Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [180,2,Mod(11,180)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(180, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("180.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 180.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: | \(1.43730723638\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.39556 | − | 0.228922i | −1.60382 | − | 0.654027i | 1.89519 | + | 0.638951i | −0.866025 | + | 0.500000i | 2.08851 | + | 1.27989i | −1.04714 | − | 0.604565i | −2.49858 | − | 1.32555i | 2.14450 | + | 2.09789i | 1.32305 | − | 0.499529i |
11.2 | −1.37234 | − | 0.341569i | 1.63013 | − | 0.585384i | 1.76666 | + | 0.937501i | 0.866025 | − | 0.500000i | −2.43705 | + | 0.246546i | 1.44910 | + | 0.836639i | −2.10425 | − | 1.89001i | 2.31465 | − | 1.90850i | −1.35927 | + | 0.390365i |
11.3 | −1.34552 | + | 0.435401i | 0.531460 | − | 1.64850i | 1.62085 | − | 1.17168i | −0.866025 | + | 0.500000i | 0.00266887 | + | 2.44949i | 3.61144 | + | 2.08506i | −1.67074 | + | 2.28224i | −2.43510 | − | 1.75222i | 0.947554 | − | 1.04983i |
11.4 | −1.29101 | + | 0.577305i | −1.30449 | + | 1.13944i | 1.33344 | − | 1.49062i | 0.866025 | − | 0.500000i | 1.02631 | − | 2.22411i | 2.51585 | + | 1.45253i | −0.860949 | + | 2.69421i | 0.403372 | − | 2.97276i | −0.829399 | + | 1.14547i |
11.5 | −1.18360 | + | 0.774003i | −0.427686 | − | 1.67842i | 0.801838 | − | 1.83223i | 0.866025 | − | 0.500000i | 1.80531 | + | 1.65555i | −4.06955 | − | 2.34956i | 0.469091 | + | 2.78926i | −2.63417 | + | 1.43567i | −0.638030 | + | 1.26211i |
11.6 | −0.981980 | − | 1.01770i | −1.63013 | + | 0.585384i | −0.0714305 | + | 1.99872i | 0.866025 | − | 0.500000i | 2.19650 | + | 1.08415i | −1.44910 | − | 0.836639i | 2.10425 | − | 1.89001i | 2.31465 | − | 1.90850i | −1.35927 | − | 0.390365i |
11.7 | −0.962433 | + | 1.03621i | 0.968388 | + | 1.43605i | −0.147445 | − | 1.99456i | −0.866025 | + | 0.500000i | −2.42005 | − | 0.378648i | 1.90150 | + | 1.09783i | 2.20868 | + | 1.76684i | −1.12445 | + | 2.78130i | 0.315389 | − | 1.37860i |
11.8 | −0.896034 | − | 1.09413i | 1.60382 | + | 0.654027i | −0.394247 | + | 1.96076i | −0.866025 | + | 0.500000i | −0.721488 | − | 2.34082i | 1.04714 | + | 0.604565i | 2.49858 | − | 1.32555i | 2.14450 | + | 2.09789i | 1.32305 | + | 0.499529i |
11.9 | −0.660227 | + | 1.25064i | −1.67907 | + | 0.425108i | −1.12820 | − | 1.65141i | −0.866025 | + | 0.500000i | 0.576911 | − | 2.38058i | −1.87435 | − | 1.08216i | 2.81019 | − | 0.320666i | 2.63857 | − | 1.42758i | −0.0535467 | − | 1.41320i |
11.10 | −0.295692 | − | 1.38296i | −0.531460 | + | 1.64850i | −1.82513 | + | 0.817857i | −0.866025 | + | 0.500000i | 2.43695 | + | 0.247538i | −3.61144 | − | 2.08506i | 1.67074 | + | 2.28224i | −2.43510 | − | 1.75222i | 0.947554 | + | 1.04983i |
11.11 | −0.236271 | + | 1.39434i | −1.35535 | − | 1.07843i | −1.88835 | − | 0.658884i | 0.866025 | − | 0.500000i | 1.82393 | − | 1.63501i | 3.33246 | + | 1.92400i | 1.36487 | − | 2.47732i | 0.673959 | + | 2.92332i | 0.492552 | + | 1.32567i |
11.12 | −0.145547 | − | 1.40670i | 1.30449 | − | 1.13944i | −1.95763 | + | 0.409483i | 0.866025 | − | 0.500000i | −1.79271 | − | 1.66919i | −2.51585 | − | 1.45253i | 0.860949 | + | 2.69421i | 0.403372 | − | 2.97276i | −0.829399 | − | 1.14547i |
11.13 | 0.0785043 | − | 1.41203i | 0.427686 | + | 1.67842i | −1.98767 | − | 0.221701i | 0.866025 | − | 0.500000i | 2.40356 | − | 0.472143i | 4.06955 | + | 2.34956i | −0.469091 | + | 2.78926i | −2.63417 | + | 1.43567i | −0.638030 | − | 1.26211i |
11.14 | 0.304404 | + | 1.38106i | 1.45681 | + | 0.936861i | −1.81468 | + | 0.840803i | 0.866025 | − | 0.500000i | −0.850407 | + | 2.29713i | 0.737635 | + | 0.425874i | −1.71360 | − | 2.25024i | 1.24458 | + | 2.72965i | 0.954154 | + | 1.04383i |
