Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [270,2,Mod(31,270)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("270.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 270.k (of order \(9\), degree \(6\), 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: | \(2.15596085457\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | 0.939693 | + | 0.342020i | −1.60992 | − | 0.638862i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | −1.29433 | − | 1.15096i | −3.84295 | + | 3.22462i | 0.500000 | + | 0.866025i | 2.18371 | + | 2.05704i | −0.500000 | + | 0.866025i |
31.2 | 0.939693 | + | 0.342020i | −0.718131 | − | 1.57616i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | −0.135743 | − | 1.72672i | 3.34524 | − | 2.80699i | 0.500000 | + | 0.866025i | −1.96858 | + | 2.26378i | −0.500000 | + | 0.866025i |
31.3 | 0.939693 | + | 0.342020i | 0.137658 | + | 1.72657i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | −0.461166 | + | 1.66953i | 1.26626 | − | 1.06252i | 0.500000 | + | 0.866025i | −2.96210 | + | 0.475354i | −0.500000 | + | 0.866025i |
31.4 | 0.939693 | + | 0.342020i | 1.69040 | − | 0.377572i | 0.766044 | + | 0.642788i | −0.173648 | + | 0.984808i | 1.71759 | + | 0.223348i | −0.268547 | + | 0.225338i | 0.500000 | + | 0.866025i | 2.71488 | − | 1.27649i | −0.500000 | + | 0.866025i |
61.1 | 0.939693 | − | 0.342020i | −1.60992 | + | 0.638862i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | −1.29433 | + | 1.15096i | −3.84295 | − | 3.22462i | 0.500000 | − | 0.866025i | 2.18371 | − | 2.05704i | −0.500000 | − | 0.866025i |
61.2 | 0.939693 | − | 0.342020i | −0.718131 | + | 1.57616i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | −0.135743 | + | 1.72672i | 3.34524 | + | 2.80699i | 0.500000 | − | 0.866025i | −1.96858 | − | 2.26378i | −0.500000 | − | 0.866025i |
61.3 | 0.939693 | − | 0.342020i | 0.137658 | − | 1.72657i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | −0.461166 | − | 1.66953i | 1.26626 | + | 1.06252i | 0.500000 | − | 0.866025i | −2.96210 | − | 0.475354i | −0.500000 | − | 0.866025i |
61.4 | 0.939693 | − | 0.342020i | 1.69040 | + | 0.377572i | 0.766044 | − | 0.642788i | −0.173648 | − | 0.984808i | 1.71759 | − | 0.223348i | −0.268547 | − | 0.225338i | 0.500000 | − | 0.866025i | 2.71488 | + | 1.27649i | −0.500000 | − | 0.866025i |
121.1 | −0.173648 | − | 0.984808i | −1.61088 | − | 0.636438i | −0.939693 | + | 0.342020i | −0.766044 | − | 0.642788i | −0.347042 | + | 1.69693i | −1.33944 | − | 0.487517i | 0.500000 | + | 0.866025i | 2.18989 | + | 2.05046i | −0.500000 | + | 0.866025i |
121.2 | −0.173648 | − | 0.984808i | −1.29249 | + | 1.15303i | −0.939693 | + | 0.342020i | −0.766044 | − | 0.642788i | 1.35995 | + | 1.07263i | 0.582091 | + | 0.211864i | 0.500000 | + | 0.866025i | 0.341047 | − | 2.98055i | −0.500000 | + | 0.866025i |
121.3 | −0.173648 | − | 0.984808i | 0.683937 | − | 1.59130i | −0.939693 | + | 0.342020i | −0.766044 | − | 0.642788i | −1.68589 | − | 0.397221i | −3.30330 | − | 1.20230i | 0.500000 | + | 0.866025i | −2.06446 | − | 2.17670i | −0.500000 | + | 0.866025i |
