Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [806,2,Mod(51,806)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(806, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("806.51");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 806 = 2 \cdot 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 806.bw (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: | \(6.43594240292\) |
Analytic rank: | \(0\) |
Dimension: | \(288\) |
Relative dimension: | \(36\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
51.1 | −0.951057 | + | 0.309017i | −2.26905 | − | 2.52003i | 0.809017 | − | 0.587785i | 1.55290 | − | 0.896568i | 2.93672 | + | 1.69552i | −0.369908 | + | 0.0388789i | −0.587785 | + | 0.809017i | −0.888400 | + | 8.45256i | −1.19984 | + | 1.33256i |
51.2 | −0.951057 | + | 0.309017i | −1.54431 | − | 1.71513i | 0.809017 | − | 0.587785i | 1.71436 | − | 0.989784i | 1.99873 | + | 1.15397i | 1.03775 | − | 0.109072i | −0.587785 | + | 0.809017i | −0.243192 | + | 2.31382i | −1.32459 | + | 1.47111i |
51.3 | −0.951057 | + | 0.309017i | −1.49867 | − | 1.66445i | 0.809017 | − | 0.587785i | −1.50762 | + | 0.870423i | 1.93966 | + | 1.11987i | −0.880871 | + | 0.0925833i | −0.587785 | + | 0.809017i | −0.210771 | + | 2.00535i | 1.16485 | − | 1.29370i |
51.4 | −0.951057 | + | 0.309017i | −1.48634 | − | 1.65075i | 0.809017 | − | 0.587785i | 2.52015 | − | 1.45501i | 1.92371 | + | 1.11065i | −3.86641 | + | 0.406376i | −0.587785 | + | 0.809017i | −0.202180 | + | 1.92361i | −1.94718 | + | 2.16256i |
51.5 | −0.951057 | + | 0.309017i | −1.38418 | − | 1.53729i | 0.809017 | − | 0.587785i | −3.14690 | + | 1.81686i | 1.79148 | + | 1.03431i | 1.86085 | − | 0.195584i | −0.587785 | + | 0.809017i | −0.133714 | + | 1.27221i | 2.43144 | − | 2.70039i |
51.6 | −0.951057 | + | 0.309017i | −0.908200 | − | 1.00866i | 0.809017 | − | 0.587785i | −1.66797 | + | 0.963006i | 1.17544 | + | 0.678642i | −3.81800 | + | 0.401288i | −0.587785 | + | 0.809017i | 0.121021 | − | 1.15144i | 1.28875 | − | 1.43131i |
51.7 | −0.951057 | + | 0.309017i | −0.814006 | − | 0.904045i | 0.809017 | − | 0.587785i | 1.99449 | − | 1.15152i | 1.05353 | + | 0.608256i | 1.37296 | − | 0.144304i | −0.587785 | + | 0.809017i | 0.158893 | − | 1.51177i | −1.54104 | + | 1.71149i |
51.8 | −0.951057 | + | 0.309017i | −0.245406 | − | 0.272550i | 0.809017 | − | 0.587785i | 0.316732 | − | 0.182866i | 0.317617 | + | 0.183376i | 4.85786 | − | 0.510581i | −0.587785 | + | 0.809017i | 0.299526 | − | 2.84979i | −0.244722 | + | 0.271791i |
51.9 | −0.951057 | + | 0.309017i | −0.0192102 | − | 0.0213351i | 0.809017 | − | 0.587785i | 2.62049 | − | 1.51294i | 0.0248629 | + | 0.0143546i | 1.57764 | − | 0.165816i | −0.587785 | + | 0.809017i | 0.313499 | − | 2.98275i | −2.02471 | + | 2.24866i |
51.10 | −0.951057 | + | 0.309017i | 0.0207746 | + | 0.0230725i | 0.809017 | − | 0.587785i | −2.48595 | + | 1.43526i | −0.0268876 | − | 0.0155236i | 2.95077 | − | 0.310139i | −0.587785 | + | 0.809017i | 0.313485 | − | 2.98261i | 1.92076 | − | 2.13322i |
