Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [62,3,Mod(3,62)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(62, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("62.3");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 62 = 2 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 62.h (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: | \(1.68937763903\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.14412 | + | 0.831254i | −2.72490 | − | 0.286398i | 0.618034 | − | 1.90211i | 2.55607 | − | 4.42725i | 3.35568 | − | 1.93741i | 11.5829 | − | 2.46202i | 0.874032 | + | 2.68999i | −1.46030 | − | 0.310396i | 0.755706 | + | 7.19006i |
3.2 | −1.14412 | + | 0.831254i | −1.73477 | − | 0.182331i | 0.618034 | − | 1.90211i | −2.54757 | + | 4.41252i | 2.13635 | − | 1.23342i | −9.72036 | + | 2.06613i | 0.874032 | + | 2.68999i | −5.82715 | − | 1.23860i | −0.753192 | − | 7.16614i |
3.3 | −1.14412 | + | 0.831254i | 4.29053 | + | 0.450953i | 0.618034 | − | 1.90211i | 1.85810 | − | 3.21833i | −5.28375 | + | 3.05058i | −1.89983 | + | 0.403821i | 0.874032 | + | 2.68999i | 9.40198 | + | 1.99845i | 0.549350 | + | 5.22671i |
3.4 | 1.14412 | − | 0.831254i | −4.29464 | − | 0.451385i | 0.618034 | − | 1.90211i | 4.65646 | − | 8.06523i | −5.28881 | + | 3.05349i | −6.98294 | + | 1.48427i | −0.874032 | − | 2.68999i | 9.43683 | + | 2.00586i | −1.37669 | − | 13.0983i |
3.5 | 1.14412 | − | 0.831254i | 0.427629 | + | 0.0449456i | 0.618034 | − | 1.90211i | 0.263285 | − | 0.456023i | 0.526621 | − | 0.304045i | 11.1409 | − | 2.36807i | −0.874032 | − | 2.68999i | −8.62248 | − | 1.83277i | −0.0778406 | − | 0.740604i |
3.6 | 1.14412 | − | 0.831254i | 3.69788 | + | 0.388663i | 0.618034 | − | 1.90211i | −1.30508 | + | 2.26046i | 4.55390 | − | 2.62920i | −9.60759 | + | 2.04216i | −0.874032 | − | 2.68999i | 4.71991 | + | 1.00325i | 0.385848 | + | 3.67110i |
11.1 | −0.437016 | − | 1.34500i | −2.55630 | + | 2.30170i | −1.61803 | + | 1.17557i | 2.12476 | + | 3.68018i | 4.21292 | + | 2.43233i | 0.717825 | + | 6.82965i | 2.28825 | + | 1.66251i | 0.296073 | − | 2.81694i | 4.02129 | − | 4.46609i |
11.2 | −0.437016 | − | 1.34500i | −1.36657 | + | 1.23046i | −1.61803 | + | 1.17557i | −4.33815 | − | 7.51390i | 2.25218 | + | 1.30030i | −1.19735 | − | 11.3920i | 2.28825 | + | 1.66251i | −0.587288 | + | 5.58767i | −8.21033 | + | 9.11849i |
11.3 | −0.437016 | − | 1.34500i | 3.50932 | − | 3.15980i | −1.61803 | + | 1.17557i | −3.00929 | − | 5.21225i | −5.78355 | − | 3.33914i | 1.44405 | + | 13.7393i | 2.28825 | + | 1.66251i | 1.39019 | − | 13.2268i | −5.69535 | + | 6.32533i |
11.4 | 0.437016 | + | 1.34500i | −3.57045 | + | 3.21485i | −1.61803 | + | 1.17557i | −3.26484 | − | 5.65487i | −5.88431 | − | 3.39731i | 0.973704 | + | 9.26418i | −2.28825 | − | 1.66251i | 1.47211 | − | 14.0062i | 6.17900 | − | 6.86247i |
