Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [231,2,Mod(2,231)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 10, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("231.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 231.be (of order \(30\), degree \(8\), 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: | \(1.84454428669\) |
Analytic rank: | \(0\) |
Dimension: | \(224\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −1.69754 | − | 1.88531i | −1.50120 | − | 0.863943i | −0.463691 | + | 4.41173i | 0.0606689 | − | 0.285425i | 0.919548 | + | 4.29680i | 0.885230 | + | 2.49326i | 4.99976 | − | 3.63254i | 1.50720 | + | 2.59390i | −0.641101 | + | 0.370140i |
2.2 | −1.65113 | − | 1.83377i | 1.72117 | − | 0.193857i | −0.427411 | + | 4.06654i | −0.660086 | + | 3.10546i | −3.19736 | − | 2.83614i | −2.64526 | − | 0.0509795i | 4.17018 | − | 3.02981i | 2.92484 | − | 0.667320i | 6.78458 | − | 3.91708i |
2.3 | −1.62295 | − | 1.80247i | 0.590856 | − | 1.62816i | −0.405869 | + | 3.86159i | 0.153087 | − | 0.720219i | −3.89363 | + | 1.57742i | 1.64166 | − | 2.07484i | 3.69461 | − | 2.68429i | −2.30178 | − | 1.92401i | −1.54662 | + | 0.892944i |
2.4 | −1.41473 | − | 1.57122i | −1.63740 | + | 0.564721i | −0.258206 | + | 2.45666i | −0.705701 | + | 3.32006i | 3.20379 | + | 1.77379i | −0.181035 | − | 2.63955i | 0.804266 | − | 0.584333i | 2.36218 | − | 1.84935i | 6.21493 | − | 3.58819i |
2.5 | −1.36136 | − | 1.51195i | 0.270437 | + | 1.71081i | −0.223616 | + | 2.12757i | −0.0600850 | + | 0.282678i | 2.21849 | − | 2.73792i | −2.04626 | + | 1.67715i | 0.229262 | − | 0.166568i | −2.85373 | + | 0.925333i | 0.509191 | − | 0.293981i |
2.6 | −1.06973 | − | 1.18806i | −0.998701 | + | 1.41513i | −0.0580986 | + | 0.552771i | 0.161799 | − | 0.761203i | 2.74960 | − | 0.327297i | 1.95474 | + | 1.78298i | −1.86786 | + | 1.35708i | −1.00519 | − | 2.82659i | −1.07744 | + | 0.622058i |
2.7 | −1.06933 | − | 1.18762i | 1.50618 | + | 0.855229i | −0.0578990 | + | 0.550872i | 0.711086 | − | 3.34540i | −0.594929 | − | 2.70329i | −0.575806 | − | 2.58233i | −1.86963 | + | 1.35837i | 1.53717 | + | 2.57626i | −4.73343 | + | 2.73285i |
2.8 | −1.03170 | − | 1.14582i | 0.391245 | − | 1.68728i | −0.0394378 | + | 0.375225i | 0.582427 | − | 2.74010i | −2.33696 | + | 1.29247i | −2.49909 | + | 0.868634i | −2.02413 | + | 1.47062i | −2.69386 | − | 1.32028i | −3.74054 | + | 2.15960i |
2.9 | −0.830242 | − | 0.922077i | −0.464723 | − | 1.66854i | 0.0481326 | − | 0.457951i | −0.859357 | + | 4.04296i | −1.15269 | + | 1.81380i | −0.152025 | + | 2.64138i | −2.46985 | + | 1.79445i | −2.56807 | + | 1.55082i | 4.44139 | − | 2.56424i |
2.10 | −0.675892 | − | 0.750655i | 1.68081 | − | 0.418192i | 0.102405 | − | 0.974320i | −0.347823 | + | 1.63638i | −1.44996 | − | 0.979053i | 2.62714 | − | 0.313255i | −2.43498 | + | 1.76911i | 2.65023 | − | 1.40580i | 1.46344 | − | 0.844920i |
2.11 | −0.590583 | − | 0.655909i | −1.48384 | − | 0.893427i | 0.127629 | − | 1.21431i | 0.627181 | − | 2.95066i | 0.290326 | + | 1.50091i | 2.53863 | − | 0.745226i | −2.29995 | + | 1.67101i | 1.40358 | + | 2.65141i | −2.30576 | + | 1.33123i |
2.12 | −0.475341 | − | 0.527920i | −1.72047 | + | 0.199922i | 0.156307 | − | 1.48716i | −0.0305586 | + | 0.143767i | 0.923355 | + | 0.813242i | −2.63481 | − | 0.240370i | −2.00883 | + | 1.45950i | 2.92006 | − | 0.687921i | 0.0904232 | − | 0.0522059i |
