Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,2,Mod(2,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(52))
chi = DirichletCharacter(H, H._module([26, 47]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("471.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 471.s (of order \(52\), degree \(24\), 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: | \(3.76095393520\) |
Analytic rank: | \(0\) |
Dimension: | \(1200\) |
Relative dimension: | \(50\) over \(\Q(\zeta_{52})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{52}]$ |
$q$-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 | −2.64211 | + | 0.484185i | 0.497232 | − | 1.65914i | 4.87629 | − | 1.84933i | −0.627219 | − | 2.01282i | −0.510409 | + | 4.62440i | 0.231121 | − | 0.104019i | −7.39084 | + | 4.46792i | −2.50552 | − | 1.64996i | 2.63176 | + | 5.01439i |
2.2 | −2.63103 | + | 0.482154i | −1.15550 | − | 1.29028i | 4.81980 | − | 1.82791i | 1.16001 | + | 3.72259i | 3.66226 | + | 2.83763i | −2.99849 | + | 1.34951i | −7.22155 | + | 4.36558i | −0.329644 | + | 2.98183i | −4.84687 | − | 9.23493i |
2.3 | −2.58276 | + | 0.473308i | −0.893751 | + | 1.48365i | 4.57659 | − | 1.73567i | −0.115271 | − | 0.369916i | 1.60612 | − | 4.25492i | −0.994360 | + | 0.447525i | −6.50457 | + | 3.93215i | −1.40242 | − | 2.65202i | 0.472800 | + | 0.900846i |
2.4 | −2.31579 | + | 0.424384i | 1.59880 | − | 0.666207i | 3.31274 | − | 1.25636i | −0.0565505 | − | 0.181477i | −3.41976 | + | 2.22130i | −2.19753 | + | 0.989028i | −3.10880 | + | 1.87934i | 2.11234 | − | 2.13027i | 0.207975 | + | 0.396263i |
2.5 | −2.27735 | + | 0.417339i | 1.73033 | − | 0.0771921i | 3.14210 | − | 1.19164i | 0.837635 | + | 2.68807i | −3.90835 | + | 0.897928i | 1.67927 | − | 0.755779i | −2.69561 | + | 1.62955i | 2.98808 | − | 0.267135i | −3.02942 | − | 5.77208i |
2.6 | −2.25488 | + | 0.413222i | −1.68169 | + | 0.414612i | 3.04369 | − | 1.15432i | 0.325864 | + | 1.04573i | 3.62069 | − | 1.62981i | 2.00200 | − | 0.901027i | −2.46252 | + | 1.48865i | 2.65619 | − | 1.39450i | −1.16690 | − | 2.22335i |
2.7 | −2.20517 | + | 0.404112i | 1.03114 | + | 1.39167i | 2.82943 | − | 1.07306i | −0.562752 | − | 1.80593i | −2.83623 | − | 2.65217i | −2.85622 | + | 1.28548i | −1.96860 | + | 1.19006i | −0.873498 | + | 2.87002i | 1.97076 | + | 3.75497i |
2.8 | −2.14879 | + | 0.393780i | −1.33115 | − | 1.10817i | 2.59219 | − | 0.983089i | −0.374862 | − | 1.20298i | 3.29674 | + | 1.85703i | 3.78902 | − | 1.70530i | −1.44392 | + | 0.872879i | 0.543937 | + | 2.95028i | 1.27921 | + | 2.43733i |
2.9 | −1.89926 | + | 0.348052i | 0.522736 | + | 1.65129i | 1.61602 | − | 0.612874i | 0.859972 | + | 2.75975i | −1.56755 | − | 2.95428i | −0.504691 | + | 0.227143i | 0.448912 | − | 0.271377i | −2.45349 | + | 1.72637i | −2.59385 | − | 4.94216i |
2.10 | −1.87532 | + | 0.343664i | −0.909746 | − | 1.47389i | 1.52867 | − | 0.579748i | −0.603303 | − | 1.93607i | 2.21258 | + | 2.45137i | −3.07153 | + | 1.38238i | 0.595672 | − | 0.360097i | −1.34473 | + | 2.68174i | 1.79674 | + | 3.42340i |