11.5 | 0.437016 | + | 1.34500i | −0.426688 | + | 0.384192i | −1.61803 | + | 1.17557i | 2.75524 | + | 4.77222i | −0.703206 | − | 0.405996i | 0.186434 | + | 1.77380i | −2.28825 | − | 1.66251i | −0.906297 | + | 8.62284i | −5.21454 | + | 5.79133i |
11.6 | 0.437016 | + | 1.34500i | 3.58359 | − | 3.22668i | −1.61803 | + | 1.17557i | −0.136597 | − | 0.236594i | 5.90597 | + | 3.40981i | −0.133803 | − | 1.27305i | −2.28825 | − | 1.66251i | 1.48991 | − | 14.1755i | 0.258522 | − | 0.287118i |
13.1 | −1.14412 | + | 0.831254i | −0.368929 | − | 0.828629i | 0.618034 | − | 1.90211i | −4.73666 | − | 8.20413i | 1.11090 | + | 0.641379i | 1.75890 | + | 1.95345i | 0.874032 | + | 2.68999i | 5.47166 | − | 6.07689i | 12.2390 | + | 5.44917i |
13.2 | −1.14412 | + | 0.831254i | −0.297611 | − | 0.668445i | 0.618034 | − | 1.90211i | 3.04757 | + | 5.27854i | 0.896151 | + | 0.517393i | 3.96158 | + | 4.39978i | 0.874032 | + | 2.68999i | 5.66393 | − | 6.29043i | −7.87459 | − | 3.50600i |
13.3 | −1.14412 | + | 0.831254i | 2.14469 | + | 4.81705i | 0.618034 | − | 1.90211i | 0.501474 | + | 0.868579i | −6.45798 | − | 3.72851i | −2.25571 | − | 2.50522i | 0.874032 | + | 2.68999i | −12.5821 | + | 13.9738i | −1.29576 | − | 0.576909i |
13.4 | 1.14412 | − | 0.831254i | −1.14843 | − | 2.57943i | 0.618034 | − | 1.90211i | −1.15365 | − | 1.99818i | −3.45811 | − | 1.99654i | 1.42640 | + | 1.58417i | −0.874032 | − | 2.68999i | 0.687638 | − | 0.763699i | −2.98092 | − | 1.32719i |
13.5 | 1.14412 | − | 0.831254i | 0.750095 | + | 1.68474i | 0.618034 | − | 1.90211i | 4.26853 | + | 7.39331i | 2.25865 | + | 1.30403i | −6.71028 | − | 7.45252i | −0.874032 | − | 2.68999i | 3.74647 | − | 4.16087i | 11.0294 | + | 4.91062i |
13.6 | 1.14412 | − | 0.831254i | 1.87649 | + | 4.21466i | 0.618034 | − | 1.90211i | −2.55443 | − | 4.42440i | 5.65038 | + | 3.26225i | 6.21587 | + | 6.90342i | −0.874032 | − | 2.68999i | −8.21998 | + | 9.12921i | −6.60039 | − | 2.93868i |
17.1 | −0.437016 | + | 1.34500i | −2.55630 | − | 2.30170i | −1.61803 | − | 1.17557i | 2.12476 | − | 3.68018i | 4.21292 | − | 2.43233i | 0.717825 | − | 6.82965i | 2.28825 | − | 1.66251i | 0.296073 | + | 2.81694i | 4.02129 | + | 4.46609i |
17.2 | −0.437016 | + | 1.34500i | −1.36657 | − | 1.23046i | −1.61803 | − | 1.17557i | −4.33815 | + | 7.51390i | 2.25218 | − | 1.30030i | −1.19735 | + | 11.3920i | 2.28825 | − | 1.66251i | −0.587288 | − | 5.58767i | −8.21033 | − | 9.11849i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.h | odd | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 62.3.h.a | ✓ | 48 |
31.h | odd | 30 | 1 | inner | 62.3.h.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
62.3.h.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
62.3.h.a | ✓ | 48 | 31.h | odd | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(62, [\chi])\).