Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [670,2,Mod(29,670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(670, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("670.29");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 670 = 2 \cdot 5 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 670.h (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: | \(5.34997693543\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(26\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −0.866025 | − | 0.500000i | − | 3.40294i | 0.500000 | + | 0.866025i | 1.60001 | + | 1.56204i | −1.70147 | + | 2.94703i | −3.81692 | + | 2.20370i | − | 1.00000i | −8.57997 | −0.604635 | − | 2.15277i | ||||
29.2 | −0.866025 | − | 0.500000i | − | 2.72042i | 0.500000 | + | 0.866025i | −1.66450 | − | 1.49313i | −1.36021 | + | 2.35596i | 0.550668 | − | 0.317928i | − | 1.00000i | −4.40071 | 0.694933 | + | 2.12534i | ||||
29.3 | −0.866025 | − | 0.500000i | − | 2.35660i | 0.500000 | + | 0.866025i | −1.42939 | + | 1.71955i | −1.17830 | + | 2.04088i | 3.13521 | − | 1.81012i | − | 1.00000i | −2.55358 | 2.09766 | − | 0.774481i | ||||
29.4 | −0.866025 | − | 0.500000i | − | 2.00096i | 0.500000 | + | 0.866025i | 1.72962 | − | 1.41719i | −1.00048 | + | 1.73288i | −0.879107 | + | 0.507552i | − | 1.00000i | −1.00384 | −2.20649 | + | 0.362514i | ||||
29.5 | −0.866025 | − | 0.500000i | − | 1.19283i | 0.500000 | + | 0.866025i | −1.06930 | + | 1.96382i | −0.596414 | + | 1.03302i | −0.996756 | + | 0.575477i | − | 1.00000i | 1.57716 | 1.90795 | − | 1.16607i | ||||
29.6 | −0.866025 | − | 0.500000i | − | 1.04987i | 0.500000 | + | 0.866025i | −0.0350949 | − | 2.23579i | −0.524935 | + | 0.909214i | −0.779249 | + | 0.449900i | − | 1.00000i | 1.89777 | −1.08750 | + | 1.95380i | ||||
29.7 | −0.866025 | − | 0.500000i | − | 0.0712992i | 0.500000 | + | 0.866025i | 0.722984 | − | 2.11596i | −0.0356496 | + | 0.0617469i | 3.87744 | − | 2.23864i | − | 1.00000i | 2.99492 | −1.68410 | + | 1.47098i | ||||
29.8 | −0.866025 | − | 0.500000i | 0.433437i | 0.500000 | + | 0.866025i | −2.23506 | − | 0.0672889i | 0.216718 | − | 0.375367i | 0.301725 | − | 0.174201i | − | 1.00000i | 2.81213 | 1.90197 | + | 1.17580i | |||||
29.9 | −0.866025 | − | 0.500000i | 0.719082i | 0.500000 | + | 0.866025i | 0.192333 | − | 2.22778i | 0.359541 | − | 0.622744i | −3.77588 | + | 2.18000i | − | 1.00000i | 2.48292 | −1.28046 | + | 1.83315i | |||||
29.10 | −0.866025 | − | 0.500000i | 1.76407i | 0.500000 | + | 0.866025i | 0.428350 | + | 2.19466i | 0.882037 | − | 1.52773i | −2.08791 | + | 1.20545i | − | 1.00000i | −0.111955 | 0.726366 | − | 2.11480i | |||||
29.11 | −0.866025 | − | 0.500000i | 1.90212i | 0.500000 | + | 0.866025i | −2.20047 | − | 0.397387i | 0.951061 | − | 1.64729i | −2.70090 | + | 1.55936i | − | 1.00000i | −0.618066 | 1.70697 | + | 1.44438i | |||||