2.11 | −1.76476 | + | 0.323404i | −0.748594 | + | 1.56192i | 1.13976 | − | 0.432253i | −1.29547 | − | 4.15732i | 0.815956 | − | 2.99852i | 1.29997 | − | 0.585071i | 1.19919 | − | 0.724937i | −1.87921 | − | 2.33849i | 3.63069 | + | 6.91770i |
2.12 | −1.45245 | + | 0.266172i | −1.71334 | + | 0.253888i | 0.168739 | − | 0.0639944i | −0.893202 | − | 2.86638i | 2.42097 | − | 0.824803i | −2.00969 | + | 0.904488i | 2.29931 | − | 1.38998i | 2.87108 | − | 0.869993i | 2.06028 | + | 3.92554i |
2.13 | −1.36706 | + | 0.250523i | 1.10695 | − | 1.33216i | −0.0639385 | + | 0.0242487i | −0.772673 | − | 2.47960i | −1.17953 | + | 2.09846i | 2.24254 | − | 1.00929i | 2.46011 | − | 1.48719i | −0.549313 | − | 2.94928i | 1.67749 | + | 3.19619i |
2.14 | −1.34852 | + | 0.247125i | 1.12953 | − | 1.31307i | −0.112603 | + | 0.0427047i | 0.441900 | + | 1.41811i | −1.19870 | + | 2.04984i | −4.30939 | + | 1.93950i | 2.48780 | − | 1.50393i | −0.448318 | − | 2.96631i | −0.946359 | − | 1.80314i |
2.15 | −1.22356 | + | 0.224226i | 1.58808 | + | 0.691369i | −0.423212 | + | 0.160503i | −0.208958 | − | 0.670569i | −2.09814 | − | 0.489842i | 2.87237 | − | 1.29275i | 2.61091 | − | 1.57835i | 2.04402 | + | 2.19590i | 0.406031 | + | 0.773628i |
2.16 | −1.17172 | + | 0.214726i | −0.496480 | − | 1.65937i | −0.543213 | + | 0.206013i | 0.486414 | + | 1.56096i | 0.938044 | + | 1.83771i | 0.350360 | − | 0.157684i | 2.63112 | − | 1.59057i | −2.50702 | + | 1.64769i | −0.905119 | − | 1.72456i |
2.17 | −1.16594 | + | 0.213666i | −0.896153 | + | 1.48220i | −0.556274 | + | 0.210967i | 0.649823 | + | 2.08535i | 0.728163 | − | 1.91963i | 3.50250 | − | 1.57635i | 2.63231 | − | 1.59129i | −1.39382 | − | 2.65655i | −1.20322 | − | 2.29255i |
2.18 | −1.12589 | + | 0.206326i | −1.71242 | − | 0.260028i | −0.644984 | + | 0.244610i | 0.953289 | + | 3.05921i | 1.98164 | − | 0.0605555i | −0.513805 | + | 0.231245i | 2.63482 | − | 1.59281i | 2.86477 | + | 0.890555i | −1.70449 | − | 3.24764i |
2.19 | −0.932577 | + | 0.170901i | 1.58987 | + | 0.687239i | −1.02954 | + | 0.390453i | −0.457736 | − | 1.46893i | −1.60013 | − | 0.369192i | −2.14157 | + | 0.963841i | 2.51614 | − | 1.52106i | 2.05540 | + | 2.18525i | 0.677916 | + | 1.29166i |
2.20 | −0.467978 | + | 0.0857602i | −0.751208 | + | 1.56067i | −1.65838 | + | 0.628942i | −0.232282 | − | 0.745421i | 0.217706 | − | 0.794782i | −3.20493 | + | 1.44242i | 1.53646 | − | 0.928823i | −1.87137 | − | 2.34477i | 0.172631 | + | 0.328920i |
See next 80 embeddings (of 1200 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
157.j | odd | 52 | 1 | inner |
471.s | even | 52 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 471.2.s.a | ✓ | 1200 |
3.b | odd | 2 | 1 | inner | 471.2.s.a | ✓ | 1200 |
157.j | odd | 52 | 1 | inner | 471.2.s.a | ✓ | 1200 |
471.s | even | 52 | 1 | inner | 471.2.s.a | ✓ | 1200 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
471.2.s.a | ✓ | 1200 | 1.a | even | 1 | 1 | trivial |
471.2.s.a | ✓ | 1200 | 3.b | odd | 2 | 1 | inner |
471.2.s.a | ✓ | 1200 | 157.j | odd | 52 | 1 | inner |
471.2.s.a | ✓ | 1200 | 471.s | even | 52 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(471, [\chi])\).