Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(538,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.538");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.f (of order \(4\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(60\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
538.1 | −1.91401 | − | 1.91401i | − | 0.380921i | 5.32688i | 1.26088i | −0.729087 | + | 0.729087i | −2.25350 | − | 2.25350i | 6.36769 | − | 6.36769i | 2.85490 | 2.41334 | − | 2.41334i | |||||||
538.2 | −1.91322 | − | 1.91322i | − | 0.958783i | 5.32085i | − | 2.33392i | −1.83437 | + | 1.83437i | −0.592146 | − | 0.592146i | 6.35353 | − | 6.35353i | 2.08073 | −4.46532 | + | 4.46532i | ||||||
538.3 | −1.89459 | − | 1.89459i | 0.936891i | 5.17893i | 2.40716i | 1.77502 | − | 1.77502i | 3.07176 | + | 3.07176i | 6.02277 | − | 6.02277i | 2.12224 | 4.56057 | − | 4.56057i | ||||||||
538.4 | −1.88881 | − | 1.88881i | 3.04380i | 5.13522i | − | 3.85384i | 5.74917 | − | 5.74917i | −1.76642 | − | 1.76642i | 5.92184 | − | 5.92184i | −6.26473 | −7.27917 | + | 7.27917i | |||||||
538.5 | −1.73949 | − | 1.73949i | − | 3.20201i | 4.05165i | − | 0.618387i | −5.56987 | + | 5.56987i | 2.25367 | + | 2.25367i | 3.56882 | − | 3.56882i | −7.25288 | −1.07568 | + | 1.07568i | ||||||
538.6 | −1.73598 | − | 1.73598i | − | 2.48629i | 4.02728i | 3.17645i | −4.31616 | + | 4.31616i | 0.00734923 | + | 0.00734923i | 3.51933 | − | 3.51933i | −3.18164 | 5.51426 | − | 5.51426i | |||||||
538.7 | −1.70917 | − | 1.70917i | 2.11174i | 3.84252i | − | 0.00911084i | 3.60931 | − | 3.60931i | 1.54773 | + | 1.54773i | 3.14918 | − | 3.14918i | −1.45943 | −0.0155720 | + | 0.0155720i | |||||||
538.8 | −1.56726 | − | 1.56726i | 2.67240i | 2.91258i | 1.99974i | 4.18834 | − | 4.18834i | −1.05898 | − | 1.05898i | 1.43025 | − | 1.43025i | −4.14173 | 3.13411 | − | 3.13411i | ||||||||
538.9 | −1.52526 | − | 1.52526i | − | 2.50389i | 2.65282i | − | 3.63890i | −3.81907 | + | 3.81907i | −2.43657 | − | 2.43657i | 0.995712 | − | 0.995712i | −3.26945 | −5.55025 | + | 5.55025i | ||||||
538.10 | −1.49884 | − | 1.49884i | 0.414070i | 2.49306i | − | 2.38044i | 0.620625 | − | 0.620625i | 0.200846 | + | 0.200846i | 0.739017 | − | 0.739017i | 2.82855 | −3.56790 | + | 3.56790i | |||||||
538.11 | −1.35252 | − | 1.35252i | − | 1.50607i | 1.65860i | 1.33418i | −2.03698 | + | 2.03698i | −0.596912 | − | 0.596912i | −0.461747 | + | 0.461747i | 0.731760 | 1.80450 | − | 1.80450i | |||||||
538.12 | −1.31077 | − | 1.31077i | − | 1.31199i | 1.43623i | 0.632792i | −1.71972 | + | 1.71972i | 2.27656 | + | 2.27656i | −0.738974 | + | 0.738974i | 1.27868 | 0.829444 | − | 0.829444i | |||||||
538.13 | −1.30522 | − | 1.30522i | 1.39984i | 1.40720i | − | 3.41385i | 1.82710 | − | 1.82710i | 1.63770 | + | 1.63770i | −0.773736 | + | 0.773736i | 1.04045 | −4.45582 | + | 4.45582i | |||||||
538.14 | −1.29221 | − | 1.29221i | 0.775410i | 1.33963i | − | 0.709411i | 1.00199 | − | 1.00199i | −2.04981 | − | 2.04981i | −0.853342 | + | 0.853342i | 2.39874 | −0.916710 | + | 0.916710i | |||||||
538.15 | −1.06517 | − | 1.06517i | 1.66434i | 0.269167i | 3.94941i | 1.77280 | − | 1.77280i | 0.741657 | + | 0.741657i | −1.84363 | + | 1.84363i | 0.229976 | 4.20679 | − | 4.20679i | ||||||||
538.16 | −1.02458 | − | 1.02458i | − | 2.81698i | 0.0995254i | 1.35679i | −2.88621 | + | 2.88621i | −3.70381 | − | 3.70381i | −1.94719 | + | 1.94719i | −4.93535 | 1.39014 | − | 1.39014i | |||||||
538.17 | −0.903249 | − | 0.903249i | − | 2.65283i | − | 0.368284i | − | 3.20713i | −2.39617 | + | 2.39617i | 1.19454 | + | 1.19454i | −2.13915 | + | 2.13915i | −4.03751 | −2.89684 | + | 2.89684i | |||||
538.18 | −0.777546 | − | 0.777546i | 0.433435i | − | 0.790844i | 3.54542i | 0.337016 | − | 0.337016i | −2.52725 | − | 2.52725i | −2.17001 | + | 2.17001i | 2.81213 | 2.75673 | − | 2.75673i | |||||||
538.19 | −0.774186 | − | 0.774186i | − | 1.40798i | − | 0.801272i | − | 0.371549i | −1.09004 | + | 1.09004i | 3.30218 | + | 3.30218i | −2.16871 | + | 2.16871i | 1.01760 | −0.287648 | + | 0.287648i | |||||
538.20 | −0.756136 | − | 0.756136i | 0.596334i | − | 0.856517i | 2.30668i | 0.450910 | − | 0.450910i | 2.13477 | + | 2.13477i | −2.15992 | + | 2.15992i | 2.64439 | 1.74417 | − | 1.74417i | |||||||
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
61.d | odd | 4 | 1 | inner |
671.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.f.a | ✓ | 120 |
11.b | odd | 2 | 1 | inner | 671.2.f.a | ✓ | 120 |
61.d | odd | 4 | 1 | inner | 671.2.f.a | ✓ | 120 |
671.f | even | 4 | 1 | inner | 671.2.f.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.f.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
671.2.f.a | ✓ | 120 | 11.b | odd | 2 | 1 | inner |
671.2.f.a | ✓ | 120 | 61.d | odd | 4 | 1 | inner |
671.2.f.a | ✓ | 120 | 671.f | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).