121.4 | −0.173648 | − | 0.984808i | 1.71943 | + | 0.208681i | −0.939693 | + | 0.342020i | −0.766044 | − | 0.642788i | −0.0930655 | − | 1.72955i | 4.56065 | + | 1.65994i | 0.500000 | + | 0.866025i | 2.91290 | + | 0.717628i | −0.500000 | + | 0.866025i |
151.1 | −0.766044 | − | 0.642788i | −1.67972 | − | 0.422540i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | 1.01514 | + | 1.40339i | 0.192018 | − | 1.08899i | 0.500000 | − | 0.866025i | 2.64292 | + | 1.41950i | −0.500000 | − | 0.866025i |
151.2 | −0.766044 | − | 0.642788i | −1.08744 | + | 1.34814i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | 1.69959 | − | 0.333742i | −0.314431 | + | 1.78323i | 0.500000 | − | 0.866025i | −0.634956 | − | 2.93204i | −0.500000 | − | 0.866025i |
151.3 | −0.766044 | − | 0.642788i | 1.09919 | − | 1.33858i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | −1.70245 | + | 0.318864i | −0.158856 | + | 0.900919i | 0.500000 | − | 0.866025i | −0.583569 | − | 2.94269i | −0.500000 | − | 0.866025i |
151.4 | −0.766044 | − | 0.642788i | 1.16797 | + | 1.27900i | 0.173648 | + | 0.984808i | 0.939693 | + | 0.342020i | −0.0725903 | − | 1.73053i | 0.781269 | − | 4.43080i | 0.500000 | − | 0.866025i | −0.271692 | + | 2.98767i | −0.500000 | − | 0.866025i |
211.1 | −0.766044 | + | 0.642788i | −1.67972 | + | 0.422540i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | 1.01514 | − | 1.40339i | 0.192018 | + | 1.08899i | 0.500000 | + | 0.866025i | 2.64292 | − | 1.41950i | −0.500000 | + | 0.866025i |
211.2 | −0.766044 | + | 0.642788i | −1.08744 | − | 1.34814i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | 1.69959 | + | 0.333742i | −0.314431 | − | 1.78323i | 0.500000 | + | 0.866025i | −0.634956 | + | 2.93204i | −0.500000 | + | 0.866025i |
211.3 | −0.766044 | + | 0.642788i | 1.09919 | + | 1.33858i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | −1.70245 | − | 0.318864i | −0.158856 | − | 0.900919i | 0.500000 | + | 0.866025i | −0.583569 | + | 2.94269i | −0.500000 | + | 0.866025i |
211.4 | −0.766044 | + | 0.642788i | 1.16797 | − | 1.27900i | 0.173648 | − | 0.984808i | 0.939693 | − | 0.342020i | −0.0725903 | + | 1.73053i | 0.781269 | + | 4.43080i | 0.500000 | + | 0.866025i | −0.271692 | − | 2.98767i | −0.500000 | + | 0.866025i |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 270.2.k.e | ✓ | 24 |
3.b | odd | 2 | 1 | 810.2.k.e | 24 | ||
27.e | even | 9 | 1 | inner | 270.2.k.e | ✓ | 24 |
27.e | even | 9 | 1 | 7290.2.a.s | 12 | ||
27.f | odd | 18 | 1 | 810.2.k.e | 24 | ||
27.f | odd | 18 | 1 | 7290.2.a.v | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
270.2.k.e | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
270.2.k.e | ✓ | 24 | 27.e | even | 9 | 1 | inner |
810.2.k.e | 24 | 3.b | odd | 2 | 1 | ||
810.2.k.e | 24 | 27.f | odd | 18 | 1 | ||
7290.2.a.s | 12 | 27.e | even | 9 | 1 | ||
7290.2.a.v | 12 | 27.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 3 T_{7}^{23} - 9 T_{7}^{22} + 29 T_{7}^{21} + 303 T_{7}^{20} - 1899 T_{7}^{19} + \cdots + 2483776 \) acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\).