11.15 | 0.416164 | − | 1.35159i | −0.968388 | − | 1.43605i | −1.65361 | − | 1.12497i | −0.866025 | + | 0.500000i | −2.34396 | + | 0.711237i | −1.90150 | − | 1.09783i | −2.20868 | + | 1.76684i | −1.12445 | + | 2.78130i | 0.315389 | + | 1.37860i |
11.16 | 0.538420 | + | 1.30771i | −0.398623 | + | 1.68556i | −1.42021 | + | 1.40819i | −0.866025 | + | 0.500000i | −2.41884 | + | 0.386254i | 0.891819 | + | 0.514892i | −2.60618 | − | 1.09902i | −2.68220 | − | 1.34380i | −1.12014 | − | 0.863299i |
11.17 | 0.752973 | − | 1.19709i | 1.67907 | − | 0.425108i | −0.866065 | − | 1.80276i | −0.866025 | + | 0.500000i | 0.755401 | − | 2.33010i | 1.87435 | + | 1.08216i | −2.81019 | − | 0.320666i | 2.63857 | − | 1.42758i | −0.0535467 | + | 1.41320i |
11.18 | 1.04050 | + | 0.957789i | 0.561951 | − | 1.63836i | 0.165280 | + | 1.99316i | 0.866025 | − | 0.500000i | 2.15391 | − | 1.16648i | 0.811773 | + | 0.468677i | −1.73705 | + | 2.23218i | −2.36842 | − | 1.84135i | 1.37999 | + | 0.309220i |
11.19 | 1.08940 | − | 0.901786i | 1.35535 | + | 1.07843i | 0.373566 | − | 1.96480i | 0.866025 | − | 0.500000i | 2.44903 | − | 0.0473955i | −3.33246 | − | 1.92400i | −1.36487 | − | 2.47732i | 0.673959 | + | 2.92332i | 0.492552 | − | 1.32567i |
11.20 | 1.18642 | + | 0.769686i | 1.70656 | + | 0.296046i | 0.815168 | + | 1.82634i | −0.866025 | + | 0.500000i | 1.79683 | + | 1.66475i | −3.55496 | − | 2.05246i | −0.438576 | + | 2.79422i | 2.82471 | + | 1.01044i | −1.41231 | − | 0.0733593i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
36.h | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 180.2.q.a | ✓ | 48 |
3.b | odd | 2 | 1 | 540.2.q.a | 48 | ||
4.b | odd | 2 | 1 | inner | 180.2.q.a | ✓ | 48 |
5.b | even | 2 | 1 | 900.2.r.f | 48 | ||
5.c | odd | 4 | 1 | 900.2.o.b | 48 | ||
5.c | odd | 4 | 1 | 900.2.o.c | 48 | ||
9.c | even | 3 | 1 | 540.2.q.a | 48 | ||
9.c | even | 3 | 1 | 1620.2.e.b | 48 | ||
9.d | odd | 6 | 1 | inner | 180.2.q.a | ✓ | 48 |
9.d | odd | 6 | 1 | 1620.2.e.b | 48 | ||
12.b | even | 2 | 1 | 540.2.q.a | 48 | ||
20.d | odd | 2 | 1 | 900.2.r.f | 48 | ||
20.e | even | 4 | 1 | 900.2.o.b | 48 | ||
20.e | even | 4 | 1 | 900.2.o.c | 48 | ||
36.f | odd | 6 | 1 | 540.2.q.a | 48 | ||
36.f | odd | 6 | 1 | 1620.2.e.b | 48 | ||
36.h | even | 6 | 1 | inner | 180.2.q.a | ✓ | 48 |
36.h | even | 6 | 1 | 1620.2.e.b | 48 | ||
45.h | odd | 6 | 1 | 900.2.r.f | 48 | ||
45.l | even | 12 | 1 | 900.2.o.b | 48 | ||
45.l | even | 12 | 1 | 900.2.o.c | 48 | ||
180.n | even | 6 | 1 | 900.2.r.f | 48 | ||
180.v | odd | 12 | 1 | 900.2.o.b | 48 | ||
180.v | odd | 12 | 1 | 900.2.o.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
180.2.q.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
180.2.q.a | ✓ | 48 | 4.b | odd | 2 | 1 | inner |
180.2.q.a | ✓ | 48 | 9.d | odd | 6 | 1 | inner |
180.2.q.a | ✓ | 48 | 36.h | even | 6 | 1 | inner |
540.2.q.a | 48 | 3.b | odd | 2 | 1 | ||
540.2.q.a | 48 | 9.c | even | 3 | 1 | ||
540.2.q.a | 48 | 12.b | even | 2 | 1 | ||
540.2.q.a | 48 | 36.f | odd | 6 | 1 | ||
900.2.o.b | 48 | 5.c | odd | 4 | 1 | ||
900.2.o.b | 48 | 20.e | even | 4 | 1 | ||
900.2.o.b | 48 | 45.l | even | 12 | 1 | ||
900.2.o.b | 48 | 180.v | odd | 12 | 1 | ||
900.2.o.c | 48 | 5.c | odd | 4 | 1 | ||
900.2.o.c | 48 | 20.e | even | 4 | 1 | ||
900.2.o.c | 48 | 45.l | even | 12 | 1 | ||
900.2.o.c | 48 | 180.v | odd | 12 | 1 | ||
900.2.r.f | 48 | 5.b | even | 2 | 1 | ||
900.2.r.f | 48 | 20.d | odd | 2 | 1 | ||
900.2.r.f | 48 | 45.h | odd | 6 | 1 | ||
900.2.r.f | 48 | 180.n | even | 6 | 1 | ||
1620.2.e.b | 48 | 9.c | even | 3 | 1 | ||
1620.2.e.b | 48 | 9.d | odd | 6 | 1 | ||
1620.2.e.b | 48 | 36.f | odd | 6 | 1 | ||
1620.2.e.b | 48 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(180, [\chi])\).