51.11 | −0.951057 | + | 0.309017i | 0.435381 | + | 0.483539i | 0.809017 | − | 0.587785i | −0.679269 | + | 0.392176i | −0.563494 | − | 0.325333i | −2.57472 | + | 0.270614i | −0.587785 | + | 0.809017i | 0.269332 | − | 2.56252i | 0.524834 | − | 0.582887i |
51.12 | −0.951057 | + | 0.309017i | 0.595758 | + | 0.661657i | 0.809017 | − | 0.587785i | −0.596759 | + | 0.344539i | −0.771063 | − | 0.445174i | 0.119536 | − | 0.0125638i | −0.587785 | + | 0.809017i | 0.230724 | − | 2.19519i | 0.461083 | − | 0.512085i |
51.13 | −0.951057 | + | 0.309017i | 0.609657 | + | 0.677092i | 0.809017 | − | 0.587785i | 2.79623 | − | 1.61441i | −0.789051 | − | 0.455559i | −5.08752 | + | 0.534719i | −0.587785 | + | 0.809017i | 0.226813 | − | 2.15798i | −2.16050 | + | 2.39948i |
51.14 | −0.951057 | + | 0.309017i | 1.21331 | + | 1.34752i | 0.809017 | − | 0.587785i | −2.50460 | + | 1.44603i | −1.57034 | − | 0.906635i | −1.91881 | + | 0.201676i | −0.587785 | + | 0.809017i | −0.0300990 | + | 0.286373i | 1.93517 | − | 2.14922i |
51.15 | −0.951057 | + | 0.309017i | 1.45957 | + | 1.62102i | 0.809017 | − | 0.587785i | 0.177016 | − | 0.102200i | −1.88906 | − | 1.09065i | −0.247505 | + | 0.0260138i | −0.587785 | + | 0.809017i | −0.183768 | + | 1.74844i | −0.136770 | + | 0.151899i |
51.16 | −0.951057 | + | 0.309017i | 1.86338 | + | 2.06949i | 0.809017 | − | 0.587785i | 2.20430 | − | 1.27265i | −2.41168 | − | 1.39239i | 3.67617 | − | 0.386381i | −0.587785 | + | 0.809017i | −0.497029 | + | 4.72891i | −1.70314 | + | 1.89153i |
51.17 | −0.951057 | + | 0.309017i | 1.91163 | + | 2.12308i | 0.809017 | − | 0.587785i | 0.151504 | − | 0.0874707i | −2.47414 | − | 1.42845i | −1.12852 | + | 0.118613i | −0.587785 | + | 0.809017i | −0.539558 | + | 5.13355i | −0.117059 | + | 0.130007i |
51.18 | −0.951057 | + | 0.309017i | 2.05991 | + | 2.28776i | 0.809017 | − | 0.587785i | −3.45909 | + | 1.99711i | −2.66604 | − | 1.53924i | 2.64665 | − | 0.278174i | −0.587785 | + | 0.809017i | −0.677036 | + | 6.44157i | 2.67265 | − | 2.96828i |
51.19 | 0.951057 | − | 0.309017i | −2.26905 | − | 2.52003i | 0.809017 | − | 0.587785i | −1.55290 | + | 0.896568i | −2.93672 | − | 1.69552i | 0.369908 | − | 0.0388789i | 0.587785 | − | 0.809017i | −0.888400 | + | 8.45256i | −1.19984 | + | 1.33256i |
51.20 | 0.951057 | − | 0.309017i | −1.54431 | − | 1.71513i | 0.809017 | − | 0.587785i | −1.71436 | + | 0.989784i | −1.99873 | − | 1.15397i | −1.03775 | + | 0.109072i | 0.587785 | − | 0.809017i | −0.243192 | + | 2.31382i | −1.32459 | + | 1.47111i |
See next 80 embeddings (of 288 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
31.g | even | 15 | 1 | inner |
403.bz | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 806.2.bw.a | ✓ | 288 |
13.b | even | 2 | 1 | inner | 806.2.bw.a | ✓ | 288 |
31.g | even | 15 | 1 | inner | 806.2.bw.a | ✓ | 288 |
403.bz | even | 30 | 1 | inner | 806.2.bw.a | ✓ | 288 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
806.2.bw.a | ✓ | 288 | 1.a | even | 1 | 1 | trivial |
806.2.bw.a | ✓ | 288 | 13.b | even | 2 | 1 | inner |
806.2.bw.a | ✓ | 288 | 31.g | even | 15 | 1 | inner |
806.2.bw.a | ✓ | 288 | 403.bz | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).