Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,3,Mod(56,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("165.56");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 165.e (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: | \(4.49592436194\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
56.1 | − | 3.91781i | −2.98166 | + | 0.331216i | −11.3492 | 2.23607i | 1.29764 | + | 11.6816i | 1.80014 | 28.7928i | 8.78059 | − | 1.97515i | 8.76048 | |||||||||||
56.2 | − | 3.60682i | 0.198737 | + | 2.99341i | −9.00917 | − | 2.23607i | 10.7967 | − | 0.716809i | −9.02564 | 18.0672i | −8.92101 | + | 1.18980i | −8.06510 | ||||||||||
56.3 | − | 3.53441i | −0.569949 | − | 2.94536i | −8.49206 | − | 2.23607i | −10.4101 | + | 2.01443i | 0.0644394 | 15.8768i | −8.35032 | + | 3.35741i | −7.90318 | ||||||||||
56.4 | − | 3.30976i | 1.46910 | + | 2.61567i | −6.95453 | 2.23607i | 8.65725 | − | 4.86238i | 13.2372 | 9.77881i | −4.68347 | + | 7.68538i | 7.40086 | |||||||||||
56.5 | − | 3.05080i | 2.70308 | − | 1.30130i | −5.30736 | − | 2.23607i | −3.97000 | − | 8.24653i | 8.10434 | 3.98849i | 5.61324 | − | 7.03502i | −6.82179 | ||||||||||
56.6 | − | 2.73142i | −1.09395 | − | 2.79343i | −3.46068 | 2.23607i | −7.63005 | + | 2.98805i | −7.40152 | − | 1.47311i | −6.60653 | + | 6.11177i | 6.10765 | ||||||||||
56.7 | − | 2.57595i | −2.05309 | + | 2.18743i | −2.63553 | − | 2.23607i | 5.63470 | + | 5.28865i | 8.07832 | − | 3.51481i | −0.569667 | − | 8.98195i | −5.76000 | |||||||||
56.8 | − | 1.85137i | 1.43262 | − | 2.63583i | 0.572437 | 2.23607i | −4.87989 | − | 2.65231i | 5.32431 | − | 8.46526i | −4.89519 | − | 7.55229i | 4.13978 | ||||||||||
56.9 | − | 1.66350i | 2.17775 | + | 2.06335i | 1.23276 | − | 2.23607i | 3.43238 | − | 3.62269i | 1.72484 | − | 8.70471i | 0.485191 | + | 8.98691i | −3.71971 | |||||||||
56.10 | − | 1.38379i | −2.86837 | − | 0.878884i | 2.08514 | − | 2.23607i | −1.21619 | + | 3.96921i | −9.90208 | − | 8.42053i | 7.45513 | + | 5.04193i | −3.09424 | |||||||||
56.11 | − | 1.32767i | 1.89125 | − | 2.32877i | 2.23729 | − | 2.23607i | −3.09184 | − | 2.51095i | −13.0586 | − | 8.28106i | −1.84637 | − | 8.80857i | −2.96876 | |||||||||
56.12 | − | 0.880395i | 2.93851 | + | 0.604290i | 3.22491 | 2.23607i | 0.532014 | − | 2.58705i | 0.198790 | − | 6.36077i | 8.26967 | + | 3.55142i | 1.96862 | ||||||||||
56.13 | − | 0.278106i | −0.532850 | − | 2.95230i | 3.92266 | − | 2.23607i | −0.821052 | + | 0.148189i | 9.87562 | − | 2.20334i | −8.43214 | + | 3.14626i | −0.621864 | |||||||||
56.14 | − | 0.258153i | −2.71117 | − | 1.28435i | 3.93336 | 2.23607i | −0.331558 | + | 0.699896i | 2.97982 | − | 2.04802i | 5.70089 | + | 6.96418i | 0.577247 | ||||||||||
56.15 | 0.258153i | −2.71117 | + | 1.28435i | 3.93336 | − | 2.23607i | −0.331558 | − | 0.699896i | 2.97982 | 2.04802i | 5.70089 | − | 6.96418i | 0.577247 | |||||||||||
56.16 | 0.278106i | −0.532850 | + | 2.95230i | 3.92266 | 2.23607i | −0.821052 | − | 0.148189i | 9.87562 | 2.20334i | −8.43214 | − | 3.14626i | −0.621864 | ||||||||||||
56.17 | 0.880395i | 2.93851 | − | 0.604290i | 3.22491 | − | 2.23607i | 0.532014 | + | 2.58705i | 0.198790 | 6.36077i | 8.26967 | − | 3.55142i | 1.96862 | |||||||||||
56.18 | 1.32767i | 1.89125 | + | 2.32877i | 2.23729 | 2.23607i | −3.09184 | + | 2.51095i | −13.0586 | 8.28106i | −1.84637 | + | 8.80857i | −2.96876 | ||||||||||||
56.19 | 1.38379i | −2.86837 | + | 0.878884i | 2.08514 | 2.23607i | −1.21619 | − | 3.96921i | −9.90208 | 8.42053i | 7.45513 | − | 5.04193i | −3.09424 | ||||||||||||
56.20 | 1.66350i | 2.17775 | − | 2.06335i | 1.23276 | 2.23607i | 3.43238 | + | 3.62269i | 1.72484 | 8.70471i | 0.485191 | − | 8.98691i | −3.71971 | ||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 165.3.e.a | ✓ | 28 |
3.b | odd | 2 | 1 | inner | 165.3.e.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.3.e.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
165.3.e.a | ✓ | 28 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(165, [\chi])\).