29.12 | −0.866025 | − | 0.500000i | 2.24783i | 0.500000 | + | 0.866025i | 1.83882 | − | 1.27230i | 1.12391 | − | 1.94667i | −0.323048 | + | 0.186512i | − | 1.00000i | −2.05272 | −2.22861 | + | 0.182431i | |||||
29.13 | −0.866025 | − | 0.500000i | 2.72838i | 0.500000 | + | 0.866025i | −1.87831 | − | 1.21324i | 1.36419 | − | 2.36285i | 2.29856 | − | 1.32708i | − | 1.00000i | −4.44406 | 1.02005 | + | 1.98985i | |||||
29.14 | 0.866025 | + | 0.500000i | − | 2.72838i | 0.500000 | + | 0.866025i | −1.87831 | + | 1.21324i | 1.36419 | − | 2.36285i | −2.29856 | + | 1.32708i | 1.00000i | −4.44406 | −2.23328 | + | 0.111537i | |||||
29.15 | 0.866025 | + | 0.500000i | − | 2.24783i | 0.500000 | + | 0.866025i | 1.83882 | + | 1.27230i | 1.12391 | − | 1.94667i | 0.323048 | − | 0.186512i | 1.00000i | −2.05272 | 0.956317 | + | 2.02125i | |||||
29.16 | 0.866025 | + | 0.500000i | − | 1.90212i | 0.500000 | + | 0.866025i | −2.20047 | + | 0.397387i | 0.951061 | − | 1.64729i | 2.70090 | − | 1.55936i | 1.00000i | −0.618066 | −2.10436 | − | 0.756089i | |||||
29.17 | 0.866025 | + | 0.500000i | − | 1.76407i | 0.500000 | + | 0.866025i | 0.428350 | − | 2.19466i | 0.882037 | − | 1.52773i | 2.08791 | − | 1.20545i | 1.00000i | −0.111955 | 1.46829 | − | 1.68645i | |||||
29.18 | 0.866025 | + | 0.500000i | − | 0.719082i | 0.500000 | + | 0.866025i | 0.192333 | + | 2.22778i | 0.359541 | − | 0.622744i | 3.77588 | − | 2.18000i | 1.00000i | 2.48292 | −0.947326 | + | 2.02548i | |||||
29.19 | 0.866025 | + | 0.500000i | − | 0.433437i | 0.500000 | + | 0.866025i | −2.23506 | + | 0.0672889i | 0.216718 | − | 0.375367i | −0.301725 | + | 0.174201i | 1.00000i | 2.81213 | −1.96926 | − | 1.05925i | |||||
29.20 | 0.866025 | + | 0.500000i | 0.0712992i | 0.500000 | + | 0.866025i | 0.722984 | + | 2.11596i | −0.0356496 | + | 0.0617469i | −3.87744 | + | 2.23864i | 1.00000i | 2.99492 | −0.431858 | + | 2.19397i | ||||||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
67.c | even | 3 | 1 | inner |
335.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 670.2.h.c | ✓ | 52 |
5.b | even | 2 | 1 | inner | 670.2.h.c | ✓ | 52 |
67.c | even | 3 | 1 | inner | 670.2.h.c | ✓ | 52 |
335.i | even | 6 | 1 | inner | 670.2.h.c | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
670.2.h.c | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
670.2.h.c | ✓ | 52 | 5.b | even | 2 | 1 | inner |
670.2.h.c | ✓ | 52 | 67.c | even | 3 | 1 | inner |
670.2.h.c | ✓ | 52 | 335.i | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{26} + 51 T_{3}^{24} + 1129 T_{3}^{22} + 14301 T_{3}^{20} + 114897 T_{3}^{18} + 612675 T_{3}^{16} + 2202906 T_{3}^{14} + 5313734 T_{3}^{12} + 8389455 T_{3}^{10} + 8263965 T_{3}^{8} + 4677865 T_{3}^{6} + \cdots + 625 \)
acting on \(S_{2}^{\mathrm{new}}(670, [\chi])\).