2.13 | −0.109613 | − | 0.121738i | −0.460773 | + | 1.66964i | 0.206252 | − | 1.96236i | 0.280949 | − | 1.32176i | 0.253764 | − | 0.126921i | −0.638360 | − | 2.56759i | −0.526557 | + | 0.382566i | −2.57538 | − | 1.53865i | −0.191704 | + | 0.110680i |
2.14 | −0.0572311 | − | 0.0635616i | 1.72188 | − | 0.187403i | 0.208292 | − | 1.98177i | 0.496749 | − | 2.33702i | −0.110457 | − | 0.0987204i | −0.857429 | + | 2.50296i | −0.276277 | + | 0.200727i | 2.92976 | − | 0.645370i | −0.176974 | + | 0.102176i |
2.15 | 0.0572311 | + | 0.0635616i | 1.01290 | + | 1.40501i | 0.208292 | − | 1.98177i | −0.496749 | + | 2.33702i | −0.0313351 | + | 0.144791i | −0.857429 | + | 2.50296i | 0.276277 | − | 0.200727i | −0.948078 | + | 2.84625i | −0.176974 | + | 0.102176i |
2.16 | 0.109613 | + | 0.121738i | 0.932465 | − | 1.45963i | 0.206252 | − | 1.96236i | −0.280949 | + | 1.32176i | 0.279902 | − | 0.0464781i | −0.638360 | − | 2.56759i | 0.526557 | − | 0.382566i | −1.26102 | − | 2.72210i | −0.191704 | + | 0.110680i |
2.17 | 0.475341 | + | 0.527920i | −1.00265 | − | 1.41234i | 0.156307 | − | 1.48716i | 0.0305586 | − | 0.143767i | 0.268999 | − | 1.20066i | −2.63481 | − | 0.240370i | 2.00883 | − | 1.45950i | −0.989382 | + | 2.83216i | 0.0904232 | − | 0.0522059i |
2.18 | 0.590583 | + | 0.655909i | −1.65683 | − | 0.504891i | 0.127629 | − | 1.21431i | −0.627181 | + | 2.95066i | −0.647333 | − | 1.38491i | 2.53863 | − | 0.745226i | 2.29995 | − | 1.67101i | 2.49017 | + | 1.67304i | −2.30576 | + | 1.33123i |
2.19 | 0.675892 | + | 0.750655i | 0.813902 | + | 1.52891i | 0.102405 | − | 0.974320i | 0.347823 | − | 1.63638i | −0.597572 | + | 1.64434i | 2.62714 | − | 0.313255i | 2.43498 | − | 1.76911i | −1.67513 | + | 2.48877i | 1.46344 | − | 0.844920i |
2.20 | 0.830242 | + | 0.922077i | −1.55093 | + | 0.771117i | 0.0481326 | − | 0.457951i | 0.859357 | − | 4.04296i | −1.99867 | − | 0.789862i | −0.152025 | + | 2.64138i | 2.46985 | − | 1.79445i | 1.81076 | − | 2.39189i | 4.44139 | − | 2.56424i |
See next 80 embeddings (of 224 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
11.d | odd | 10 | 1 | inner |
21.h | odd | 6 | 1 | inner |
33.f | even | 10 | 1 | inner |
77.o | odd | 30 | 1 | inner |
231.be | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 231.2.be.a | ✓ | 224 |
3.b | odd | 2 | 1 | inner | 231.2.be.a | ✓ | 224 |
7.c | even | 3 | 1 | inner | 231.2.be.a | ✓ | 224 |
11.d | odd | 10 | 1 | inner | 231.2.be.a | ✓ | 224 |
21.h | odd | 6 | 1 | inner | 231.2.be.a | ✓ | 224 |
33.f | even | 10 | 1 | inner | 231.2.be.a | ✓ | 224 |
77.o | odd | 30 | 1 | inner | 231.2.be.a | ✓ | 224 |
231.be | even | 30 | 1 | inner | 231.2.be.a | ✓ | 224 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
231.2.be.a | ✓ | 224 | 1.a | even | 1 | 1 | trivial |
231.2.be.a | ✓ | 224 | 3.b | odd | 2 | 1 | inner |
231.2.be.a | ✓ | 224 | 7.c | even | 3 | 1 | inner |
231.2.be.a | ✓ | 224 | 11.d | odd | 10 | 1 | inner |
231.2.be.a | ✓ | 224 | 21.h | odd | 6 | 1 | inner |
231.2.be.a | ✓ | 224 | 33.f | even | 10 | 1 | inner |
231.2.be.a | ✓ | 224 | 77.o | odd | 30 | 1 | inner |
231.2.be.a | ✓ | 224 | 231.be | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).