0.881135 | − | 2.05514i | |||||||||
179.9 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.861447 | − | 2.06347i | 0 | − | 3.19371i | − | 1.00000i | 0 | −1.77777 | + | 1.35629i | ||||||||
179.10 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.35629 | − | 1.77777i | 0 | 3.19371i | − | 1.00000i | 0 | −2.06347 | + | 0.861447i | |||||||||
179.11 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −1.54820 | − | 1.61341i | 0 | 1.47489i | − | 1.00000i | 0 | 0.534078 | + | 2.17135i | |||||||||
179.12 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 2.17135 | + | 0.534078i | 0 | − | 1.47489i | − | 1.00000i | 0 | −1.61341 | − | 1.54820i | ||||||||
179.13 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.0434002 | − | 2.23565i | 0 | − | 1.06029i | − | 1.00000i | 0 | −1.08024 | + | 1.95783i | ||||||||
179.14 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.95783 | − | 1.08024i | 0 | 1.06029i | − | 1.00000i | 0 | −2.23565 | − | 0.0434002i | |||||||||
179.15 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0.762301 | − | 2.10212i | 0 | − | 4.86395i | − | 1.00000i | 0 | −1.71123 | + | 1.43934i | ||||||||
179.16 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.43934 | − | 1.71123i | 0 | 4.86395i | − | 1.00000i | 0 | −2.10212 | + | 0.762301i | |||||||||
179.17 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −0.627210 | − | 2.14630i | 0 | − | 0.317852i | 1.00000i | 0 | 0.529970 | − | 2.17236i | |||||||||
179.18 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 2.17236 | − | 0.529970i | 0 | 0.317852i | 1.00000i | 0 | 2.14630 | + | 0.627210i | ||||||||||
179.19 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | −2.17135 | − | 0.534078i | 0 | − | 1.47489i | 1.00000i | 0 | −1.61341 | − | 1.54820i | |||||||||
179.20 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 1.54820 | + | 1.61341i | 0 | 1.47489i | 1.00000i | 0 | 0.534078 | + | 2.17135i | ||||||||||
See all 64 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 |
19.d | odd | 6 | 1 | inner |
57.f | even | 6 | 1 | inner |
95.h | odd | 6 | 1 | inner |
285.q | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1710.2.q.b | ✓ | 64 |
3.b | odd | 2 | 1 | inner | 1710.2.q.b | ✓ | 64 |
5.b | even | 2 | 1 | inner | 1710.2.q.b | ✓ | 64 |
15.d | odd | 2 | 1 | inner | 1710.2.q.b | ✓ | 64 |
19.d | odd | 6 | 1 | inner | 1710.2.q.b | ✓ | 64 |
57.f | even | 6 | 1 | inner | 1710.2.q.b | ✓ | 64 |
95.h | odd | 6 | 1 | inner | 1710.2.q.b | ✓ | 64 |
285.q | even | 6 | 1 | inner | 1710.2.q.b | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1710.2.q.b | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
1710.2.q.b | ✓ | 64 | 3.b | odd | 2 | 1 | inner |
1710.2.q.b | ✓ | 64 | 5.b | even | 2 | 1 | inner |
1710.2.q.b | ✓ | 64 | 15.d | odd | 2 | 1 | inner |
1710.2.q.b | ✓ | 64 | 19.d | odd | 6 | 1 | inner |
1710.2.q.b | ✓ | 64 | 57.f | even | 6 | 1 | inner |
1710.2.q.b | ✓ | 64 | 95.h | odd | 6 | 1 | inner |
1710.2.q.b | ✓ | 64 | 285.q | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 56T_{7}^{14} + 1100T_{7}^{12} + 9232T_{7}^{10} + 33142T_{7}^{8} + 52152T_{7}^{6} + 31212T_{7}^{4} + 3456T_{7}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\).