Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [670,3,Mod(669,670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(670, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("670.669");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 670 = 2 \cdot 5 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 670.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: | \(18.2561777121\) |
Analytic rank: | \(0\) |
Dimension: | \(68\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
669.1 | −1.41421 | −1.77494 | 2.00000 | −3.68387 | + | 3.38070i | 2.51014 | −10.8562 | −2.82843 | −5.84960 | 5.20977 | − | 4.78103i | ||||||||||||||
669.2 | −1.41421 | −0.0124828 | 2.00000 | 4.79667 | − | 1.41135i | 0.0176534 | −9.71384 | −2.82843 | −8.99984 | −6.78352 | + | 1.99596i | ||||||||||||||
669.3 | −1.41421 | −2.51077 | 2.00000 | −1.59553 | + | 4.73860i | 3.55077 | 12.6787 | −2.82843 | −2.69602 | 2.25641 | − | 6.70139i | ||||||||||||||
669.4 | −1.41421 | −3.99608 | 2.00000 | 4.47252 | − | 2.23529i | 5.65132 | 2.78599 | −2.82843 | 6.96868 | −6.32511 | + | 3.16118i | ||||||||||||||
669.5 | −1.41421 | 5.26432 | 2.00000 | −3.36480 | + | 3.69839i | −7.44488 | 5.58736 | −2.82843 | 18.7131 | 4.75855 | − | 5.23032i | ||||||||||||||
669.6 | −1.41421 | −4.71040 | 2.00000 | 2.12438 | + | 4.52626i | 6.66152 | −9.43300 | −2.82843 | 13.1879 | −3.00433 | − | 6.40109i | ||||||||||||||
669.7 | −1.41421 | 4.35599 | 2.00000 | −3.67481 | + | 3.39055i | −6.16030 | −9.33106 | −2.82843 | 9.97462 | 5.19696 | − | 4.79496i | ||||||||||||||
669.8 | −1.41421 | −0.138900 | 2.00000 | 0.630186 | − | 4.96013i | 0.196434 | 8.09234 | −2.82843 | −8.98071 | −0.891217 | + | 7.01468i | ||||||||||||||
669.9 | −1.41421 | 1.81538 | 2.00000 | −4.91531 | − | 0.916390i | −2.56734 | 6.65615 | −2.82843 | −5.70439 | 6.95129 | + | 1.29597i | ||||||||||||||
669.10 | −1.41421 | 2.60604 | 2.00000 | 4.19101 | − | 2.72680i | −3.68550 | 7.60396 | −2.82843 | −2.20856 | −5.92698 | + | 3.85628i | ||||||||||||||
669.11 | −1.41421 | 4.88427 | 2.00000 | 3.13203 | + | 3.89749i | −6.90740 | 5.12422 | −2.82843 | 14.8561 | −4.42935 | − | 5.51188i | ||||||||||||||
669.12 | −1.41421 | −3.95933 | 2.00000 | −4.95609 | − | 0.661209i | 5.59934 | −0.640014 | −2.82843 | 6.67633 | 7.00897 | + | 0.935091i | ||||||||||||||
669.13 | −1.41421 | −0.496387 | 2.00000 | −2.14002 | − | 4.51888i | 0.701998 | −2.51848 | −2.82843 | −8.75360 | 3.02645 | + | 6.39067i | ||||||||||||||
669.14 | −1.41421 | 1.48160 | 2.00000 | −0.518984 | + | 4.97299i | −2.09530 | −3.34558 | −2.82843 | −6.80485 | 0.733954 | − | 7.03287i | ||||||||||||||
669.15 | −1.41421 | −2.06083 | 2.00000 | 4.62161 | − | 1.90807i | 2.91446 | 4.51265 | −2.82843 | −4.75297 | −6.53594 | + | 2.69843i | ||||||||||||||
669.16 | −1.41421 | −5.38418 | 2.00000 | −2.05371 | + | 4.55876i | 7.61437 | 4.57967 | −2.82843 | 19.9893 | 2.90439 | − | 6.44706i | ||||||||||||||
669.17 | −1.41421 | 3.22250 | 2.00000 | 2.93470 | − | 4.04815i | −4.55730 | −7.54023 | −2.82843 | 1.38449 | −4.15030 | + | 5.72495i | ||||||||||||||
669.18 | −1.41421 | 3.22250 | 2.00000 | 2.93470 | + | 4.04815i | −4.55730 | −7.54023 | −2.82843 | 1.38449 | −4.15030 | − | 5.72495i | ||||||||||||||
669.19 | −1.41421 | −5.38418 | 2.00000 | −2.05371 | − | 4.55876i | 7.61437 | 4.57967 | −2.82843 | 19.9893 | 2.90439 | + | 6.44706i | ||||||||||||||
669.20 | −1.41421 | −2.06083 | 2.00000 | 4.62161 | + | 1.90807i | 2.91446 | 4.51265 | −2.82843 | −4.75297 | −6.53594 | − | 2.69843i | ||||||||||||||
See all 68 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
67.b | odd | 2 | 1 | inner |
335.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 670.3.d.a | ✓ | 68 |
5.b | even | 2 | 1 | inner | 670.3.d.a | ✓ | 68 |
67.b | odd | 2 | 1 | inner | 670.3.d.a | ✓ | 68 |
335.d | odd | 2 | 1 | inner | 670.3.d.a | ✓ | 68 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
670.3.d.a | ✓ | 68 | 1.a | even | 1 | 1 | trivial |
670.3.d.a | ✓ | 68 | 5.b | even | 2 | 1 | inner |
670.3.d.a | ✓ | 68 | 67.b | odd | 2 | 1 | inner |
670.3.d.a | ✓ | 68 | 335.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(670, [\chi])\).