Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,3,Mod(191,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, 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("380.191");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.b (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: | \(10.3542500457\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
191.1 | −2.00000 | − | 0.00104961i | 3.00750i | 4.00000 | + | 0.00419842i | 2.23607 | 0.00315669 | − | 6.01499i | − | 0.387511i | −7.99999 | − | 0.0125953i | −0.0450421 | −4.47214 | − | 0.00234699i | |||||||
191.2 | −2.00000 | + | 0.00104961i | − | 3.00750i | 4.00000 | − | 0.00419842i | 2.23607 | 0.00315669 | + | 6.01499i | 0.387511i | −7.99999 | + | 0.0125953i | −0.0450421 | −4.47214 | + | 0.00234699i | |||||||
191.3 | −1.99914 | − | 0.0586031i | 4.03802i | 3.99313 | + | 0.234312i | −2.23607 | 0.236641 | − | 8.07258i | − | 12.0049i | −7.96910 | − | 0.702432i | −7.30564 | 4.47022 | + | 0.131040i | |||||||
191.4 | −1.99914 | + | 0.0586031i | − | 4.03802i | 3.99313 | − | 0.234312i | −2.23607 | 0.236641 | + | 8.07258i | 12.0049i | −7.96910 | + | 0.702432i | −7.30564 | 4.47022 | − | 0.131040i | |||||||
191.5 | −1.98112 | − | 0.274141i | 4.09700i | 3.84969 | + | 1.08622i | −2.23607 | 1.12316 | − | 8.11666i | 4.68200i | −7.32894 | − | 3.20729i | −7.78542 | 4.42992 | + | 0.612999i | ||||||||
191.6 | −1.98112 | + | 0.274141i | − | 4.09700i | 3.84969 | − | 1.08622i | −2.23607 | 1.12316 | + | 8.11666i | − | 4.68200i | −7.32894 | + | 3.20729i | −7.78542 | 4.42992 | − | 0.612999i | ||||||
191.7 | −1.93566 | − | 0.503224i | 0.768825i | 3.49353 | + | 1.94814i | 2.23607 | 0.386891 | − | 1.48818i | − | 10.1723i | −5.78193 | − | 5.52895i | 8.40891 | −4.32826 | − | 1.12524i | |||||||
191.8 | −1.93566 | + | 0.503224i | − | 0.768825i | 3.49353 | − | 1.94814i | 2.23607 | 0.386891 | + | 1.48818i | 10.1723i | −5.78193 | + | 5.52895i | 8.40891 | −4.32826 | + | 1.12524i | |||||||
191.9 | −1.80036 | − | 0.871032i | − | 0.512340i | 2.48261 | + | 3.13635i | −2.23607 | −0.446265 | + | 0.922398i | 3.99381i | −1.73774 | − | 7.80899i | 8.73751 | 4.02573 | + | 1.94769i | |||||||
191.10 | −1.80036 | + | 0.871032i | 0.512340i | 2.48261 | − | 3.13635i | −2.23607 | −0.446265 | − | 0.922398i | − | 3.99381i | −1.73774 | + | 7.80899i | 8.73751 | 4.02573 | − | 1.94769i | |||||||
191.11 | −1.70466 | − | 1.04601i | − | 1.85712i | 1.81172 | + | 3.56618i | −2.23607 | −1.94256 | + | 3.16575i | − | 8.17539i | 0.641893 | − | 7.97421i | 5.55112 | 3.81173 | + | 2.33895i | ||||||
191.12 | −1.70466 | + | 1.04601i | 1.85712i | 1.81172 | − | 3.56618i | −2.23607 | −1.94256 | − | 3.16575i | 8.17539i | 0.641893 | + | 7.97421i | 5.55112 | 3.81173 | − | 2.33895i | ||||||||
191.13 | −1.68205 | − | 1.08200i | 2.49923i | 1.65856 | + | 3.63994i | 2.23607 | 2.70416 | − | 4.20382i | 6.10491i | 1.14864 | − | 7.91711i | 2.75385 | −3.76117 | − | 2.41942i | ||||||||
191.14 | −1.68205 | + | 1.08200i | − | 2.49923i | 1.65856 | − | 3.63994i | 2.23607 | 2.70416 | + | 4.20382i | − | 6.10491i | 1.14864 | + | 7.91711i | 2.75385 | −3.76117 | + | 2.41942i | ||||||
191.15 | −1.65600 | − | 1.12146i | 5.88343i | 1.48464 | + | 3.71428i | 2.23607 | 6.59806 | − | 9.74294i | − | 1.76155i | 1.70687 | − | 7.81579i | −25.6148 | −3.70292 | − | 2.50767i | |||||||
191.16 | −1.65600 | + | 1.12146i | − | 5.88343i | 1.48464 | − | 3.71428i | 2.23607 | 6.59806 | + | 9.74294i | 1.76155i | 1.70687 | + | 7.81579i | −25.6148 | −3.70292 | + | 2.50767i | |||||||
191.17 | −1.32631 | − | 1.49696i | 0.452282i | −0.481798 | + | 3.97088i | 2.23607 | 0.677050 | − | 0.599867i | − | 5.37605i | 6.58327 | − | 4.54539i | 8.79544 | −2.96572 | − | 3.34731i | |||||||
191.18 | −1.32631 | + | 1.49696i | − | 0.452282i | −0.481798 | − | 3.97088i | 2.23607 | 0.677050 | + | 0.599867i | 5.37605i | 6.58327 | + | 4.54539i | 8.79544 | −2.96572 | + | 3.34731i | |||||||
191.19 | −1.30085 | − | 1.51914i | 4.10739i | −0.615584 | + | 3.95235i | −2.23607 | 6.23970 | − | 5.34309i | 8.55175i | 6.80496 | − | 4.20625i | −7.87063 | 2.90879 | + | 3.39690i | ||||||||
191.20 | −1.30085 | + | 1.51914i | − | 4.10739i | −0.615584 | − | 3.95235i | −2.23607 | 6.23970 | + | 5.34309i | − | 8.55175i | 6.80496 | + | 4.20625i | −7.87063 | 2.90879 | − | 3.39690i | ||||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.3.b.a | ✓ | 72 |
4.b | odd | 2 | 1 | inner | 380.3.b.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.3.b.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
380.3.b.a | ✓ | 72 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).